### The Mizar article:

### Meet -- Continuous Lattices

**by****Artur Kornilowicz**

- Received October 10, 1996
Copyright (c) 1996 Association of Mizar Users

- MML identifier: WAYBEL_2
- [ MML identifier index ]

environ vocabulary FUNCT_1, RELAT_1, RELAT_2, LATTICE3, ORDERS_1, SUBSET_1, QUANTAL1, BOOLE, WAYBEL_0, YELLOW_0, LATTICES, ORDINAL2, FUNCT_5, TARSKI, MCART_1, PRE_TOPC, SEQM_3, REALSET1, BHSP_3, FINSET_1, WELLORD1, CAT_1, YELLOW_2, FINSUB_1, WELLORD2, YELLOW_1, FILTER_2, LATTICE2, WAYBEL_1, WAYBEL_2; notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, RELSET_1, FINSUB_1, FUNCT_1, TOLER_1, FINSET_1, PARTFUN1, FUNCT_2, PRE_TOPC, STRUCT_0, REALSET1, ORDERS_1, ORDERS_3, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_2, WAYBEL_1, YELLOW_3, YELLOW_4; constructors FINSUB_1, ORDERS_3, YELLOW_3, YELLOW_4, WAYBEL_1, TOLER_1, MEMBERED, PRE_TOPC; clusters STRUCT_0, LATTICE3, FINSET_1, WAYBEL_0, YELLOW_0, YELLOW_2, YELLOW_3, YELLOW_4, WAYBEL_1, RELSET_1, FINSUB_1, SUBSET_1, RELAT_1, FUNCT_2, PARTFUN1; requirements SUBSET, BOOLE; definitions WAYBEL_1, YELLOW_0, LATTICE3, ORDERS_3, TARSKI, WAYBEL_0, XBOOLE_0; theorems BORSUK_1, FINSUB_1, FUNCT_1, FUNCT_2, LATTICE3, MCART_1, ORDERS_1, ORDERS_3, PRE_TOPC, RELAT_1, STRUCT_0, TARSKI, WAYBEL_0, YELLOW_0, ZFMISC_1, YELLOW_1, YELLOW_2, WAYBEL_1, YELLOW_3, YELLOW_4, WELLORD1, FUNCT_5, REALSET1, TEX_1, XBOOLE_1; schemes FINSET_1, SUPINF_2, YELLOW_3, FUNCT_2; begin :: Preliminaries definition let X, Y be non empty set, f be Function of X, Y, Z be non empty Subset of X; cluster f.:Z -> non empty; coherence proof consider x being Element of Z; A1: dom f = X by FUNCT_2:def 1; then f.x in rng f by FUNCT_1:def 5; hence thesis by A1,FUNCT_1:def 12; end; end; definition cluster reflexive connected -> with_infima with_suprema (non empty RelStr); coherence proof let L be non empty RelStr such that A1: L is reflexive connected; thus L is with_infima proof let x, y be Element of L; per cases by A1,WAYBEL_0:def 29; suppose A2: x <= y; take z = x; thus z <= x & z <= y by A1,A2,YELLOW_0:def 1; thus thesis; suppose A3: y <= x; take z = y; thus z <= x & z <= y by A1,A3,YELLOW_0:def 1; thus thesis; end; let x, y be Element of L; per cases by A1,WAYBEL_0:def 29; suppose A4: x <= y; take z = y; thus z >= x & z >= y by A1,A4,YELLOW_0:def 1; thus thesis; suppose A5: y <= x; take z = x; thus z >= x & z >= y by A1,A5,YELLOW_0:def 1; thus thesis; end; end; definition let C be Chain; cluster [#]C -> directed; coherence; end; theorem Th1: for L being up-complete Semilattice for D being non empty directed Subset of L, x being Element of L holds ex_sup_of {x} "/\" D,L proof let L be up-complete Semilattice, D be non empty directed Subset of L, x be Element of L; reconsider xx = {x} as non empty directed Subset of L by WAYBEL_0:5; ex_sup_of xx "/\" D,L by WAYBEL_0:75; hence ex_sup_of {x} "/\" D,L; end; theorem for L being up-complete sup-Semilattice for D being non empty directed Subset of L, x being Element of L holds ex_sup_of {x} "\/" D,L proof let L be up-complete sup-Semilattice, D be non empty directed Subset of L, x be Element of L; reconsider xx = {x} as non empty directed Subset of L by WAYBEL_0:5; ex_sup_of xx "\/" D,L by WAYBEL_0:75; hence ex_sup_of {x} "\/" D,L; end; theorem Th3: for L being up-complete sup-Semilattice for A, B being non empty directed Subset of L holds A is_<=_than sup (A "\/" B) proof let L be up-complete sup-Semilattice, A, B be non empty directed Subset of L; A1: A "\/" B = { x "\/" y where x, y is Element of L : x in A & y in B } by YELLOW_4:def 3; ex_sup_of A "\/" B,L by WAYBEL_0:75; then A2: A "\/" B is_<=_than sup (A "\/" B) by YELLOW_0:def 9; let x be Element of L such that A3: x in A; consider b being Element of B; x "\/" b in A "\/" B by A1,A3; then A4: x "\/" b <= sup (A "\/" B) by A2,LATTICE3:def 9; ex xx being Element of L st x <= xx & b <= xx & for c being Element of L st x <= c & b <= c holds xx <= c by LATTICE3:def 10; then x <= x "\/" b by LATTICE3:def 13; hence x <= sup (A "\/" B) by A4,YELLOW_0:def 2; end; theorem Th4: for L being up-complete sup-Semilattice for A, B being non empty directed Subset of L holds sup (A "\/" B) = sup A "\/" sup B proof let L be up-complete sup-Semilattice, A, B be non empty directed Subset of L; A1: ex_sup_of A "\/" B,L by WAYBEL_0:75; A2: ex_sup_of A,L & ex_sup_of B,L by WAYBEL_0:75; then A is_<=_than sup A & B is_<=_than sup B by YELLOW_0:30; then A "\/" B is_<=_than sup A "\/" sup B by YELLOW_4:27; then A3: sup (A "\/" B) <= sup A "\/" sup B by A1,YELLOW_0:def 9; A is_<=_than sup (A "\/" B) & B is_<=_than sup (A "\/" B) by Th3; then sup A <= sup (A "\/" B) & sup B <= sup (A "\/" B) by A2,YELLOW_0:30; then A4: sup A "\/" sup B <= sup (A "\/" B) "\/" sup (A "\/" B) by YELLOW_3:3; sup (A "\/" B) <= sup (A "\/" B); then sup (A "\/" B) "\/" sup (A "\/" B) = sup (A "\/" B) by YELLOW_0:24; hence thesis by A3,A4,ORDERS_1:25; end; theorem Th5: for L being up-complete Semilattice for D being non empty directed Subset of [:L,L:] holds { sup ({x} "/\" proj2 D) where x is Element of L : x in proj1 D } = { sup X where X is non empty directed Subset of L: ex x being Element of L st X = {x} "/\" proj2 D & x in proj1 D } proof let L be up-complete Semilattice, D be non empty directed Subset of [:L,L:]; defpred P[set] means ex x being Element of L st $1 = {x} "/\" proj2 D & x in proj1 D; reconsider D1 = proj1 D, D2 = proj2 D as non empty directed Subset of L by YELLOW_3:21,22; thus { sup ({x} "/\" proj2 D) where x is Element of L : x in proj1 D } c= { sup X where X is non empty directed Subset of L: P[X] } proof let q be set; assume q in {sup ({x} "/\" proj2 D) where x is Element of L : x in proj1 D}; then consider x being Element of L such that A1: q = sup ({x} "/\" D2) & x in D1; reconsider xx = {x} as non empty directed Subset of L by WAYBEL_0:5; xx "/\" D2 is non empty directed; hence q in { sup X where X is non empty directed Subset of L: P[X] } by A1; end; let q be set; assume q in { sup X where X is non empty directed Subset of L: P[X] }; then consider X being non empty directed Subset of L such that A2: q = sup X & P[X]; thus q in { sup ({x} "/\" proj2 D) where x is Element of L : x in proj1 D} by A2; end; theorem Th6: for L being Semilattice, D being non empty directed Subset of [:L,L:] holds union {X where X is non empty directed Subset of L: ex x being Element of L st X = {x} "/\" proj2 D & x in proj1 D} = proj1 D "/\" proj2 D proof let L be Semilattice, D be non empty directed Subset of [:L,L:]; set D1 = proj1 D, D2 = proj2 D; defpred P[set] means ex x being Element of L st $1 = {x} "/\" proj2 D & x in proj1 D; A1: D1 "/\" D2 = { x "/\" y where x, y is Element of L : x in D1 & y in D2 } by YELLOW_4:def 4; thus union {X where X is non empty directed Subset of L: P[X]} c= D1 "/\" D2 proof let q be set; assume q in union {X where X is non empty directed Subset of L: P[X]}; then consider w being set such that A2: q in w & w in {X where X is non empty directed Subset of L: P[X]} by TARSKI:def 4; consider e being non empty directed Subset of L such that A3: w = e & P[e] by A2; consider p being Element of L such that A4: e = {p} "/\" D2 & p in D1 by A3; {p} "/\" D2 = { p "/\" y where y is Element of L : y in D2 } by YELLOW_4:42; then consider y being Element of L such that A5: q = p "/\" y & y in D2 by A2,A3,A4; thus q in D1 "/\" D2 by A1,A4,A5; end; let q be set; assume q in D1 "/\" D2; then consider x, y being Element of L such that A6: q = x "/\" y & x in D1 & y in D2 by A1; {x} "/\" D2 = { x "/\" d where d is Element of L : d in D2 } by YELLOW_4:42; then A7: q in {x} "/\" D2 by A6; reconsider xx = {x} as non empty directed Subset of L by WAYBEL_0:5; reconsider T = D2 as non empty directed Subset of L by YELLOW_3:21,22; xx "/\" T is non empty directed; then {x} "/\" D2 in {X where X is non empty directed Subset of L: P[X]} by A6; hence q in union {X where X is non empty directed Subset of L: P[X]} by A7,TARSKI:def 4; end; theorem Th7: for L being up-complete Semilattice for D being non empty directed Subset of [:L,L:] holds ex_sup_of union {X where X is non empty directed Subset of L: ex x being Element of L st X = {x} "/\" proj2 D & x in proj1 D},L proof let L be up-complete Semilattice, D be non empty directed Subset of [:L,L:]; reconsider D1 = proj1 D, D2 = proj2 D as non empty directed Subset of L by YELLOW_3:21,22; set A = {X where X is non empty directed Subset of L: ex x being Element of L st X = {x} "/\" D2 & x in D1}; union A c= the carrier of L proof let q be set; assume q in union A; then consider r being set such that A1: q in r & r in A by TARSKI:def 4; consider s being non empty directed Subset of L such that A2: r = s & ex x being Element of L st s = {x} "/\" D2 & x in D1 by A1; thus q in the carrier of L by A1,A2; end; then reconsider S = union A as Subset of L; S = D1 "/\" D2 by Th6; hence thesis by WAYBEL_0:75; end; theorem Th8: for L being up-complete Semilattice for D being non empty directed Subset of [:L,L:] holds ex_sup_of {sup X where X is non empty directed Subset of L: ex x being Element of L st X = {x} "/\" proj2 D & x in proj1 D},L proof let L be up-complete Semilattice, D be non empty directed Subset of [:L,L:]; reconsider D1 = proj1 D, D2 = proj2 D as non empty directed Subset of L by YELLOW_3:21,22; defpred P[set] means ex x being Element of L st $1 = {x} "/\" D2 & x in D1; set A = {sup X where X is non empty directed Subset of L: P[X]}; A c= the carrier of L proof let q be set; assume q in A; then consider X being non empty directed Subset of L such that A1: q = sup X & P[X]; thus q in the carrier of L by A1; end; then reconsider A as Subset of L; set R = {X where X is non empty directed Subset of L: P[X]}; union R c= the carrier of L proof let q be set; assume q in union R; then consider r being set such that A2: q in r & r in R by TARSKI:def 4; consider s being non empty directed Subset of L such that A3: r = s & ex x being Element of L st s = {x} "/\" D2 & x in D1 by A2; thus q in the carrier of L by A2,A3; end; then reconsider UR = union R as Subset of L; set a = sup UR; A4: ex_sup_of UR,L by Th7; A5: A is_<=_than a proof let b be Element of L; assume b in A; then consider X being non empty directed Subset of L such that A6: b = sup X & P[X]; A7: ex_sup_of X,L by WAYBEL_0:75; X in R by A6; then X c= UR by ZFMISC_1:92; hence b <= a by A4,A6,A7,YELLOW_0:34; end; for b being Element of L st A is_<=_than b holds a <= b proof let b be Element of L such that A8: A is_<=_than b; UR is_<=_than b proof let k be Element of L; assume k in UR; then consider k1 being set such that A9: k in k1 and A10: k1 in R by TARSKI:def 4; consider s being non empty directed Subset of L such that A11: k1 = s and A12: P[s] by A10; consider x being Element of L such that A13: s = {x} "/\" D2 & x in D1 by A12; {x} "/\" D2 = {x "/\" d2 where d2 is Element of L: d2 in D2} by YELLOW_4:42; then consider d2 being Element of L such that A14: k = x "/\" d2 & d2 in D2 by A9,A11,A13; sup s in A by A12; then A15: sup s <= b by A8,LATTICE3:def 9; ex_sup_of s,L by A13,Th1; then x "/\" d2 <= sup s by A9,A11,A14,YELLOW_4:1; hence k <= b by A14,A15,YELLOW_0:def 2; end; hence a <= b by A4,YELLOW_0:def 9; end; hence thesis by A5,YELLOW_0:15; end; theorem Th9: for L being up-complete Semilattice for D being non empty directed Subset of [:L,L:] holds "\/" ({ sup X where X is non empty directed Subset of L: ex x being Element of L st X = {x} "/\" proj2 D & x in proj1 D },L) <= "\/" (union {X where X is non empty directed Subset of L: ex x being Element of L st X = {x} "/\" proj2 D & x in proj1 D},L) proof let L be up-complete Semilattice, D be non empty directed Subset of [:L,L:]; defpred P[set] means ex x being Element of L st $1 = {x} "/\" proj2 D & x in proj1 D; A1: ex_sup_of { sup X where X is non empty directed Subset of L: P[X] },L by Th8; "\/"(union {X where X is non empty directed Subset of L: P[X]},L) is_>=_than { sup X where X is non empty directed Subset of L: P[X] } proof let a be Element of L; assume a in { sup X where X is non empty directed Subset of L: P[X] }; then consider q being non empty directed Subset of L such that A2: a = sup q & P[q]; A3: ex_sup_of q,L by WAYBEL_0:75; A4: ex_sup_of union {X where X is non empty directed Subset of L: P[X]},L by Th7; q in {X where X is non empty directed Subset of L: P[X]} by A2; then q c= union {X where X is non empty directed Subset of L: P[X]} by ZFMISC_1:92; hence a <= "\/"(union {X where X is non empty directed Subset of L: P[X]},L) by A2,A3,A4,YELLOW_0:34; end; hence thesis by A1,YELLOW_0:def 9; end; theorem Th10: for L being up-complete Semilattice for D being non empty directed Subset of [:L,L:] holds "\/" ({ sup X where X is non empty directed Subset of L: ex x being Element of L st X = {x} "/\" proj2 D & x in proj1 D},L) = "\/" (union {X where X is non empty directed Subset of L: ex x being Element of L st X = {x} "/\" proj2 D & x in proj1 D},L) proof let L be up-complete Semilattice, D be non empty directed Subset of [:L,L:]; defpred P[set] means ex x being Element of L st $1 = {x} "/\" proj2 D & x in proj1 D; A1: "\/" ({ sup X where X is non empty directed Subset of L: P[X] },L) <= "\/"(union {X where X is non empty directed Subset of L: P[X]},L) by Th9; A2: ex_sup_of union {X where X is non empty directed Subset of L: P[X]},L by Th7; union {X where X is non empty directed Subset of L: P[X]} is_<=_than "\/" ({ "\/"(X,L) where X is non empty directed Subset of L: P[X] },L) proof let a be Element of L; assume a in union {X where X is non empty directed Subset of L: P[X]}; then consider b being set such that A3: a in b & b in {X where X is non empty directed Subset of L: P[X]} by TARSKI:def 4; consider c being non empty directed Subset of L such that A4: b = c & P[c] by A3; ex_sup_of c,L by WAYBEL_0:75; then A5: a <= "\/"(c,L) by A3,A4,YELLOW_4:1; A6: ex_sup_of {"\/"(X,L) where X is non empty directed Subset of L:P[X]},L by Th8; "\/"(c,L) in { "\/" (X,L) where X is non empty directed Subset of L: P[X] } by A4; then "\/"(c,L) <= "\/" ({ "\/"(X,L) where X is non empty directed Subset of L: P[X] },L) by A6,YELLOW_4:1; hence a <= "\/" ({ "\/"(X,L) where X is non empty directed Subset of L: P[X] },L) by A5,YELLOW_0:def 2; end; then "\/"(union {X where X is non empty directed Subset of L: P[X]},L) <= "\/" ({ "\/"(X,L) where X is non empty directed Subset of L: P[X] },L) by A2,YELLOW_0:def 9; hence thesis by A1,ORDERS_1:25; end; definition let S, T be up-complete (non empty reflexive RelStr); cluster [:S,T:] -> up-complete; coherence proof let X be non empty directed Subset of [:S,T:]; reconsider X1 = proj1 X as non empty directed Subset of S by YELLOW_3:21,22; reconsider X2 = proj2 X as non empty directed Subset of T by YELLOW_3:21,22; consider x being Element of S such that A1: x is_>=_than X1 and A2: for z being Element of S st z is_>=_than X1 holds x <= z by WAYBEL_0:def 39; consider y being Element of T such that A3: y is_>=_than X2 and A4: for z being Element of T st z is_>=_than X2 holds y <= z by WAYBEL_0:def 39; take [x,y]; thus [x,y] is_>=_than X by A1,A3,YELLOW_3:31; let z be Element of [:S,T:] such that A5: z is_>=_than X; the carrier of [:S,T:] = [:the carrier of S, the carrier of T:] by YELLOW_3:def 2; then A6: z = [z`1,z`2] by MCART_1:23; then z`1 is_>=_than X1 & z`2 is_>=_than X2 by A5,YELLOW_3:31; then x <= z`1 & y <= z`2 by A2,A4; hence [x,y] <= z by A6,YELLOW_3:11; end; end; theorem for S, T being non empty reflexive antisymmetric RelStr st [:S,T:] is up-complete holds S is up-complete & T is up-complete proof let S, T be non empty reflexive antisymmetric RelStr such that A1: [:S,T:] is up-complete; thus S is up-complete proof let X be non empty directed Subset of S; consider D being non empty directed Subset of T; consider x being Element of [:S,T:] such that A2: x is_>=_than [:X,D:] and A3: for y being Element of [:S,T:] st y is_>=_than [:X,D:] holds x <= y by A1,WAYBEL_0:def 39; take x`1; the carrier of [:S,T:] = [:the carrier of S, the carrier of T:] by YELLOW_3:def 2; then A4: x = [x`1,x`2] by MCART_1:23; hence x`1 is_>=_than X by A2,YELLOW_3:29; let y be Element of S such that A5: y is_>=_than X; ex_sup_of [:X,D:],[:S,T:] by A1,WAYBEL_0:75; then ex_sup_of D,T by YELLOW_3:39; then sup D is_>=_than D by YELLOW_0:def 9; then [y,sup D] is_>=_than [:X,D:] by A5,YELLOW_3:30; then x <= [y,sup D] by A3; hence x`1 <= y by A4,YELLOW_3:11; end; let X be non empty directed Subset of T; consider D being non empty directed Subset of S; consider x being Element of [:S,T:] such that A6: x is_>=_than [:D,X:] and A7: for y being Element of [:S,T:] st y is_>=_than [:D,X:] holds x <= y by A1,WAYBEL_0:def 39; take x`2; the carrier of [:S,T:] = [:the carrier of S, the carrier of T:] by YELLOW_3:def 2; then A8: x = [x`1,x`2] by MCART_1:23; hence x`2 is_>=_than X by A6,YELLOW_3:29; let y be Element of T such that A9: y is_>=_than X; ex_sup_of [:D,X:],[:S,T:] by A1,WAYBEL_0:75; then ex_sup_of D,S by YELLOW_3:39; then sup D is_>=_than D by YELLOW_0:def 9; then [sup D,y] is_>=_than [:D,X:] by A9,YELLOW_3:30; then x <= [sup D,y] by A7; hence x`2 <= y by A8,YELLOW_3:11; end; theorem Th12: for L being up-complete antisymmetric (non empty reflexive RelStr) for D being non empty directed Subset of [:L,L:] holds sup D = [sup proj1 D,sup proj2 D] proof let L be up-complete antisymmetric (non empty reflexive RelStr), D be non empty directed Subset of [:L,L:]; reconsider D1 = proj1 D, D2 = proj2 D as non empty directed Subset of L by YELLOW_3:21,22; reconsider C = the carrier of L as non empty set; D c= the carrier of [:L,L:]; then D c= [:the carrier of L, the carrier of L:] by YELLOW_3:def 2; then reconsider D' = D as non empty Subset of [:C,C:]; A1: ex_sup_of D1,L & ex_sup_of D2,L by WAYBEL_0:75; A2: ex_sup_of D,[:L,L:] by WAYBEL_0:75; A3: ex_sup_of [:D1,D2:],[:L,L:] by WAYBEL_0:75; the carrier of [:L,L:] = [:C,C:] by YELLOW_3:def 2; then consider d1, d2 being set such that A4: d1 in C & d2 in C & sup D = [d1,d2] by ZFMISC_1:def 2; reconsider d1, d2 as Element of L by A4; A5: D1 is_<=_than d1 proof let b be Element of L; assume b in D1; then consider x being set such that A6: [b,x] in D by FUNCT_5:def 1; reconsider x as Element of D2 by A6,FUNCT_5:4; D is_<=_than [d1,d2] by A2,A4,YELLOW_0:def 9; then [b,x] <= [d1,d2] by A6,LATTICE3:def 9; hence b <= d1 by YELLOW_3:11; end; D2 is_<=_than d2 proof let b be Element of L; assume b in D2; then consider x being set such that A7: [x,b] in D by FUNCT_5:def 2; reconsider x as Element of D1 by A7,FUNCT_5:4; D is_<=_than [d1,d2] by A2,A4,YELLOW_0:def 9; then [x,b] <= [d1,d2] by A7,LATTICE3:def 9; hence b <= d2 by YELLOW_3:11; end; then sup D1 <= d1 & sup D2 <= d2 by A1,A5,YELLOW_0:def 9; then A8: [sup D1,sup D2] <= sup D by A4,YELLOW_3:11; reconsider D1, D2 as non empty Subset of L; D' c= [:D1,D2:] by YELLOW_3:1; then sup D <= sup [:D1,D2:] by A2,A3,YELLOW_0:34; then sup D <= [sup proj1 D,sup proj2 D] by A1,YELLOW_3:43; hence [sup proj1 D,sup proj2 D] = sup D by A8,ORDERS_1:25; end; theorem Th13: for S1, S2 being RelStr, D being Subset of S1 for f being map of S1,S2 st f is monotone holds f.:(downarrow D) c= downarrow (f.:D) proof let S1, S2 be RelStr, D be Subset of S1, f be map of S1,S2 such that A1: f is monotone; let q be set; assume A2: q in f.:(downarrow D); then consider x being set such that A3: x in dom f & x in downarrow D & q = f.x by FUNCT_1:def 12; reconsider s1 = S1, s2 = S2 as non empty RelStr by A2,A3,STRUCT_0:def 1; reconsider f1 = f as map of s1,s2; reconsider x as Element of s1 by A3; consider y being Element of s1 such that A4: x <= y & y in D by A3,WAYBEL_0:def 15; f1.x is Element of s2; then reconsider q1 = q, fy = f1.y as Element of s2 by A3; A5: q1 <= fy by A1,A3,A4,ORDERS_3:def 5; the carrier of s2 <> {}; then dom f = the carrier of s1 by FUNCT_2:def 1; then f.y in f.:D by A4,FUNCT_1:def 12; hence q in downarrow (f.:D) by A5,WAYBEL_0:def 15; end; theorem Th14: for S1, S2 being RelStr, D being Subset of S1 for f being map of S1,S2 st f is monotone holds f.:(uparrow D) c= uparrow (f.:D) proof let S1, S2 be RelStr, D be Subset of S1, f be map of S1,S2 such that A1: f is monotone; let q be set; assume A2: q in f.:(uparrow D); then consider x being set such that A3: x in dom f & x in uparrow D & q = f.x by FUNCT_1:def 12; reconsider s1 = S1, s2 = S2 as non empty RelStr by A2,A3,STRUCT_0:def 1; reconsider f1 = f as map of s1,s2; reconsider x as Element of s1 by A3; consider y being Element of s1 such that A4: y <= x & y in D by A3,WAYBEL_0:def 16; f1.x is Element of s2; then reconsider q1 = q, fy = f1.y as Element of s2 by A3; A5: fy <= q1 by A1,A3,A4,ORDERS_3:def 5; the carrier of s2 <> {}; then dom f = the carrier of s1 by FUNCT_2:def 1; then f.y in f.:D by A4,FUNCT_1:def 12; hence q in uparrow (f.:D) by A5,WAYBEL_0:def 16; end; definition cluster trivial -> distributive complemented (non empty reflexive RelStr); coherence proof let L be non empty reflexive RelStr such that A1: L is trivial; thus L is distributive proof let x,y,z be Element of L; thus x "/\" (y "\/" z) = (x "/\" y) "\/" (x "/\" z) by A1,REALSET1:def 20; end; let x be Element of L; take y = x; thus x "\/" y = Top L by A1,REALSET1:def 20; thus x "/\" y = Bottom L by A1,REALSET1:def 20; end; end; definition cluster strict non empty trivial LATTICE; existence proof consider B being strict trivial (non empty reflexive RelStr); take B; thus thesis; end; end; theorem Th15: for H being distributive complete LATTICE for a being Element of H, X being finite Subset of H holds sup ({a} "/\" X) = a "/\" sup X proof let H be distributive complete LATTICE, a be Element of H, X be finite Subset of H; defpred P[set] means ex A being Subset of H st A = $1 & a "/\" sup A = sup({a} "/\" A); A1: X is finite; A2: P[{}] proof {} c= the carrier of H by XBOOLE_1:2; then reconsider A = {} as Subset of H; take A; thus A = {}; A3: Bottom H <= a by YELLOW_0:44; A4: {a} "/\" {}H = {} by YELLOW_4:36; thus a "/\" sup A = a "/\" Bottom H by YELLOW_0:def 11 .= Bottom H by A3,YELLOW_0:25 .= sup({a} "/\" A) by A4,YELLOW_0:def 11; end; A5: for x, B being set st x in X & B c= X & P[B] holds P[B \/ {x}] proof let x, B be set such that A6: x in X & B c= X and A7: P[B]; consider A being Subset of H such that A8: A = B & a "/\" sup A = sup({a} "/\" A) by A7; reconsider x1 = x as Element of H by A6; B c= the carrier of H & {x1} c= the carrier of H by A6,XBOOLE_1:1; then B \/ {x} c= the carrier of H by XBOOLE_1:8; then reconsider C = B \/ {x} as Subset of H; take C; thus C = B \/ {x}; A9: ex_sup_of {a} "/\" A,H & ex_sup_of {a "/\" x1},H by YELLOW_0:17; A10: {a} "/\" C = ({a} "/\" A) \/ ({a} "/\" {x1}) by A8,YELLOW_4:43 .= ({a} "/\" A) \/ {a "/\" x1} by YELLOW_4:46; ex_sup_of B,H & ex_sup_of {x},H by YELLOW_0:17; hence a "/\" sup C = a "/\" ("\/"(B, H) "\/" "\/"({x}, H)) by YELLOW_2:3 .= sup({a} "/\" A) "\/" (a "/\" "\/"({x}, H)) by A8,WAYBEL_1:def 3 .= sup({a} "/\" A) "\/" (a "/\" x1) by YELLOW_0:39 .= sup({a} "/\" A) "\/" sup{a "/\" x1} by YELLOW_0:39 .= sup({a} "/\" C) by A9,A10,YELLOW_2:3; end; P[X] from Finite(A1,A2,A5); hence sup({a} "/\" X) = a "/\" sup X; end; theorem for H being distributive complete LATTICE for a being Element of H, X being finite Subset of H holds inf ({a} "\/" X) = a "\/" inf X proof let H be distributive complete LATTICE, a be Element of H, X be finite Subset of H; defpred P[set] means ex A being Subset of H st A = $1 & a "\/" inf A = inf({a} "\/" A); A1: X is finite; A2: P[{}] proof {} c= the carrier of H by XBOOLE_1:2; then reconsider A = {} as Subset of H; take A; thus A = {}; A3: a <= Top H by YELLOW_0:45; A4: {a} "\/" {}H = {} by YELLOW_4:9; thus a "\/" inf A = a "\/" Top H by YELLOW_0:def 12 .= Top H by A3,YELLOW_0:24 .= inf({a} "\/" A) by A4,YELLOW_0:def 12; end; A5: for x, B being set st x in X & B c= X & P[B] holds P[B \/ {x}] proof let x, B be set such that A6: x in X & B c= X and A7: P[B]; consider A being Subset of H such that A8: A = B & a "\/" inf A = inf({a} "\/" A) by A7; reconsider x1 = x as Element of H by A6; B c= the carrier of H & {x1} c= the carrier of H by A6,XBOOLE_1:1; then B \/ {x} c= the carrier of H by XBOOLE_1:8; then reconsider C = B \/ {x} as Subset of H; take C; thus C = B \/ {x}; A9: ex_inf_of {a} "\/" A,H & ex_inf_of {a "\/" x1},H by YELLOW_0:17; A10: {a} "\/" C = ({a} "\/" A) \/ ({a} "\/" {x1}) by A8,YELLOW_4:16 .= ({a} "\/" A) \/ {a "\/" x1} by YELLOW_4:19; ex_inf_of B,H & ex_inf_of {x},H by YELLOW_0:17; hence a "\/" inf C = a "\/" ("/\"(B, H) "/\" "/\"({x}, H)) by YELLOW_2:4 .= inf({a} "\/" A) "/\" (a "\/" "/\"({x}, H)) by A8,WAYBEL_1:6 .= inf({a} "\/" A) "/\" (a "\/" x1) by YELLOW_0:39 .= inf({a} "\/" A) "/\" inf{a "\/" x1} by YELLOW_0:39 .= inf({a} "\/" C) by A9,A10,YELLOW_2:4; end; P[X] from Finite(A1,A2,A5); hence inf({a} "\/" X) = a "\/" inf X; end; theorem Th17: for H being complete distributive LATTICE, a being Element of H for X being finite Subset of H holds a "/\" preserves_sup_of X proof let H be complete distributive LATTICE, a be Element of H, X be finite Subset of H; assume ex_sup_of X,H; thus ex_sup_of a "/\".:X,H by YELLOW_0:17; thus sup (a"/\".:X) = "\/"({a"/\"y where y is Element of H: y in X},H) by WAYBEL_1:64 .= sup({a} "/\" X) by YELLOW_4:42 .= a "/\" sup X by Th15 .= a"/\".sup X by WAYBEL_1:def 18; end; begin :: The Properties of Nets scheme ExNet { L() -> non empty RelStr, N() -> prenet of L(), F(set) -> Element of L()} : ex M being prenet of L() st the RelStr of M = the RelStr of N() & for i being Element of M holds (the mapping of M).i = F((the mapping of N()).i) proof deffunc G(Element of N()) = F((the mapping of N()).$1); A1: for i being Element of N() holds G(i) in the carrier of L(); consider f being Function of the carrier of N(), the carrier of L() such that A2: for i being Element of N() holds f.i = G(i) from FunctR_ealEx(A1); set M = NetStr (# the carrier of N(), the InternalRel of N(), f #); M is directed proof A3: the RelStr of M = the RelStr of N(); A4: [#]N() is directed by WAYBEL_0:def 6; [#]N() = the carrier of N() by PRE_TOPC:12 .= [#]M by PRE_TOPC:12; hence [#]M is directed by A3,A4,WAYBEL_0:3; end; then reconsider M as prenet of L(); take M; thus thesis by A2; end; theorem Th18: for L being non empty RelStr for N being prenet of L st N is eventually-directed holds rng netmap (N,L) is directed proof let L be non empty RelStr, N be prenet of L such that A1: N is eventually-directed; set f = netmap (N,L); let x, y be Element of L such that A2: x in rng f and A3: y in rng f; consider a being set such that A4: a in dom f & f.a = x by A2,FUNCT_1:def 5; consider b being set such that A5: b in dom f & f.b = y by A3,FUNCT_1:def 5; A6: dom f = the carrier of N by FUNCT_2:def 1; reconsider a, b as Element of N by A4,A5; consider ja being Element of N such that A7: for k being Element of N st ja <= k holds N.a <= N.k by A1,WAYBEL_0:11; consider jb being Element of N such that A8: for k being Element of N st jb <= k holds N.b <= N.k by A1,WAYBEL_0:11; A9: [#]N = the carrier of N by PRE_TOPC:12; [#]N is directed by WAYBEL_0:def 6; then consider c being Element of N such that A10: c in [#]N & ja <= c & jb <= c by A9,WAYBEL_0:def 1; take z = f.c; thus z in rng f by A6,FUNCT_1:def 5; the mapping of N = netmap(N,L) by WAYBEL_0:def 7; then N.a = f.a & N.b = f.b & N.c = f.c by WAYBEL_0:def 8; hence x <= z & y <= z by A4,A5,A7,A8,A10; end; theorem Th19: for L being non empty reflexive RelStr, D being non empty directed Subset of L for n being Function of D, the carrier of L holds NetStr (#D,(the InternalRel of L)|_2 D,n#) is prenet of L proof let L be non empty reflexive RelStr, D be non empty directed Subset of L, n be Function of D, the carrier of L; set N = NetStr (#D,(the InternalRel of L)|_2 D,n#); N is directed proof let x, y be Element of N; assume x in [#]N & y in [#]N; reconsider x1 = x, y1 = y as Element of D; consider z1 being Element of L such that A1: z1 in D & x1 <= z1 & y1 <= z1 by WAYBEL_0:def 1; reconsider z = z1 as Element of N by A1; take z; D = [#]N by PRE_TOPC:12; hence z in [#]N; N is SubRelStr of L proof thus the carrier of N c= the carrier of L; thus the InternalRel of N c= the InternalRel of L by WELLORD1:15; end; then reconsider N as SubRelStr of L; N is full proof thus the InternalRel of N = (the InternalRel of L)|_2 the carrier of N; end; hence x <= z & y <= z by A1,YELLOW_0:61; end; hence NetStr (#D,(the InternalRel of L)|_2 D,n#) is prenet of L; end; theorem Th20: for L being non empty reflexive RelStr, D being non empty directed Subset of L for n being Function of D, the carrier of L, N being prenet of L st n = id D & N = NetStr (#D,(the InternalRel of L)|_2 D,n#) holds N is eventually-directed proof let L be non empty reflexive RelStr, D be non empty directed Subset of L, n be Function of D, the carrier of L, N be prenet of L such that A1: n = id D and A2: N = NetStr (#D,(the InternalRel of L)|_2 D,n#); for i being Element of N ex j being Element of N st for k being Element of N st j <= k holds N.i <= N.k proof let i be Element of N; take j = i; let k be Element of N such that A3: j <= k; n.j in the carrier of L & n.k in the carrier of L by A2,FUNCT_2:7; then reconsider nj = n.j, nk = n.k as Element of L; k in the carrier of N; then reconsider k1 = k as Element of L by A2; A4: N is SubRelStr of L proof thus the carrier of N c= the carrier of L by A2; thus the InternalRel of N c= the InternalRel of L by A2,WELLORD1:15; end; A5: nj = j & nk = k by A1,A2,FUNCT_1:35; then nj <= k1 by A3,A4,YELLOW_0:60; then N.i <= nk by A2,A5,WAYBEL_0:def 8; hence N.i <= N.k by A2,WAYBEL_0:def 8; end; hence N is eventually-directed by WAYBEL_0:11; end; definition let L be non empty RelStr, N be NetStr over L; func sup N -> Element of L equals :Def1: Sup the mapping of N; coherence; end; definition let L be non empty RelStr, J be set, f be Function of J,the carrier of L; func FinSups f -> prenet of L means :Def2: ex g being Function of Fin J, the carrier of L st for x being Element of Fin J holds g.x = sup (f.:x) & it = NetStr (# Fin J, RelIncl Fin J, g #); existence proof deffunc F(Element of Fin J) = sup (f.:$1); A1: for x being Element of Fin J holds F(x) in the carrier of L; consider g being Function of Fin J, the carrier of L such that A2: for x being Element of Fin J holds g.x = F(x) from FunctR_ealEx(A1); set M = NetStr (# Fin J, RelIncl Fin J, g #); M is directed proof let x, y be Element of M such that x in [#]M & y in [#]M; reconsider x1 = x, y1 = y as Element of Fin J; reconsider z = x1 \/ y1 as Element of M; take z; z in the carrier of M; hence z in [#]M by PRE_TOPC:12; A3: InclPoset Fin J = RelStr(#Fin J, RelIncl Fin J#) by YELLOW_1:def 1; then reconsider x2 = x, y2 = y, z1 = z as Element of InclPoset Fin J; x c= z & y c= z by XBOOLE_1:7; then x2 <= z1 & y2 <= z1 by YELLOW_1:3; hence x <= z & y <= z by A3,YELLOW_0:1; end; then reconsider M as prenet of L; take M; thus thesis by A2; end; uniqueness proof let A, B be prenet of L such that A4: ex g being Function of Fin J, the carrier of L st for x being Element of Fin J holds g.x = sup (f.:x) & A = NetStr (#Fin J, RelIncl Fin J, g#) and A5: ex g being Function of Fin J, the carrier of L st for x being Element of Fin J holds g.x = sup (f.:x) & B = NetStr (#Fin J, RelIncl Fin J, g#); consider g1 being Function of Fin J, the carrier of L such that A6: for x being Element of Fin J holds g1.x = sup (f.:x) & A = NetStr (#Fin J, RelIncl Fin J, g1#) by A4; consider g2 being Function of Fin J, the carrier of L such that A7: for x being Element of Fin J holds g2.x = sup (f.:x) & B = NetStr (#Fin J, RelIncl Fin J, g2#) by A5; for x being set st x in Fin J holds g1.x = g2.x proof let x be set; assume A8: x in Fin J; hence g1.x = sup (f.:x) by A6 .= g2.x by A7,A8; end; hence A = B by A6,A7,FUNCT_2:18; end; end; theorem for L being non empty RelStr, J, x being set for f being Function of J,the carrier of L holds x is Element of FinSups f iff x is Element of Fin J proof let L be non empty RelStr, J, x be set, f be Function of J,the carrier of L; consider g being Function of Fin J, the carrier of L such that A1: for x being Element of Fin J holds g.x = sup (f.:x) & FinSups f = NetStr (# Fin J, RelIncl Fin J, g #) by Def2; thus thesis by A1; end; definition let L be complete antisymmetric (non empty reflexive RelStr), J be set, f be Function of J,the carrier of L; cluster FinSups f -> monotone; coherence proof let x, y be Element of FinSups f such that A1: x <= y; consider g being Function of Fin J, the carrier of L such that A2: for x being Element of Fin J holds g.x = sup (f.:x) & FinSups f = NetStr (# Fin J, RelIncl Fin J, g #) by Def2; A3: ex_sup_of f.:x,L by YELLOW_0:17; A4: ex_sup_of f.:y,L by YELLOW_0:17; g.x = sup (f.:x) & g.y = sup (f.:y) by A2; then reconsider fx = g.x as Element of L; A5: InclPoset Fin J = RelStr(#Fin J, RelIncl Fin J#) by YELLOW_1:def 1; then reconsider x1 = x, y1 = y as Element of InclPoset Fin J by A2; let a, b be Element of L such that A6: a = netmap(FinSups f,L).x & b = netmap(FinSups f,L).y; A7: g = netmap(FinSups f,L) by A2,WAYBEL_0:def 7; x1 <= y1 by A1,A2,A5,YELLOW_0:1; then x c= y by YELLOW_1:3; then f.:x c= f.:y by RELAT_1:156; then sup (f.:x) <= sup (f.:y) by A3,A4,YELLOW_0:34; then fx <= sup (f.:y) by A2; hence a <= b by A2,A6,A7; end; end; definition let L be non empty RelStr, x be Element of L, N be non empty NetStr over L; func x "/\" N -> strict NetStr over L means :Def3: the RelStr of it = the RelStr of N & for i being Element of it ex y being Element of L st y = (the mapping of N).i & (the mapping of it).i = x "/\" y; existence proof defpred P[set,set] means ex y being Element of L st y = (the mapping of N).$1 & $2 = x "/\" y; A1: for e being Element of N ex u being Element of L st P[e,u] proof let e be Element of N; take x "/\" (the mapping of N).e, (the mapping of N).e; thus thesis; end; ex g being Function of the carrier of N, the carrier of L st for i being Element of N holds P[i,g.i] from FuncExD(A1); then consider g being Function of the carrier of N, the carrier of L such that A2: for i being Element of N ex y being Element of L st y = (the mapping of N).i & g.i = x "/\" y; take NetStr (#the carrier of N,the InternalRel of N,g#); thus thesis by A2; end; uniqueness proof let A, B be strict NetStr over L such that A3: the RelStr of A = the RelStr of N & for i being Element of A ex y being Element of L st y = (the mapping of N).i & (the mapping of A).i = x "/\" y and A4: the RelStr of B = the RelStr of N & for i being Element of B ex y being Element of L st y = (the mapping of N).i & (the mapping of B).i = x "/\" y; reconsider C = the carrier of A as non empty set by A3; reconsider f1 = the mapping of A, f2 = the mapping of B as Function of C, the carrier of L by A3,A4; for c being Element of C holds f1.c = f2.c proof let c be Element of C; consider ya being Element of L such that A5: ya = (the mapping of N).c & f1.c = x "/\" ya by A3; consider yb being Element of L such that A6: yb = (the mapping of N).c & f2.c = x "/\" yb by A3,A4; thus f1.c = f2.c by A5,A6; end; hence A = B by A3,A4,FUNCT_2:113; end; end; theorem Th22: for L being non empty RelStr, N being non empty NetStr over L for x being Element of L, y being set holds y is Element of N iff y is Element of x "/\" N proof let L be non empty RelStr, N be non empty NetStr over L, x be Element of L, y be set; the RelStr of x "/\" N = the RelStr of N by Def3; hence y is Element of N iff y is Element of x "/\" N; end; definition let L be non empty RelStr, x be Element of L, N be non empty NetStr over L; cluster x "/\" N -> non empty; coherence proof the RelStr of x "/\" N = the RelStr of N by Def3; hence thesis by STRUCT_0:def 1; end; end; definition let L be non empty RelStr, x be Element of L, N be prenet of L; cluster x "/\" N -> directed; coherence proof A1: the RelStr of x "/\" N = the RelStr of N by Def3; A2: [#]N is directed by WAYBEL_0:def 6; [#]N = the carrier of N by PRE_TOPC:12 .= [#](x "/\" N) by A1,PRE_TOPC:12; hence [#](x "/\" N) is directed by A1,A2,WAYBEL_0:3; end; end; theorem Th23: for L being non empty RelStr, x being Element of L for F being non empty NetStr over L holds rng (the mapping of x "/\" F) = {x} "/\" rng the mapping of F proof let L be non empty RelStr, x be Element of L, F be non empty NetStr over L; set f = the mapping of F, h = the mapping of x "/\" F, A = rng the mapping of F; A1: the RelStr of x "/\" F = the RelStr of F by Def3; A2: {x} "/\" A = {x "/\" y where y is Element of L: y in A} by YELLOW_4:42; A3: dom f = the carrier of F by FUNCT_2:def 1; A4: dom h = the carrier of F by A1,FUNCT_2:def 1; thus rng the mapping of x "/\" F c= {x} "/\" A proof let q be set; assume q in rng h; then consider a being set such that A5: a in dom h & q = h.a by FUNCT_1:def 5; reconsider a as Element of x "/\" F by A5; consider y being Element of L such that A6: y = f.a & h.a = x "/\" y by Def3; y in A by A1,A3,A6,FUNCT_1:def 5; hence q in {x} "/\" A by A2,A5,A6; end; let q be set; assume q in {x} "/\" A; then consider y being Element of L such that A7: q = x "/\" y & y in A by A2; consider z being set such that A8: z in dom f & y = f.z by A7,FUNCT_1:def 5; reconsider z as Element of x "/\" F by A1,A8; consider w being Element of L such that A9: w = f.z & h.z = x "/\" w by Def3; thus q in rng h by A4,A7,A8,A9,FUNCT_1:def 5; end; theorem Th24: for L being non empty RelStr, J being set for f being Function of J,the carrier of L st for x being set holds ex_sup_of f.:x,L holds rng netmap(FinSups f,L) c= finsups rng f proof let L be non empty RelStr, J be set, f be Function of J,the carrier of L such that A1: for x being set holds ex_sup_of f.:x,L; set h = netmap(FinSups f,L); A2: finsups rng f = {"\/"(Y,L) where Y is finite Subset of rng f: ex_sup_of Y,L} by WAYBEL_0:def 27; consider g being Function of Fin J, the carrier of L such that A3: for x being Element of Fin J holds g.x = sup (f.:x) & FinSups f = NetStr (# Fin J, RelIncl Fin J, g #) by Def2; let q be set; assume q in rng h; then consider x being set such that A4: x in dom h & h.x = q by FUNCT_1:def 5; reconsider x as Element of Fin J by A3,A4; g = h by A3,WAYBEL_0:def 7; then A5: h.x = sup (f.:x) by A3; A6: f.:x is finite Subset of rng f by RELAT_1:144; ex_sup_of f.:x,L by A1; hence q in finsups rng f by A2,A4,A5,A6; end; theorem Th25: for L being non empty reflexive antisymmetric RelStr, J being set for f being Function of J,the carrier of L holds rng f c= rng netmap (FinSups f,L) proof let L be non empty reflexive antisymmetric RelStr, J be set, f be Function of J,the carrier of L; per cases; suppose A1: J is non empty; let a be set; assume a in rng f; then consider b being set such that A2: b in dom f & a = f.b by FUNCT_1:def 5; consider g being Function of Fin J, the carrier of L such that A3: for x being Element of Fin J holds g.x = sup (f.:x) & FinSups f = NetStr (# Fin J, RelIncl Fin J, g #) by Def2; A4: g = netmap (FinSups f,L) by A3,WAYBEL_0:def 7; reconsider b as Element of J by A2; {b} c= J by A1,ZFMISC_1:37; then reconsider x = {b} as Element of Fin J by FINSUB_1:def 5; dom g = Fin J by FUNCT_2:def 1; then A5: x in dom g; f.b in rng f by A2,FUNCT_1:def 5; then reconsider fb = f.b as Element of L; g.{b} = sup (f.:x) by A3 .= sup {fb} by A2,FUNCT_1:117 .= a by A2,YELLOW_0:39; hence a in rng netmap (FinSups f,L) by A4,A5,FUNCT_1:def 5; suppose A6: J is empty; dom f = J by FUNCT_2:def 1; then rng f = {} by A6,RELAT_1:65; hence thesis by XBOOLE_1:2; end; theorem Th26: for L being non empty reflexive antisymmetric RelStr, J being set for f being Function of J,the carrier of L st ex_sup_of rng f,L & ex_sup_of rng netmap (FinSups f,L),L & for x being Element of Fin J holds ex_sup_of f.:x,L holds Sup f = sup FinSups f proof let L be non empty reflexive antisymmetric RelStr, J be set, f be Function of J,the carrier of L such that A1: ex_sup_of rng f,L & ex_sup_of rng netmap (FinSups f,L),L and A2: for x being Element of Fin J holds ex_sup_of f.:x,L; set h = netmap (FinSups f,L); rng f c= rng h by Th25; then A3: "\/"(rng f,L) <= sup rng h by A1,YELLOW_0:34; rng h is_<=_than "\/"(rng f,L) proof let a be Element of L; assume a in rng h; then consider x being set such that A4: x in dom h & a = h.x by FUNCT_1:def 5; consider g being Function of Fin J, the carrier of L such that A5: for x being Element of Fin J holds g.x = sup (f.:x) & FinSups f = NetStr (# Fin J, RelIncl Fin J, g #) by Def2; reconsider x as Element of Fin J by A4,A5; A6: g = h by A5,WAYBEL_0:def 7; A7: ex_sup_of f.:x,L by A2; f.:x c= rng f by RELAT_1:144; then sup(f.:x) <= "\/"(rng f,L) by A1,A7,YELLOW_0:34; hence a <= "\/"(rng f,L) by A4,A5,A6; end; then A8: sup rng h <= "\/"(rng f,L) by A1,YELLOW_0:def 9; thus Sup f = "\/"(rng f,L) by YELLOW_2:def 5 .= sup rng netmap (FinSups f,L) by A3,A8,ORDERS_1:25 .= Sup netmap (FinSups f,L) by YELLOW_2:def 5 .= Sup the mapping of FinSups f by WAYBEL_0:def 7 .= sup FinSups f by Def1; end; theorem Th27: for L being with_infima antisymmetric transitive RelStr, N being prenet of L for x being Element of L st N is eventually-directed holds x "/\" N is eventually-directed proof let L be with_infima antisymmetric transitive RelStr, N be prenet of L, x be Element of L such that A1: N is eventually-directed; A2: the RelStr of x "/\" N = the RelStr of N by Def3; for i being Element of x "/\" N ex j being Element of x "/\" N st for k being Element of x "/\" N st j <= k holds (x "/\" N).i <= (x "/\" N).k proof let i1 be Element of x "/\" N; reconsider i = i1 as Element of N by Th22; consider j being Element of N such that A3: for k being Element of N st j <= k holds N.i <= N.k by A1,WAYBEL_0:11; reconsider j1 = j as Element of x "/\" N by Th22; take j1; let k1 be Element of x "/\" N; reconsider k = k1 as Element of N by Th22; assume j1 <= k1; then j <= k by A2,YELLOW_0:1; then A4: N.i <= N.k by A3; consider yi being Element of L such that A5: yi = (the mapping of N).i1 & (the mapping of x "/\" N).i1 = x "/\" yi by Def3; consider yk being Element of L such that A6: yk = (the mapping of N).k1 & (the mapping of x "/\" N).k1 = x "/\" yk by Def3; A7: (x "/\" N).i1 = x "/\" yi by A5,WAYBEL_0:def 8 .= x "/\" (N.i) by A5,WAYBEL_0:def 8; (x "/\" N).k1 = x "/\" yk by A6,WAYBEL_0:def 8 .= x "/\" (N.k) by A6,WAYBEL_0:def 8; hence (x "/\" N).i1 <= (x "/\" N).k1 by A4,A7,WAYBEL_1:2; end; hence x "/\" N is eventually-directed by WAYBEL_0:11; end; theorem Th28: for L being up-complete Semilattice st for x being Element of L, E being non empty directed Subset of L st x <= sup E holds x <= sup ({x} "/\" E) holds for D being non empty directed Subset of L, x being Element of L holds x "/\" sup D = sup ({x} "/\" D) proof let L be up-complete Semilattice such that A1: for x being Element of L, E being non empty directed Subset of L st x <= sup E holds x <= sup ({x} "/\" E); let D be non empty directed Subset of L, x be Element of L; ex w being Element of L st x >= w & sup D >= w & for c being Element of L st x >= c & sup D >= c holds w >= c by LATTICE3:def 11; then x "/\" sup D <= sup D by LATTICE3:def 14; then A2: x "/\" sup D <= sup ({x "/\" sup D} "/\" D) by A1; ex_sup_of D,L by WAYBEL_0:75; then A3: sup D is_>=_than D by YELLOW_0:30; reconsider T = {x "/\" sup D} as non empty directed Subset of L by WAYBEL_0:5; A4: ex_sup_of T "/\" D,L by WAYBEL_0:75; {x "/\" sup D} "/\" D is_<=_than x "/\" sup D by YELLOW_4:52; then sup ({x "/\" sup D} "/\" D) <= x "/\" sup D by A4,YELLOW_0:30; hence x "/\" sup D = sup ({x "/\" sup D} "/\" D) by A2,ORDERS_1:25 .= sup ({x} "/\" {sup D} "/\" D) by YELLOW_4:46 .= sup ({x} "/\" ({sup D} "/\" D)) by YELLOW_4:41 .= sup ({x} "/\" D) by A3,YELLOW_4:51; end; theorem for L being with_suprema Poset st for X being directed Subset of L, x being Element of L holds x "/\" sup X = sup ({x} "/\" X) holds for X being Subset of L, x being Element of L st ex_sup_of X,L holds x "/\" sup X = sup ({x} "/\" finsups X) proof let L be with_suprema Poset such that A1: for X being directed Subset of L, x being Element of L holds x "/\" sup X = sup ({x} "/\" X); let X be Subset of L, x be Element of L; assume ex_sup_of X,L; hence x "/\" sup X = x "/\" sup finsups X by WAYBEL_0:55 .= sup ({x} "/\" finsups X) by A1; end; theorem for L being up-complete LATTICE st for X being Subset of L, x being Element of L holds x "/\" sup X = sup ({x} "/\" finsups X) holds for X being non empty directed Subset of L, x being Element of L holds x "/\" sup X = sup ({x} "/\" X) proof let L be up-complete LATTICE such that A1: for X being Subset of L, x being Element of L holds x "/\" sup X = sup ({x} "/\" finsups X); let X be non empty directed Subset of L, x be Element of L; A2: {x} "/\" finsups X = {x "/\" y where y is Element of L : y in finsups X} by YELLOW_4:42; reconsider T = {x} as non empty directed Subset of L by WAYBEL_0:5; A3: ex_sup_of T "/\" X,L by WAYBEL_0:75; A4: ex_sup_of T "/\" finsups X,L by WAYBEL_0:75; {x} "/\" finsups X is_<=_than sup ({x} "/\" X) proof let q be Element of L; assume q in {x} "/\" finsups X; then consider y being Element of L such that A5: q = x "/\" y & y in finsups X by A2; finsups X = {"\/"(Y,L) where Y is finite Subset of X: ex_sup_of Y,L} by WAYBEL_0:def 27; then consider Y being finite Subset of X such that A6: y = "\/"(Y,L) & ex_sup_of Y,L by A5; consider z being Element of L such that A7: z in X & z is_>=_than Y by WAYBEL_0:1; A8: "\/"(Y,L) <= z by A6,A7,YELLOW_0:30; x in {x} by TARSKI:def 1; then x "/\" z in {x} "/\" X by A7,YELLOW_4:37; then A9: x "/\" z <= sup ({x} "/\" X) by A3,YELLOW_4:1; x <= x; then x "/\" y <= x "/\" z by A6,A8,YELLOW_3:2; hence q <= sup ({x} "/\" X) by A5,A9,YELLOW_0:def 2; end; then sup ({x} "/\" finsups X) <= sup ({x} "/\" X) by A4,YELLOW_0:30; then A10: x "/\" sup X <= sup ({x} "/\" X) by A1; ex_sup_of X,L by WAYBEL_0:75; then sup ({x} "/\" X) <= x "/\" sup X by A3,YELLOW_4:53; hence x "/\" sup X = sup ({x} "/\" X) by A10,ORDERS_1:25; end; begin :: On the inf and sup operation definition let L be non empty RelStr; func inf_op L -> map of [:L,L:], L means :Def4: for x, y being Element of L holds it.[x,y] = x "/\" y; existence proof deffunc F(Element of L, Element of L) = $1 "/\" $2; A1: for x, y being Element of L holds F(x,y) is Element of L; consider f being map of [:L,L:], L such that A2: for x, y being Element of L holds f.[x,y] = F(x,y) from Kappa2DS(A1); take f; thus thesis by A2; end; uniqueness proof let f, g be map of [:L,L:], L such that A3: for x, y being Element of L holds f.[x,y] = x "/\" y and A4: for x, y being Element of L holds g.[x,y] = x "/\" y; now let x, y be Element of L; thus f.[x,y] = x "/\" y by A3 .= g.[x,y] by A4; end; hence f = g by YELLOW_3:13; end; end; theorem Th31: for L being non empty RelStr, x being Element of [:L,L:] holds (inf_op L).x = x`1 "/\" x`2 proof let L be non empty RelStr, x be Element of [:L,L:]; the carrier of [:L,L:] = [:the carrier of L, the carrier of L:] by YELLOW_3:def 2; then consider a, b being set such that A1: a in the carrier of L & b in the carrier of L & x = [a,b] by ZFMISC_1:def 2; thus (inf_op L).x = (inf_op L).[x`1,x`2] by A1,MCART_1:8 .= x`1 "/\" x`2 by Def4; end; definition let L be with_infima transitive antisymmetric RelStr; cluster inf_op L -> monotone; coherence proof set f = inf_op L; let x, y be Element of [:L,L:]; assume x <= y; then A1: x`1 <= y`1 & x`2 <= y`2 by YELLOW_3:12; let a, b be Element of L such that A2: a = f.x & b = f.y; f.x = x`1 "/\" x`2 & f.y = y`1 "/\" y`2 by Th31; hence a <= b by A1,A2,YELLOW_3:2; end; end; theorem Th32: for S being non empty RelStr, D1, D2 being Subset of S holds (inf_op S).:[:D1,D2:] = D1 "/\" D2 proof let S be non empty RelStr, D1, D2 be Subset of S; set f = inf_op S; thus f.:[:D1,D2:] c= D1 "/\" D2 proof let q be set; assume A1: q in f.:[:D1,D2:]; then reconsider q1 = q as Element of S; consider c being Element of [:S,S:] such that A2: c in [:D1,D2:] & q1 = f.c by A1,FUNCT_2:116; consider x, y being set such that A3: x in D1 & y in D2 & c = [x,y] by A2,ZFMISC_1:def 2; reconsider x, y as Element of S by A3; q = x "/\" y by A2,A3,Def4; then q in { a "/\" b where a, b is Element of S : a in D1 & b in D2 } by A3; hence q in D1 "/\" D2 by YELLOW_4:def 4; end; reconsider X = [:D1,D2:] as set; let q be set; assume q in D1 "/\" D2; then q in { x "/\" y where x, y is Element of S : x in D1 & y in D2 } by YELLOW_4:def 4; then consider x, y being Element of S such that A4: q = x "/\" y & x in D1 & y in D2; A5: q = f.[x,y] by A4,Def4; [x,y] in X by A4,ZFMISC_1:106; hence thesis by A5,FUNCT_2:43; end; theorem Th33: for L being up-complete Semilattice for D being non empty directed Subset of [:L,L:] holds sup ((inf_op L).:D) = sup (proj1 D "/\" proj2 D) proof let L be up-complete Semilattice, D be non empty directed Subset of [:L,L:]; reconsider C = the carrier of L as non empty set; D c= the carrier of [:L,L:]; then D c= [:the carrier of L,the carrier of L:] by YELLOW_3:def 2; then reconsider D' = D as non empty Subset of [:C,C:]; reconsider D1 = proj1 D, D2 = proj2 D as non empty directed Subset of L by YELLOW_3:21,22; set f = inf_op L; A1: D' c= [:D1,D2:] by YELLOW_3:1; A2: f.:D is directed by YELLOW_2:17; then A3: ex_sup_of f.:D,L by WAYBEL_0:75; A4: ex_sup_of D1 "/\" D2,L by WAYBEL_0:75; downarrow (f.:D) is directed by A2,WAYBEL_0:30; then A5: ex_sup_of downarrow (f.:D),L by WAYBEL_0:75; [:D1,D2:] c= downarrow D by YELLOW_3:48; then A6: f.:[:D1,D2:] c= f.:(downarrow D) by RELAT_1:156; A7: f.:(downarrow D) c= downarrow (f.:D) by Th13; f.:D' c= f.:[:D1,D2:] by A1,RELAT_1:156; then f.:D' c= D1 "/\" D2 by Th32; then A8: sup (f.:D) <= sup (D1 "/\" D2) by A3,A4,YELLOW_0:34; A9: f.:[:D1,D2:] c= downarrow (f.:D) by A6,A7,XBOOLE_1:1; f.:[:D1,D2:] = D1 "/\" D2 by Th32; then sup (D1 "/\" D2) <= sup (downarrow (f.: D)) by A4,A5,A9,YELLOW_0:34; then sup (D1 "/\" D2) <= sup(f.:D) by A3,WAYBEL_0:33; hence thesis by A8,ORDERS_1:25; end; definition let L be non empty RelStr; func sup_op L -> map of [:L,L:], L means :Def5: for x, y being Element of L holds it.[x,y] = x "\/" y; existence proof deffunc F(Element of L, Element of L) = $1 "\/" $2; A1: for x, y being Element of L holds F(x,y) is Element of L; consider f being map of [:L,L:], L such that A2: for x, y being Element of L holds f.[x,y] = F(x,y) from Kappa2DS(A1); take f; thus thesis by A2; end; uniqueness proof let f, g be map of [:L,L:], L such that A3: for x, y being Element of L holds f.[x,y] = x "\/" y and A4: for x, y being Element of L holds g.[x,y] = x "\/" y; now let x, y be Element of L; thus f.[x,y] = x "\/" y by A3 .= g.[x,y] by A4; end; hence f = g by YELLOW_3:13; end; end; theorem Th34: for L being non empty RelStr, x being Element of [:L,L:] holds (sup_op L).x = x`1 "\/" x`2 proof let L be non empty RelStr, x be Element of [:L,L:]; the carrier of [:L,L:] = [:the carrier of L, the carrier of L:] by YELLOW_3:def 2; then consider a, b being set such that A1: a in the carrier of L & b in the carrier of L & x = [a,b] by ZFMISC_1:def 2; thus (sup_op L).x = (sup_op L).[x`1,x`2] by A1,MCART_1:8 .= x`1 "\/" x`2 by Def5; end; definition let L be with_suprema transitive antisymmetric RelStr; cluster sup_op L -> monotone; coherence proof set f = sup_op L; let x, y be Element of [:L,L:]; assume x <= y; then A1: x`1 <= y`1 & x`2 <= y`2 by YELLOW_3:12; let a, b be Element of L such that A2: a = f.x & b = f.y; f.x = x`1 "\/" x`2 & f.y = y`1 "\/" y`2 by Th34; hence a <= b by A1,A2,YELLOW_3:3; end; end; theorem Th35: for S being non empty RelStr, D1, D2 being Subset of S holds (sup_op S).:[:D1,D2:] = D1 "\/" D2 proof let S be non empty RelStr, D1, D2 be Subset of S; set f = sup_op S; thus f.:[:D1,D2:] c= D1 "\/" D2 proof let q be set; assume A1: q in f.:[:D1,D2:]; then reconsider q1 = q as Element of S; consider c being Element of [:S,S:] such that A2: c in [:D1,D2:] & q1 = f.c by A1,FUNCT_2:116; consider x, y being set such that A3: x in D1 & y in D2 & c = [x,y] by A2,ZFMISC_1:def 2; reconsider x, y as Element of S by A3; q = x "\/" y by A2,A3,Def5; then q in { a "\/" b where a, b is Element of S : a in D1 & b in D2 } by A3; hence q in D1 "\/" D2 by YELLOW_4:def 3; end; reconsider X = [:D1,D2:] as set; let q be set; assume q in D1 "\/" D2; then q in { x "\/" y where x, y is Element of S : x in D1 & y in D2 } by YELLOW_4:def 3; then consider x, y being Element of S such that A4: q = x "\/" y & x in D1 & y in D2; A5: q = f.[x,y] by A4,Def5; [x,y] in X by A4,ZFMISC_1:106; hence thesis by A5,FUNCT_2:43; end; theorem for L being complete (non empty Poset) for D being non empty filtered Subset of [:L,L:] holds inf ((sup_op L).:D) = inf (proj1 D "\/" proj2 D) proof let L be complete (non empty Poset), D be non empty filtered Subset of [:L,L:]; reconsider C = the carrier of L as non empty set; D c= the carrier of [:L,L:]; then D c= [:the carrier of L,the carrier of L:] by YELLOW_3:def 2; then reconsider D' = D as non empty Subset of [:C,C:]; set D1 = proj1 D, D2 = proj2 D, f = sup_op L; A1: D' c= [:D1,D2:] by YELLOW_3:1; A2: ex_inf_of f.:D',L by YELLOW_0:17; A3: ex_inf_of D1 "\/" D2,L by YELLOW_0:17; A4: ex_inf_of uparrow (f.:D),L by YELLOW_0:17; [:D1,D2:] c= uparrow D by YELLOW_3:49; then A5: f.:[:D1,D2:] c= f.:(uparrow D) by RELAT_1:156; A6: f.:(uparrow D) c= uparrow (f.:D) by Th14; f.:D' c= f.:[:D1,D2:] by A1,RELAT_1:156; then f.:D' c= D1 "\/" D2 by Th35; then A7: inf (f.:D) >= inf (D1 "\/" D2) by A2,A3,YELLOW_0:35; A8: f.:[:D1,D2:] c= uparrow (f.:D) by A5,A6,XBOOLE_1:1; f.:[:D1,D2:] = D1 "\/" D2 by Th35; then inf (D1 "\/" D2) >= inf (uparrow (f.:D)) by A3,A4,A8,YELLOW_0:35; then inf (D1 "\/" D2) >= inf(f.:D) by A2,WAYBEL_0:38; hence thesis by A7,ORDERS_1:25; end; begin :: Meet-continuous lattices :: Def 4.1, s.30 definition let R be non empty reflexive RelStr; attr R is satisfying_MC means :Def6: for x being Element of R, D being non empty directed Subset of R holds x "/\" sup D = sup ({x} "/\" D); end; definition let R be non empty reflexive RelStr; attr R is meet-continuous means :Def7: R is up-complete satisfying_MC; end; definition cluster trivial -> satisfying_MC (non empty reflexive RelStr); coherence proof let IT be non empty reflexive RelStr; assume IT is trivial; then reconsider I = IT as trivial (non empty reflexive RelStr); let x be Element of IT, D be non empty directed Subset of IT; consider a being Element of I such that A1: the carrier of I = {a} by TEX_1:def 1; thus x "/\" sup D = sup ({x} "/\" D) by A1,REALSET1:def 20; end; end; definition cluster meet-continuous -> up-complete satisfying_MC (non empty reflexive RelStr); coherence by Def7; cluster up-complete satisfying_MC -> meet-continuous (non empty reflexive RelStr); coherence by Def7; end; definition cluster strict non empty trivial LATTICE; existence proof consider B being strict trivial non empty reflexive RelStr; take B; thus thesis; end; end; theorem Th37: for S being non empty reflexive RelStr st for X being Subset of S, x being Element of S holds x "/\" sup X = "\/"({x"/\"y where y is Element of S: y in X},S) holds S is satisfying_MC proof let S be non empty reflexive RelStr such that A1: for X being Subset of S, x being Element of S holds x "/\" sup X = "\/"({x"/\"y where y is Element of S: y in X},S); let y be Element of S, D be non empty directed Subset of S; thus sup ({y} "/\" D) = "\/"({y"/\"z where z is Element of S: z in D},S) by YELLOW_4:42 .= y "/\" sup D by A1; end; :: Th 4.2, s.30, 1 => 2 theorem Th38: for L being up-complete Semilattice st SupMap L is meet-preserving holds for I1, I2 being Ideal of L holds (sup I1) "/\" (sup I2) = sup (I1 "/\" I2) proof let L be up-complete Semilattice such that A1: SupMap L is meet-preserving; let I1, I2 be Ideal of L; reconsider x = I1, y = I2 as Element of InclPoset(Ids L) by YELLOW_2:43; set f = SupMap L; A2: f preserves_inf_of {x,y} by A1,WAYBEL_0:def 34; reconsider fx = f.x as Element of L; thus (sup I1) "/\" (sup I2) = fx "/\" (sup I2) by YELLOW_2:def 3 .= f.x "/\" f.y by YELLOW_2:def 3 .= f.(x "/\" y) by A2,YELLOW_3:8 .= f.(I1 "/\" I2) by YELLOW_4:58 .= sup (I1 "/\" I2) by YELLOW_2:def 3; end; definition let L be up-complete sup-Semilattice; cluster SupMap L -> join-preserving; coherence proof set f = SupMap L; A1: dom f = the carrier of InclPoset(Ids L) by FUNCT_2:def 1; let x, y be Element of InclPoset(Ids L); assume ex_sup_of {x,y},InclPoset(Ids L); f.:{x,y} = {f.x,f.y} by A1,FUNCT_1:118; hence ex_sup_of f.:{x,y},L by YELLOW_0:20; reconsider x1 = x, y1 = y as Ideal of L by YELLOW_2:43; A2: ex_sup_of x1 "\/" y1,L by WAYBEL_0:75; thus sup (f.:{x,y}) = sup {f.x,f.y} by A1,FUNCT_1:118 .= f.x "\/" f.y by YELLOW_0:41 .= f.x "\/" sup y1 by YELLOW_2:def 3 .= sup x1 "\/" sup y1 by YELLOW_2:def 3 .= sup (x1 "\/" y1) by Th4 .= sup downarrow (x1 "\/" y1) by A2,WAYBEL_0:33 .= f.(downarrow (x1 "\/" y1)) by YELLOW_2:def 3 .= f.(x "\/" y) by YELLOW_4:30 .= f.sup {x,y} by YELLOW_0:41; end; end; :: Th 4.2, s.30, 2 => 1 theorem Th39: for L being up-complete Semilattice st for I1, I2 being Ideal of L holds (sup I1) "/\" (sup I2) = sup (I1 "/\" I2) holds SupMap L is meet-preserving proof let L be up-complete Semilattice such that A1: for I1, I2 being Ideal of L holds (sup I1) "/\" (sup I2) = sup (I1 "/\" I2); set f = SupMap L; A2: dom f = the carrier of InclPoset(Ids L) by FUNCT_2:def 1; let x, y be Element of InclPoset(Ids L); assume ex_inf_of {x,y},InclPoset(Ids L); f.:{x,y} = {f.x,f.y} by A2,FUNCT_1:118; hence ex_inf_of f.:{x,y},L by YELLOW_0:21; reconsider x1 = x, y1 = y as Ideal of L by YELLOW_2:43; thus inf (f.:{x,y}) = inf {f.x,f.y} by A2,FUNCT_1:118 .= f.x "/\" f.y by YELLOW_0:40 .= f.x "/\" (sup y1) by YELLOW_2:def 3 .= (sup x1) "/\" (sup y1) by YELLOW_2:def 3 .= sup (x1 "/\" y1) by A1 .= f.(x1 "/\" y1) by YELLOW_2:def 3 .= f.(x "/\" y) by YELLOW_4:58 .= f.inf {x,y} by YELLOW_0:40; end; :: Th 4.2, s.30, 2 => 3 theorem Th40: for L being up-complete Semilattice st for I1, I2 being Ideal of L holds (sup I1) "/\" (sup I2) = sup (I1 "/\" I2) holds for D1, D2 be directed non empty Subset of L holds (sup D1) "/\" (sup D2) = sup (D1 "/\" D2) proof let L be up-complete Semilattice such that A1: for I1, I2 being Ideal of L holds (sup I1) "/\" (sup I2) = sup (I1 "/\" I2); let D1, D2 be directed non empty Subset of L; A2: ex_sup_of D1,L & ex_sup_of downarrow D1,L by WAYBEL_0:75; A3: ex_sup_of D2,L & ex_sup_of downarrow D2,L by WAYBEL_0:75; A4: ex_sup_of D1 "/\" D2,L & ex_sup_of downarrow (D1 "/\" D2),L by WAYBEL_0:75; A5: ex_sup_of (downarrow D1 "/\" downarrow D2),L & ex_sup_of downarrow (downarrow D1 "/\" downarrow D2),L by WAYBEL_0:75; thus (sup D1) "/\" (sup D2) = (sup downarrow D1) "/\" (sup D2) by A2,WAYBEL_0:33 .= (sup downarrow D1) "/\" (sup downarrow D2) by A3,WAYBEL_0:33 .= sup (downarrow D1 "/\" downarrow D2) by A1 .= sup downarrow ((downarrow D1) "/\" (downarrow D2)) by A5,WAYBEL_0:33 .= sup downarrow (D1 "/\" D2) by YELLOW_4:62 .= sup (D1 "/\" D2) by A4,WAYBEL_0:33; end; :: Th 4.2, s.30, 4 => 7 theorem Th41: for L being non empty reflexive RelStr st L is satisfying_MC holds for x being Element of L, N being non empty prenet of L st N is eventually-directed holds x "/\" sup N = sup ({x} "/\" rng netmap (N,L)) proof let L be non empty reflexive RelStr such that A1: L is satisfying_MC; let x be Element of L, N be non empty prenet of L; assume N is eventually-directed; then A2: rng netmap (N,L) is directed by Th18; thus x "/\" sup N = x "/\" Sup the mapping of N by Def1 .= x "/\" Sup netmap (N,L) by WAYBEL_0:def 7 .= x "/\" sup rng netmap (N,L) by YELLOW_2:def 5 .= sup ({x} "/\" rng netmap (N,L)) by A1,A2,Def6; end; :: Th 4.2, s.30, 7 => 4 theorem Th42: for L being non empty reflexive RelStr st for x being Element of L, N being prenet of L st N is eventually-directed holds x "/\" sup N = sup ({x} "/\" rng netmap (N,L)) holds L is satisfying_MC proof let L be non empty reflexive RelStr such that A1: for x being Element of L, N being prenet of L st N is eventually-directed holds x "/\" sup N = sup ({x} "/\" rng netmap (N,L)); let x be Element of L, D be non empty directed Subset of L; reconsider n = id D as Function of D, the carrier of L by FUNCT_2:9; reconsider N = NetStr (#D,(the InternalRel of L)|_2 D,n#) as prenet of L by Th19; D c= D; then A2: D = n.:D by BORSUK_1:3 .= rng n by FUNCT_2:45 .= rng netmap (N,L) by WAYBEL_0:def 7; A3: N is eventually-directed by Th20; A4: Sup netmap (N,L) = Sup the mapping of N by WAYBEL_0:def 7 .= sup N by Def1; thus x "/\" sup D = x "/\" Sup netmap (N,L) by A2,YELLOW_2:def 5 .= sup ({x} "/\" D) by A1,A2,A3,A4; end; :: Th 4.2, s.30, 6 => 3 theorem Th43: for L being up-complete antisymmetric (non empty reflexive RelStr) st inf_op L is directed-sups-preserving holds for D1, D2 being non empty directed Subset of L holds (sup D1) "/\" (sup D2) = sup (D1 "/\" D2) proof let L be up-complete antisymmetric (non empty reflexive RelStr) such that A1: inf_op L is directed-sups-preserving; let D1, D2 be non empty directed Subset of L; set X = [:D1,D2:], f = inf_op L; A2: f preserves_sup_of X by A1,WAYBEL_0:def 37; A3: ex_sup_of X,[:L,L:] by WAYBEL_0:75; A4: ex_sup_of D1,L & ex_sup_of D2,L by WAYBEL_0:75; thus (sup D1) "/\" (sup D2) = f.[sup D1,sup D2] by Def4 .= f.(sup X) by A4,YELLOW_3:43 .= sup (f.:X) by A2,A3,WAYBEL_0:def 31 .= sup (D1 "/\" D2) by Th32; end; :: Th 4.2, s.30, 3 => 4 theorem Th44: for L being non empty reflexive antisymmetric RelStr st for D1, D2 being non empty directed Subset of L holds (sup D1) "/\" (sup D2) = sup (D1 "/\" D2) holds L is satisfying_MC proof let L be non empty reflexive antisymmetric RelStr such that A1: for D1, D2 being non empty directed Subset of L holds (sup D1) "/\" (sup D2) = sup (D1 "/\" D2); let x be Element of L, D be non empty directed Subset of L; A2: {x} is directed by WAYBEL_0:5; thus x "/\" sup D = sup {x} "/\" sup D by YELLOW_0:39 .= sup ({x} "/\" D) by A1,A2; end; :: Th 4.2, s.30, MC => 5 theorem Th45: for L being satisfying_MC with_infima antisymmetric (non empty reflexive RelStr) holds for x being Element of L, D being non empty directed Subset of L st x <= sup D holds x = sup ({x} "/\" D) proof let L be satisfying_MC with_infima antisymmetric (non empty reflexive RelStr); let x be Element of L, D be non empty directed Subset of L; assume x <= sup D; hence x = x "/\" sup D by YELLOW_0:25 .= sup ({x} "/\" D) by Def6; end; :: Th 4.2, s.30, 5 => 6 theorem Th46: for L being up-complete Semilattice st for x being Element of L, E being non empty directed Subset of L st x <= sup E holds x <= sup ({x} "/\" E) holds inf_op L is directed-sups-preserving proof let L be up-complete Semilattice such that A1: for x being Element of L, E being non empty directed Subset of L st x <= sup E holds x <= sup ({x} "/\" E); let D be Subset of [:L,L:]; assume D is non empty directed; then reconsider DD = D as non empty directed Subset of [:L,L:]; thus inf_op L preserves_sup_of D proof set f = inf_op L; assume ex_sup_of D,[:L,L:]; f.:DD is directed by YELLOW_2:17; hence ex_sup_of f.:D,L by WAYBEL_0:75; reconsider C = the carrier of L as non empty set; reconsider D1 = proj1 DD, D2 = proj2 DD as non empty directed Subset of L by YELLOW_3:21,22; set d2 = sup D2; {x "/\" y where x, y is Element of L : x in D1 & y in {d2}} c= C proof let q be set; assume q in {x "/\" y where x, y is Element of L : x in D1 & y in {d2}}; then consider x, y being Element of L such that A2: q = x "/\" y & x in D1 & y in {d2}; thus q in C by A2; end; then reconsider F = {x "/\" y where x, y is Element of L : x in D1 & y in {d2}} as Subset of L; A3: F = { sup ({z} "/\" D2) where z is Element of L : z in D1 } proof thus F c= { sup ({z} "/\" D2) where z is Element of L : z in D1 } proof let q be set; assume q in F; then consider x, y being Element of L such that A4: q = x "/\" y & x in D1 & y in {d2}; q = x "/\" d2 by A4,TARSKI:def 1 .= sup ({x} "/\" D2) by A1,Th28; hence q in { sup ({z} "/\" D2) where z is Element of L : z in D1 } by A4; end; let q be set; assume q in { sup ({z} "/\" D2) where z is Element of L : z in D1 }; then consider z being Element of L such that A5: q = sup ({z} "/\" D2) & z in D1; A6: q = z "/\" d2 by A1,A5,Th28; d2 in {d2} by TARSKI:def 1; hence q in F by A5,A6; end; defpred P[set] means ex x being Element of L st $1 = {x} "/\" D2 & x in D1; A7: "\/" ({ sup X where X is non empty directed Subset of L: P[X] },L) = "\/" (union {X where X is non empty directed Subset of L: P[X]},L) by Th10; thus sup (f.:D) = sup (D1 "/\" D2) by Th33 .= "\/" ({ "\/" (X,L) where X is non empty directed Subset of L: P[X] },L) by A7,Th6 .= "\/" ({ sup ({x} "/\" D2) where x is Element of L : x in D1 },L) by Th5 .= sup ({d2} "/\" D1) by A3,YELLOW_4:def 4 .= sup D1 "/\" d2 by A1,Th28 .= f.[sup D1,sup D2] by Def4 .= f.sup D by Th12; end; end; :: Th 4.2, s.30, 7 => 8 theorem Th47: for L being complete antisymmetric (non empty reflexive RelStr) st for x being Element of L, N being prenet of L st N is eventually-directed holds x "/\" sup N = sup ({x} "/\" rng netmap (N,L)) holds for x being Element of L, J being set, f being Function of J,the carrier of L holds x "/\" Sup f = sup(x "/\" FinSups f) proof let L be complete antisymmetric (non empty reflexive RelStr) such that A1: for x being Element of L, N being prenet of L st N is eventually-directed holds x "/\" sup N = sup ({x} "/\" rng netmap (N,L)); let x be Element of L, J be set, f be Function of J,the carrier of L; set F = FinSups f; A2: ex_sup_of rng f,L by YELLOW_0:17; A3: ex_sup_of rng netmap (FinSups f,L),L by YELLOW_0:17; A4: for x being Element of Fin J holds ex_sup_of f.:x,L by YELLOW_0:17; consider g being Function of Fin J, the carrier of L such that A5: for x being Element of Fin J holds g.x = sup (f.:x) & F = NetStr (# Fin J, RelIncl Fin J, g #) by Def2; A6: g = netmap (F,L) by A5,WAYBEL_0:def 7; A7: Sup netmap (F,L) = Sup the mapping of F by WAYBEL_0:def 7 .= sup F by Def1; thus x "/\" Sup f = x "/\" sup F by A2,A3,A4,Th26 .= x "/\" Sup the mapping of F by Def1 .= x "/\" Sup netmap (F,L) by WAYBEL_0:def 7 .= sup ({x} "/\" rng netmap (F,L)) by A1,A7 .= "\/" (rng the mapping of x "/\" F,L) by A5,A6,Th23 .= Sup (the mapping of x "/\" F) by YELLOW_2:def 5 .= sup (x "/\" F) by Def1; end; :: Th 4.2, s.30, 8 => 7 theorem Th48: for L being complete Semilattice st for x being Element of L, J being set, f being Function of J,the carrier of L holds x "/\" Sup f = sup (x "/\" FinSups f) holds for x being Element of L, N being prenet of L st N is eventually-directed holds x "/\" sup N = sup ({x} "/\" rng netmap (N,L)) proof let L be complete Semilattice such that A1: for x being Element of L, J being set for f being Function of J,the carrier of L holds x "/\" Sup f = sup (x "/\" FinSups f); let x be Element of L, N be prenet of L such that A2: N is eventually-directed; A3: Sup netmap (N,L) = Sup the mapping of N by WAYBEL_0:def 7 .= sup N by Def1; set f = the mapping of N; reconsider R = rng netmap (N,L) as non empty directed Subset of L by A2,Th18; A4: ex_sup_of R,L by WAYBEL_0:75; reconsider xx = {x} as non empty directed Subset of L by WAYBEL_0:5; A5: ex_sup_of xx "/\" R ,L by WAYBEL_0:75; Sup netmap (N,L) = "\/"(rng netmap(N,L),L) by YELLOW_2:def 5; then A6: sup ({x} "/\" rng netmap (N,L)) <= x "/\" Sup netmap (N,L) by A4,A5,YELLOW_4:53; Sup (the mapping of x "/\" FinSups f) = "\/"(rng the mapping of x "/\" FinSups f,L) by YELLOW_2:def 5; then A7: sup (x "/\" FinSups f) = "\/"(rng the mapping of x "/\" FinSups f,L) by Def1; set h = the mapping of FinSups f; x "/\" FinSups f is eventually-directed by Th27; then rng netmap(x "/\" FinSups f,L) is directed by Th18; then rng the mapping of x "/\" FinSups f is directed by WAYBEL_0:def 7; then A8: ex_sup_of rng the mapping of x "/\" FinSups f,L by WAYBEL_0:75; rng the mapping of x "/\" FinSups f is_<=_than sup ({x} "/\" rng netmap (N,L)) proof let a be Element of L; assume a in rng the mapping of x "/\" FinSups f; then A9: a in {x} "/\" rng h by Th23; {x} "/\" rng h = {x "/\" y where y is Element of L: y in rng h} by YELLOW_4:42; then consider y being Element of L such that A10: a = x "/\" y & y in rng h by A9; A11: finsups rng f = {"\/"(Y,L) where Y is finite Subset of rng f: ex_sup_of Y,L} by WAYBEL_0:def 27; A12: for x being set holds ex_sup_of f.:x,L by YELLOW_0:17; A13: y in rng netmap(FinSups f,L) by A10,WAYBEL_0:def 7; rng netmap(FinSups f,L) c= finsups rng f by A12,Th24; then y in finsups rng f by A13; then consider Y being finite Subset of rng f such that A14: y = "\/"(Y,L) & ex_sup_of Y,L by A11; rng netmap (N,L) is directed by A2,Th18; then rng f is directed by WAYBEL_0:def 7; then consider z being Element of L such that A15: z in rng f & z is_>=_than Y by WAYBEL_0:1; A16: "\/"(Y,L) <= z by A14,A15,YELLOW_0:30; A17: netmap(N,L) = f by WAYBEL_0:def 7; x in {x} by TARSKI:def 1; then x "/\" z in {x} "/\" rng f by A15,YELLOW_4:37; then A18: x "/\" z <= sup (xx "/\" rng f) by A5,A17,YELLOW_4:1; x <= x; then x "/\" y <= x "/\" z by A14,A16,YELLOW_3:2; then a <= sup ({x} "/\" rng f) by A10,A18,YELLOW_0:def 2; hence a <= sup ({x} "/\" rng netmap (N,L)) by WAYBEL_0:def 7; end; then A19: "\/"(rng the mapping of x "/\" FinSups f,L) <= sup ({x} "/\" rng netmap (N,L)) by A8,YELLOW_0:def 9; x "/\" Sup f = sup(x "/\" FinSups f) by A1; then x "/\" Sup netmap (N,L) <= sup ({x} "/\" rng netmap (N,L)) by A7,A19, WAYBEL_0:def 7; hence x "/\" sup N = sup ({x} "/\" rng netmap (N,L)) by A3,A6,ORDERS_1:25; end; :: Th 4.2, s.30, 4 <=> 1 theorem Th49: for L being up-complete LATTICE holds L is meet-continuous iff SupMap L is meet-preserving join-preserving proof let L be up-complete LATTICE; hereby assume L is meet-continuous; then A1: L is satisfying_MC by Def7; for D1, D2 being non empty directed Subset of L holds (sup D1) "/\" (sup D2) = sup (D1 "/\" D2) proof let D1, D2 be non empty directed Subset of L; for x being Element of L, E being non empty directed Subset of L st x <= sup E holds x <= sup ({x} "/\" E) by A1,Th45; then inf_op L is directed-sups-preserving by Th46; hence (sup D1) "/\" (sup D2) = sup (D1 "/\" D2) by Th43; end; then for I1, I2 being Ideal of L holds (sup I1) "/\" (sup I2) = sup (I1 "/\" I2); hence SupMap L is meet-preserving join-preserving by Th39; end; assume A2: SupMap L is meet-preserving join-preserving; thus L is up-complete; for I1, I2 being Ideal of L holds (sup I1) "/\" (sup I2) = sup (I1 "/\" I2 ) by A2,Th38; then for D1, D2 be non empty directed Subset of L holds (sup D1) "/\" (sup D2) = sup (D1 "/\" D2) by Th40; hence L is satisfying_MC by Th44; end; definition let L be meet-continuous LATTICE; cluster SupMap L -> meet-preserving join-preserving; coherence by Th49; end; :: Th 4.2, s.30, 4 <=> 2 theorem Th50: for L being up-complete LATTICE holds L is meet-continuous iff for I1, I2 being Ideal of L holds (sup I1) "/\" (sup I2) = sup (I1 "/\" I2) proof let L be up-complete LATTICE; hereby assume L is meet-continuous; then SupMap L is meet-preserving join-preserving by Th49; hence for I1, I2 being Ideal of L holds (sup I1) "/\" (sup I2) = sup (I1 "/\" I2) by Th38; end; assume for I1, I2 being Ideal of L holds (sup I1) "/\" (sup I2) = sup (I1 "/\" I2); then for D1, D2 be directed non empty Subset of L holds (sup D1) "/\" (sup D2) = sup (D1 "/\" D2) by Th40; hence L is up-complete & L is satisfying_MC by Th44; end; :: Th 4.2, s.30, 4 <=> 3 theorem Th51: for L being up-complete LATTICE holds L is meet-continuous iff for D1, D2 being non empty directed Subset of L holds (sup D1) "/\" (sup D2) = sup (D1 "/\" D2) proof let L be up-complete LATTICE; hereby assume L is meet-continuous; then for I1, I2 being Ideal of L holds (sup I1) "/\" (sup I2) = sup (I1 "/\" I2) by Th50; hence for D1, D2 be directed non empty Subset of L holds (sup D1) "/\" (sup D2) = sup (D1 "/\" D2) by Th40; end; assume for D1, D2 being non empty directed Subset of L holds (sup D1) "/\" (sup D2) = sup (D1 "/\" D2); hence L is up-complete & L is satisfying_MC by Th44; end; :: Th 4.2, s.30, 4 <=> 5 theorem for L being up-complete LATTICE holds L is meet-continuous iff for x being Element of L, D being non empty directed Subset of L st x <= sup D holds x = sup ({x} "/\" D) proof let L be up-complete LATTICE; hereby assume L is meet-continuous; then L is satisfying_MC by Def7; hence for x being Element of L, D being non empty directed Subset of L st x <= sup D holds x = sup ({x} "/\" D) by Th45; end; assume for x being Element of L, D being non empty directed Subset of L st x <= sup D holds x = sup ({x} "/\" D); then for x being Element of L, D being non empty directed Subset of L st x <= sup D holds x <= sup ({x} "/\" D); then inf_op L is directed-sups-preserving by Th46; then for D1, D2 being non empty directed Subset of L holds (sup D1) "/\" (sup D2) = sup (D1 "/\" D2) by Th43; hence thesis by Th51; end; :: Th 4.2, s.30, 4 <=> 6 theorem Th53: for L being up-complete Semilattice holds L is meet-continuous iff inf_op L is directed-sups-preserving proof let L be up-complete Semilattice; hereby assume L is meet-continuous; then L is satisfying_MC by Def7; then for x being Element of L, D being non empty directed Subset of L st x <= sup D holds x <= sup ({x} "/\" D) by Th45; hence inf_op L is directed-sups-preserving by Th46; end; assume inf_op L is directed-sups-preserving; then for D1, D2 being non empty directed Subset of L holds (sup D1) "/\" (sup D2) = sup (D1 "/\" D2) by Th43; hence L is up-complete & L is satisfying_MC by Th44; end; definition let L be meet-continuous Semilattice; cluster inf_op L -> directed-sups-preserving; coherence by Th53; end; :: Th 4.2, s.30, 4 <=> 7 theorem Th54: for L being up-complete Semilattice holds L is meet-continuous iff for x being Element of L, N being non empty prenet of L st N is eventually-directed holds x "/\" sup N = sup ({x} "/\" rng netmap (N,L)) proof let L be up-complete Semilattice; hereby assume L is meet-continuous; then L is satisfying_MC by Def7; hence for x being Element of L, N being non empty prenet of L st N is eventually-directed holds x "/\" sup N = sup ({x} "/\" rng netmap (N,L)) by Th41; end; assume for x being Element of L, N being non empty prenet of L st N is eventually-directed holds x "/\" sup N = sup ({x} "/\" rng netmap (N,L)); hence L is up-complete & L is satisfying_MC by Th42; end; :: Th 4.2, s.30, 4 <=> 8 theorem for L being complete Semilattice holds L is meet-continuous iff for x being Element of L, J being set for f being Function of J,the carrier of L holds x "/\" Sup f = sup(x "/\" FinSups f) proof let L be complete Semilattice; hereby assume L is meet-continuous; then for x being Element of L, N being non empty prenet of L st N is eventually-directed holds x "/\" sup N = sup ({x} "/\" rng netmap (N,L)) by Th54; hence for x being Element of L, J being set for f being Function of J,the carrier of L holds x "/\" Sup f = sup(x "/\" FinSups f) by Th47; end; assume for x being Element of L, J being set for f being Function of J,the carrier of L holds x "/\" Sup f = sup(x "/\" FinSups f); then for x being Element of L, N being prenet of L st N is eventually-directed holds x "/\" sup N = sup ({x} "/\" rng netmap (N,L)) by Th48; hence L is up-complete & L is satisfying_MC by Th42; end; Lm1: for L being meet-continuous Semilattice, x being Element of L holds x "/\" is directed-sups-preserving proof let L be meet-continuous Semilattice, x be Element of L; let X be Subset of L such that A1: X is non empty directed; assume ex_sup_of X,L; A2: x"/\".:X is directed by A1,YELLOW_2:17; reconsider X1 = X as non empty Subset of L by A1; x"/\".:X1 is non empty; hence ex_sup_of x"/\".:X,L by A2,WAYBEL_0:75; A3: for x being Element of L, E being non empty directed Subset of L st x <= sup E holds x <= sup ({x} "/\" E) by Th45; thus sup (x"/\".:X) = "\/"({x"/\"y where y is Element of L: y in X},L) by WAYBEL_1:64 .= sup({x} "/\" X) by YELLOW_4:42 .= x "/\" sup X by A1,A3,Th28 .= x"/\".sup X by WAYBEL_1:def 18; end; definition let L be meet-continuous Semilattice, x be Element of L; cluster x "/\" -> directed-sups-preserving; coherence by Lm1; end; :: Remark 4.3 s.31 theorem Th56: for H being complete (non empty Poset) holds H is Heyting iff H is meet-continuous distributive proof let H be complete (non empty Poset); hereby assume A1: H is Heyting; then for x being Element of H holds x "/\" has_an_upper_adjoint by WAYBEL_1:def 19; then A2: for X being Subset of H, x being Element of H holds x "/\" sup X = "\/"({x"/\"y where y is Element of H: y in X},H) by WAYBEL_1:67; thus H is meet-continuous proof thus H is up-complete satisfying_MC by A2,Th37; end; thus H is distributive by A1,WAYBEL_1:69; end; assume that A3: H is meet-continuous distributive; thus H is LATTICE; let a be Element of H; A4: for X being finite Subset of H holds a "/\" preserves_sup_of X by A3,Th17; for X being non empty directed Subset of H holds a "/\" preserves_sup_of X proof let X be non empty directed Subset of H; a "/\" is directed-sups-preserving by A3,Lm1; hence a "/\" preserves_sup_of X by WAYBEL_0:def 37; end; then a "/\" is sups-preserving by A4,WAYBEL_0:74; hence a "/\" has_an_upper_adjoint by WAYBEL_1:18; end; definition cluster complete Heyting -> meet-continuous distributive (non empty Poset); coherence by Th56; cluster complete meet-continuous distributive -> Heyting (non empty Poset); coherence by Th56; end;

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