Copyright (c) 1990 Association of Mizar Users
environ vocabulary RLVECT_1, RLSUB_1, FINSET_1, RLVECT_2, FINSEQ_1, FUNCT_1, ORDERS_1, SEQ_1, RELAT_1, FINSEQ_4, BOOLE, FUNCT_2, ARYTM_1, TARSKI, CARD_1, ZFMISC_1, RLVECT_3, HAHNBAN; notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, NUMBERS, FINSEQ_1, RELAT_1, ORDINAL1, FUNCT_1, FUNCT_2, STRUCT_0, RLVECT_1, FINSEQ_4, FRAENKEL, FINSET_1, REAL_1, RLSUB_1, ORDERS_1, RLSUB_2, RLVECT_2, CARD_1, NAT_1; constructors NAT_1, REAL_1, ORDERS_1, RLSUB_2, RLVECT_2, FINSEQ_4, MEMBERED, PARTFUN1, XBOOLE_0; clusters SUBSET_1, FUNCT_1, RLVECT_1, RLSUB_1, FINSET_1, RLVECT_2, RELSET_1, STRUCT_0, ARYTM_3, MEMBERED, ZFMISC_1, XBOOLE_0; requirements NUMERALS, BOOLE, SUBSET, ARITHM; definitions XBOOLE_0, RLSUB_1, RLVECT_2, TARSKI; theorems CARD_1, CARD_2, FUNCT_1, FUNCT_2, FINSEQ_1, FINSEQ_2, FINSEQ_3, FINSEQ_4, FINSET_1, ORDERS_1, ORDERS_2, RLSUB_1, RLSUB_2, RLVECT_1, RLVECT_2, TARSKI, ZFMISC_1, RELAT_1, RELSET_1, ORDINAL1, XBOOLE_0, XBOOLE_1, XCMPLX_0, XCMPLX_1; schemes FINSEQ_1, FUNCT_1, FUNCT_2, NAT_1, RLVECT_2, XBOOLE_0; begin reserve x,y,X,Y,Z for set; reserve a,b for Real; reserve k,n for Nat; reserve V for RealLinearSpace; reserve W1,W2,W3 for Subspace of V; reserve v,v1,v2,u for VECTOR of V; reserve A,B,C for Subset of V; reserve T for finite Subset of V; reserve L,L1,L2 for Linear_Combination of V; reserve l for Linear_Combination of A; reserve F,G,H for FinSequence of the carrier of V; reserve f,g for Function of the carrier of V, REAL; reserve p,q,r for FinSequence; reserve M for non empty set; reserve CF for Choice_Function of M; Lm1: f (#) (F ^ G) = (f (#) F) ^ (f (#) G) proof set H = (f (#) F) ^ (f (#) G); set I = F ^ G; A1: len H = len(f (#) F) + len(f (#) G) by FINSEQ_1:35 .= len F + len(f (#) G) by RLVECT_2:def 9 .= len F + len G by RLVECT_2:def 9 .= len I by FINSEQ_1:35; A2: len F = len(f (#) F) & len G = len(f (#) G) by RLVECT_2:def 9; now let k; assume A3: k in dom H; now per cases by A3,FINSEQ_1:38; suppose A4: k in dom(f (#) F); then A5: k in dom F by A2,FINSEQ_3:31; then A6: k in dom(F ^ G) by FINSEQ_3:24; A7: F/.k = F.k by A5,FINSEQ_4:def 4 .= (F ^ G).k by A5,FINSEQ_1:def 7 .= (F ^ G)/.k by A6,FINSEQ_4:def 4; thus H.k = (f (#) F).k by A4,FINSEQ_1:def 7 .= f.(I/.k) * I/.k by A4,A7,RLVECT_2:def 9; suppose ex n st n in dom(f (#) G) & k = len(f (#) F) + n; then consider n such that A8: n in dom(f (#) G) and A9: k = len(f (#) F) + n; A10: n in dom G by A2,A8,FINSEQ_3:31; A11: k in dom I by A1,A3,FINSEQ_3:31; A12: G/.n = G.n by A10,FINSEQ_4:def 4 .= (F ^ G).k by A2,A9,A10,FINSEQ_1:def 7 .= (F ^ G)/.k by A11,FINSEQ_4:def 4; thus H.k = (f (#) G).n by A8,A9,FINSEQ_1:def 7 .= f.(I/.k) * I/.k by A8,A12,RLVECT_2:def 9; end; hence H.k = f.(I/.k) * I/.k; end; hence thesis by A1,RLVECT_2:def 9; end; theorem Th1: Sum(L1 + L2) = Sum(L1) + Sum(L2) proof consider F such that A1: F is one-to-one and A2: rng F = Carrier(L1 + L2) and A3: Sum((L1 + L2) (#) F) = Sum(L1 + L2) by RLVECT_2:def 10; consider G such that A4: G is one-to-one and A5: rng G = Carrier(L1) and A6: Sum(L1 (#) G) = Sum(L1) by RLVECT_2:def 10; consider H such that A7: H is one-to-one and A8: rng H = Carrier(L2) and A9: Sum(L2 (#) H) = Sum(L2) by RLVECT_2:def 10; set A = Carrier(L1 + L2) \/ Carrier(L1) \/ Carrier(L2); set C1 = A \ Carrier(L1); consider p such that A10: rng p = C1 and A11: p is one-to-one by FINSEQ_4:73; reconsider p as FinSequence of the carrier of V by A10,FINSEQ_1:def 4; A12: len p = len(L1 (#) p) by RLVECT_2:def 9; now let k; assume A13: k in dom p; then k in dom(L1 (#) p) by A12,FINSEQ_3:31; then A14: (L1 (#) p).k = L1.(p/.k) * p/.k by RLVECT_2:def 9; p/.k = p.k by A13,FINSEQ_4:def 4; then p/.k in C1 by A10,A13,FUNCT_1:def 5; then not p/.k in Carrier(L1) by XBOOLE_0:def 4; hence (L1 (#) p).k = 0 * p/.k by A14,RLVECT_2:28; end; then A15: Sum(L1 (#) p) = 0 * Sum(p) by A12,RLVECT_2:5 .= 0.V by RLVECT_1:23; set GG = G ^ p; set g = L1 (#) GG; A16: Sum(g) = Sum((L1 (#) G) ^ (L1 (#) p)) by Lm1 .= Sum(L1 (#) G) + 0.V by A15,RLVECT_1:58 .= Sum(L1 (#) G) by RLVECT_1:10; set C2 = A \ Carrier(L2); consider q such that A17: rng q = C2 and A18: q is one-to-one by FINSEQ_4:73; reconsider q as FinSequence of the carrier of V by A17,FINSEQ_1:def 4; A19: len q = len(L2 (#) q) by RLVECT_2:def 9; now let k; assume A20: k in dom q; then k in dom(L2 (#) q) by A19,FINSEQ_3:31; then A21: (L2 (#) q).k = L2.(q/.k) * q/.k by RLVECT_2:def 9; q/.k = q.k by A20,FINSEQ_4:def 4; then q/.k in C2 by A17,A20,FUNCT_1:def 5; then not q/.k in Carrier(L2) by XBOOLE_0:def 4; hence (L2 (#) q).k = 0 * q/.k by A21,RLVECT_2:28; end; then A22: Sum(L2 (#) q) = 0 * Sum(q) by A19,RLVECT_2:5 .= 0.V by RLVECT_1:23; set HH = H ^ q; set h = L2 (#) HH; A23: Sum(h) = Sum((L2 (#) H) ^ (L2 (#) q)) by Lm1 .= Sum(L2 (#) H) + 0.V by A22,RLVECT_1:58 .= Sum(L2 (#) H) by RLVECT_1:10; set C3 = A \ Carrier(L1 + L2); consider r such that A24: rng r = C3 and A25: r is one-to-one by FINSEQ_4:73; reconsider r as FinSequence of the carrier of V by A24,FINSEQ_1:def 4; A26: len r = len((L1 + L2) (#) r) by RLVECT_2:def 9; now let k; assume A27: k in dom r; then k in dom((L1 + L2) (#) r) by A26,FINSEQ_3:31; then A28: ((L1 + L2) (#) r).k = (L1 + L2).(r/.k) * r/.k by RLVECT_2:def 9; r/.k = r.k by A27,FINSEQ_4:def 4; then r/.k in C3 by A24,A27,FUNCT_1:def 5; then not r/.k in Carrier((L1 + L2)) by XBOOLE_0:def 4; hence ((L1 + L2) (#) r).k = 0 * r/.k by A28,RLVECT_2:28; end; then A29: Sum((L1 + L2) (#) r) = 0 * Sum(r) by A26,RLVECT_2:5 .= 0.V by RLVECT_1:23; set FF = F ^ r; set f = (L1 + L2) (#) FF; A30: Sum(f) = Sum(((L1 + L2) (#) F) ^ ((L1 + L2) (#) r)) by Lm1 .= Sum((L1 + L2) (#) F) + 0.V by A29,RLVECT_1:58 .= Sum((L1 + L2) (#) F) by RLVECT_1:10; A31: rng G misses rng p proof assume not thesis; then A32: rng G /\ rng p <> {} by XBOOLE_0:def 7; consider x being Element of rng G /\ rng p; x in Carrier(L1) & x in C1 by A5,A10,A32,XBOOLE_0:def 3; hence thesis by XBOOLE_0:def 4; end; A33: rng H misses rng q proof assume not thesis; then A34: rng H /\ rng q <> {} by XBOOLE_0:def 7; consider x being Element of rng H /\ rng q; x in Carrier(L2) & x in C2 by A8,A17,A34,XBOOLE_0:def 3; hence thesis by XBOOLE_0:def 4; end; rng F misses rng r proof assume not thesis; then A35: rng F /\ rng r <> {} by XBOOLE_0:def 7; consider x being Element of rng F /\ rng r; x in Carrier(L1 + L2) & x in C3 by A2,A24,A35,XBOOLE_0:def 3; hence thesis by XBOOLE_0:def 4; end; then A36: FF is one-to-one & HH is one-to-one & GG is one-to-one by A1,A4,A7,A11,A18,A25,A31,A33,FINSEQ_3:98; rng GG = rng G \/ rng p & rng HH = rng H \/ rng q & rng FF = rng F \/ rng r by FINSEQ_1:44; then A37: rng GG = Carrier(L1) \/ A & rng HH = Carrier(L2) \/ A & rng FF = Carrier(L1 + L2) \/ A by A2,A5,A8,A10,A17,A24,XBOOLE_1: 39; A = Carrier(L1) \/ (Carrier(L1 + L2) \/ Carrier(L2)) & A = Carrier(L1 + L2) \/ (Carrier(L1) \/ Carrier(L2)) by XBOOLE_1:4; then Carrier(L1) c= A & Carrier(L2) c= A & Carrier(L1 + L2) c= A by XBOOLE_1:7 ; then A38: rng GG = A & rng HH = A & rng FF = A by A37,XBOOLE_1:12; then A39: len GG = len FF & len GG = len HH by A36,RLVECT_1:106; deffunc Q(Nat)=FF <- (GG.$1); consider P being FinSequence such that A40: len P = len FF and A41: for k st k in Seg(len FF) holds P.k = Q(k) from SeqLambda; A42: rng P c= Seg(len FF) proof let x; assume x in rng P; then consider y such that A43: y in dom P and A44: P.y = x by FUNCT_1:def 5; reconsider y as Nat by A43,FINSEQ_3:25; A45: y in Seg(len FF) by A40,A43,FINSEQ_1:def 3; then y in dom GG by A39,FINSEQ_1:def 3; then GG.y in rng FF by A38,FUNCT_1:def 5; then P.y = FF <- (GG.y) & FF just_once_values GG.y by A36,A41,A45,FINSEQ_4:10; then P.y in dom FF by FINSEQ_4:def 3; hence thesis by A44,FINSEQ_1:def 3; end; A46: now let x; thus x in dom GG implies x in dom P & P.x in dom FF proof assume x in dom GG; then x in Seg(len P) by A39,A40,FINSEQ_1:def 3; hence x in dom P by FINSEQ_1:def 3; then rng P c= Seg(len FF) & P.x in rng P by A42,FUNCT_1:def 5; then P.x in Seg(len FF); hence thesis by FINSEQ_1:def 3; end; assume x in dom P & P.x in dom FF; then x in Seg(len P) by FINSEQ_1:def 3; hence x in dom GG by A39,A40,FINSEQ_1:def 3; end; A47: now let x; assume A48: x in dom GG; then reconsider n = x as Nat by FINSEQ_3:25; GG.n in rng FF by A38,A48,FUNCT_1:def 5; then A49: FF just_once_values GG.n by A36,FINSEQ_4:10; n in Seg(len FF) by A39,A48,FINSEQ_1:def 3; then FF.(P.n) = FF.(FF <- (GG.n)) by A41 .= GG.n by A49,FINSEQ_4:def 3; hence GG.x = FF.(P.x); end; then A50: GG = FF * P by A46,FUNCT_1:20; Seg(len FF) c= rng P proof let x; assume A51: x in Seg(len FF); set f = FF" * GG; dom(FF") = rng GG by A36,A38,FUNCT_1:55; then A52: rng f = rng(FF") by RELAT_1:47 .= dom FF by A36,FUNCT_1:55; A53: rng P c= dom FF by A42,FINSEQ_1:def 3; f = FF " * FF * P by A50,RELAT_1:55 .= id(dom FF) * P by A36,FUNCT_1:61 .= P by A53,RELAT_1:79; hence thesis by A51,A52,FINSEQ_1:def 3; end; then A54: Seg(len FF) = rng P by A42,XBOOLE_0:def 10; A55: dom P = Seg(len FF) by A40,FINSEQ_1:def 3; then A56: P is one-to-one by A54,FINSEQ_4:75; Seg(len FF) = {} implies Seg(len FF) = {}; then reconsider P as Function of Seg(len FF), Seg(len FF) by A42,A55,FUNCT_2:def 1,RELSET_1:11; reconsider P as Permutation of Seg(len FF) by A54,A56,FUNCT_2:83; A57: len f = len FF by RLVECT_2:def 9; then A58: Seg len FF = dom f by FINSEQ_1:def 3; then reconsider Fp = f * P as FinSequence of the carrier of V by FINSEQ_2:51; dom f = Seg(len f) by FINSEQ_1:def 3; then A59: Sum(Fp) = Sum(f) by A57,RLVECT_2:9; deffunc Q(Nat)=HH <- (GG.$1); consider R being FinSequence such that A60: len R = len HH and A61: for k st k in Seg(len HH) holds R.k =Q(k) from SeqLambda; A62: rng R c= Seg(len HH) proof let x; assume x in rng R; then consider y such that A63: y in dom R and A64: R.y = x by FUNCT_1:def 5; reconsider y as Nat by A63,FINSEQ_3:25; A65: y in Seg(len HH) by A60,A63,FINSEQ_1:def 3; then y in dom GG by A39,FINSEQ_1:def 3; then GG.y in rng HH by A38,FUNCT_1:def 5; then R.y = HH <- (GG.y) & HH just_once_values GG.y by A36,A61,A65,FINSEQ_4:10; then R.y in dom HH by FINSEQ_4:def 3; hence thesis by A64,FINSEQ_1:def 3; end; A66: now let x; thus x in dom GG implies x in dom R & R.x in dom HH proof assume x in dom GG; then x in Seg(len R) by A39,A60,FINSEQ_1:def 3; hence x in dom R by FINSEQ_1:def 3; then rng R c= Seg(len HH) & R.x in rng R by A62,FUNCT_1:def 5; then R.x in Seg(len HH); hence thesis by FINSEQ_1:def 3; end; assume x in dom R & R.x in dom HH; then x in Seg(len R) by FINSEQ_1:def 3; hence x in dom GG by A39,A60,FINSEQ_1:def 3; end; A67: now let x; assume A68: x in dom GG; then reconsider n = x as Nat by FINSEQ_3:25; GG.n in rng HH by A38,A68,FUNCT_1:def 5; then A69: HH just_once_values GG.n by A36,FINSEQ_4:10; n in Seg(len HH) by A39,A68,FINSEQ_1:def 3; then HH.(R.n) = HH.(HH <- (GG.n)) by A61 .= GG.n by A69,FINSEQ_4:def 3; hence GG.x = HH.(R.x); end; then A70: GG = HH * R by A66,FUNCT_1:20; Seg(len HH) c= rng R proof let x; assume A71: x in Seg(len HH); set f = HH" * GG; dom(HH") = rng GG by A36,A38,FUNCT_1:55; then A72: rng f = rng(HH") by RELAT_1:47 .= dom HH by A36,FUNCT_1:55; A73: rng R c= dom HH by A62,FINSEQ_1:def 3; f = HH " * HH * R by A70,RELAT_1:55 .= id(dom HH) * R by A36,FUNCT_1:61 .= R by A73,RELAT_1:79; hence thesis by A71,A72,FINSEQ_1:def 3; end; then A74: Seg(len HH) = rng R by A62,XBOOLE_0:def 10; A75: dom R = Seg(len HH) by A60,FINSEQ_1:def 3; then A76: R is one-to-one by A74,FINSEQ_4:75; Seg(len HH) = {} implies Seg(len HH) = {}; then reconsider R as Function of Seg(len HH), Seg(len HH) by A62,A75,FUNCT_2:def 1,RELSET_1:11; reconsider R as Permutation of Seg(len HH) by A74,A76,FUNCT_2:83; A77: len h = len HH by RLVECT_2:def 9; then A78: Seg len HH = dom h by FINSEQ_1:def 3; then reconsider Hp = h * R as FinSequence of the carrier of V by FINSEQ_2:51; dom h = Seg(len h) by FINSEQ_1:def 3; then A79: Sum(Hp) = Sum(h) by A77,RLVECT_2:9; deffunc Q(Nat)=g/.$1 + Hp/.$1; consider I being FinSequence such that A80: len I = len GG and A81: for k st k in Seg(len GG) holds I.k = Q(k) from SeqLambda; rng I c= the carrier of V proof let x; assume x in rng I; then consider y such that A82: y in dom I and A83: I.y = x by FUNCT_1: def 5; reconsider y as Nat by A82,FINSEQ_3:25; dom I = Seg(len I) by FINSEQ_1:def 3; then I.y = g/.y + Hp/.y by A80,A81,A82; hence thesis by A83; end; then reconsider I as FinSequence of the carrier of V by FINSEQ_1:def 4; A84: len Fp = len I by A39,A57,A58,A80,FINSEQ_2:48; then A85: dom I = Seg(len I) & dom Fp = Seg(len I) by FINSEQ_1:def 3; A86: len Hp = len GG & len g = len GG by A39,A77,A78,FINSEQ_2:48,RLVECT_2: def 9; now let x; assume A87: x in dom I; then A88: x in dom g & x in dom Hp & x in dom HH & x in dom GG & x in dom Fp by A39,A80,A84,A86,FINSEQ_3:31; reconsider k = x as Nat by A87,FINSEQ_3:25; set v = GG/.k; k in dom g by A80,A86,A87,FINSEQ_3:31; then A89: g/.k = g.k by FINSEQ_4:def 4 .= L1.v * v by A88,RLVECT_2:def 9; k in dom Hp by A80,A86,A87,FINSEQ_3:31; then A90: Hp/.k = (h * R).k by FINSEQ_4:def 4 .= h.(R.k) by A88,FUNCT_1:22; k in dom R by A39,A75,A80,A87,FINSEQ_1:def 3; then A91: R.k in dom R by A74,A75,FUNCT_1:def 5; then A92: R.k in dom HH by A60,FINSEQ_3:31; reconsider j = R.k as Nat by A91,FINSEQ_3:25; HH.j = GG.k by A67,A88 .= GG/.k by A88,FINSEQ_4:def 4; then A93: h.j = L2.v * v by A92,RLVECT_2:40; A94: Fp.k = f.(P.k) by A88,FUNCT_1:22; k in dom P by A39,A55,A80,A87,FINSEQ_1:def 3; then A95: P.k in dom P by A54,A55,FUNCT_1:def 5; then A96: P.k in dom FF by A40,FINSEQ_3:31; reconsider l = P.k as Nat by A95,FINSEQ_3:25; A97: x in Seg len GG by A80,A87,FINSEQ_1:def 3; FF.l = GG.k by A47,A88 .= GG/.k by A88,FINSEQ_4:def 4; then f.l = (L1 + L2).v * v by A96,RLVECT_2:40 .= (L1.v + L2.v) * v by RLVECT_2:def 12 .= L1.v * v + L2.v * v by RLVECT_1:def 9; hence I.x = Fp.x by A81,A89,A90,A93,A94,A97; end; then A98: I = Fp by A85,FUNCT_1:9; Seg len GG = dom g by A86,FINSEQ_1:def 3; hence thesis by A3,A6,A9,A16,A23,A30,A59,A79,A80,A81,A86,A98,RLVECT_2:4; end; theorem Th2: Sum(a * L) = a * Sum(L) proof per cases; suppose A1: a <> 0; consider F such that A2: F is one-to-one and A3: rng F = Carrier(a * L) and A4: Sum(a * L) = Sum ((a * L) (#) F) by RLVECT_2:def 10; consider G such that A5: G is one-to-one and A6: rng G = Carrier(L) and A7: Sum(L) = Sum(L (#) G) by RLVECT_2:def 10; set g = L (#) G; set f = (a * L) (#) F; set l = a * L; deffunc Q(Nat) = F <- (G.$1); consider P being FinSequence such that A8: len P = len F and A9: for k st k in Seg(len F) holds P.k = Q(k) from SeqLambda; A10: Carrier(l) = Carrier(L) by A1,RLVECT_2:65; then A11: len G = len F by A2,A3,A5,A6,RLVECT_1:106; A12: rng P c= Seg(len F) proof let x; assume x in rng P; then consider y such that A13: y in dom P and A14: P.y = x by FUNCT_1:def 5; reconsider y as Nat by A13,FINSEQ_3:25; A15: y in Seg(len F) by A8,A13,FINSEQ_1:def 3; then y in dom G by A11,FINSEQ_1:def 3; then G.y in rng F by A3,A6,A10,FUNCT_1:def 5; then P.y = F <- (G.y) & F just_once_values G.y by A2,A9,A15,FINSEQ_4:10; then P.y in dom F by FINSEQ_4:def 3; hence thesis by A14,FINSEQ_1:def 3; end; A16: now let x; thus x in dom G implies x in dom P & P.x in dom F proof assume x in dom G; then x in Seg(len P) by A8,A11,FINSEQ_1:def 3; hence x in dom P by FINSEQ_1:def 3; then rng P c= Seg(len F) & P.x in rng P by A12,FUNCT_1:def 5; then P.x in Seg(len F); hence thesis by FINSEQ_1:def 3; end; assume x in dom P & P.x in dom F; then x in Seg(len P) by FINSEQ_1:def 3; hence x in dom G by A8,A11,FINSEQ_1:def 3; end; now let x; assume A17: x in dom G; then reconsider n = x as Nat by FINSEQ_3:25; G.n in rng F by A3,A6,A10,A17,FUNCT_1:def 5; then A18: F just_once_values G.n by A2,FINSEQ_4:10; n in Seg(len F) by A11,A17,FINSEQ_1:def 3; then F.(P.n) = F.(F <- (G.n)) by A9 .= G.n by A18,FINSEQ_4:def 3; hence G.x = F.(P.x); end; then A19: G = F * P by A16,FUNCT_1:20; Seg(len F) c= rng P proof let x; assume A20: x in Seg(len F); set f = F" * G; dom(F") = rng G by A2,A3,A6,A10,FUNCT_1:55; then A21: rng f = rng(F") by RELAT_1:47 .= dom F by A2,FUNCT_1:55; A22: rng P c= dom F by A12,FINSEQ_1:def 3; f = F " * F * P by A19,RELAT_1:55 .= id(dom F) * P by A2,FUNCT_1:61 .= P by A22,RELAT_1:79; hence thesis by A20,A21,FINSEQ_1:def 3; end; then A23: Seg(len F) = rng P by A12,XBOOLE_0:def 10; A24: dom P = Seg(len F) by A8,FINSEQ_1:def 3; then A25: P is one-to-one by A23,FINSEQ_4:75; Seg(len F) = {} implies Seg(len F) = {}; then reconsider P as Function of Seg(len F), Seg(len F) by A12,A24,FUNCT_2:def 1,RELSET_1:11; reconsider P as Permutation of Seg(len F) by A23,A25,FUNCT_2:83; A26: len f = len F by RLVECT_2:def 9; then A27: dom f = Seg(len F) by FINSEQ_1:def 3; then reconsider Fp = f * P as FinSequence of the carrier of V by FINSEQ_2:51; dom f = Seg(len f) by FINSEQ_1:def 3; then A28: Sum(Fp) = Sum(f) by A26,RLVECT_2:9; A29: len Fp = len f by A27,FINSEQ_2:48; then A30: len Fp = len g by A11,A26,RLVECT_2:def 9; then A31: dom Fp = dom g by FINSEQ_3:31; now let k,v; assume that A32: k in dom g and A33: v = g.k; A34: k in dom G by A11,A26,A29,A30,A32,FINSEQ_3:31; then G.k in rng G by FUNCT_1:def 5; then F just_once_values G.k by A2,A3,A6,A10,FINSEQ_4:10; then A35: F <- (G.k) in dom F by FINSEQ_4:def 3; then reconsider i = F <- (G.k) as Nat by FINSEQ_3:25; :::A36: k in dom g by A32; A36: k in Seg len F by A26,A29,A30,A32,FINSEQ_1:def 3; A37: k in dom P by A8,A26,A29,A30,A32,FINSEQ_3:31; A38: G/.k = G.k by A34,FINSEQ_4:def 4 .= F.(P.k) by A19,A37,FUNCT_1:23 .= F.i by A9,A36 .= F/.i by A35,FINSEQ_4:def 4; i in Seg(len f) by A26,A35,FINSEQ_1:def 3; then A39: i in dom f by FINSEQ_1:def 3; thus Fp.k = f.(P.k) by A37,FUNCT_1:23 .= f.(F <- (G.k)) by A9,A36 .= l.(F/.i) * F/.i by A39,RLVECT_2:def 9 .= a * L.(F/.i) * F/.i by RLVECT_2:def 13 .= a * (L.(F/.i) * F/.i) by RLVECT_1:def 9 .= a * v by A32,A33,A38,RLVECT_2:def 9; end; hence thesis by A4,A7,A28,A30,A31,RLVECT_1:56; suppose A40: a = 0; hence Sum(a * L) = Sum(ZeroLC(V)) by RLVECT_2:66 .= 0.V by RLVECT_2:48 .= a * Sum(L) by A40,RLVECT_1:23; end; theorem Th3: Sum(- L) = - Sum(L) proof thus Sum(- L) = Sum((- 1) * L) by RLVECT_2:def 14 .= (- 1) * Sum(L) by Th2 .= - Sum(L) by RLVECT_1:29; end; theorem Th4: Sum(L1 - L2) = Sum(L1) - Sum(L2) proof thus Sum(L1 - L2) = Sum(L1 + (- L2)) by RLVECT_2:def 15 .= Sum(L1) + Sum(- L2) by Th1 .= Sum(L1) + (- Sum(L2)) by Th3 .= Sum(L1) - Sum(L2) by RLVECT_1:def 11; end; definition let V; let A; attr A is linearly-independent means :Def1: for l st Sum(l) = 0.V holds Carrier(l) = {}; antonym A is linearly-dependent; end; canceled; theorem A c= B & B is linearly-independent implies A is linearly-independent proof assume that A1: A c= B and A2: B is linearly-independent; let l be Linear_Combination of A; assume A3: Sum(l) = 0.V; reconsider L = l as Linear_Combination of B by A1,RLVECT_2:33; Carrier(L) = {} by A2,A3,Def1; hence thesis; end; theorem Th7: A is linearly-independent implies not 0.V in A proof assume that A1: A is linearly-independent and A2: 0.V in A; deffunc F(Element of V) = 0; consider f such that A3: f.(0.V) = 1 and A4: for v being Element of V st v <> 0.V holds f.v = F(v) from LambdaSep1; reconsider f as Element of Funcs(the carrier of V, REAL) by FUNCT_2:11; ex T st for v st not v in T holds f.v = 0 proof take T = {0.V}; let v; assume not v in T; then v <> 0.V by TARSKI:def 1; hence thesis by A4; end; then reconsider f as Linear_Combination of V by RLVECT_2:def 5; A5: Carrier(f) = {0.V} proof thus Carrier(f) c= {0.V} proof let x; assume x in Carrier(f); then x in {v : f.v <> 0} by RLVECT_2:def 6; then consider v such that A6: v = x and A7: f.v <> 0; v = 0.V by A4,A7; hence thesis by A6,TARSKI:def 1; end; let x; assume x in {0.V}; then x = 0.V & 0 <> 1 by TARSKI:def 1; then x in {v : f.v <> 0} by A3; hence thesis by RLVECT_2:def 6; end; then Carrier(f) c= A by A2,ZFMISC_1:37; then reconsider f as Linear_Combination of A by RLVECT_2:def 8; Sum(f) = f.(0.V) * 0.V by A5,RLVECT_2:53 .= 0.V by RLVECT_1:23; hence contradiction by A1,A5,Def1; end; theorem Th8: {}(the carrier of V) is linearly-independent proof let l be Linear_Combination of {}(the carrier of V); Carrier(l) c= {} by RLVECT_2:def 8; hence thesis by XBOOLE_1:3; end; theorem Th9: {v} is linearly-independent iff v <> 0.V proof thus {v} is linearly-independent implies v <> 0.V proof assume {v} is linearly-independent; then not 0.V in {v} by Th7; hence thesis by TARSKI:def 1; end; assume A1: v <> 0.V; let l be Linear_Combination of {v}; assume A2: Sum(l) = 0.V; A3: Carrier(l) c= {v} by RLVECT_2:def 8; now per cases by A3,ZFMISC_1:39; suppose Carrier(l) = {}; hence thesis; suppose A4: Carrier(l) = {v}; A5: 0.V = l.v * v by A2,RLVECT_2:50; now assume v in Carrier(l); then v in {u : l.u <> 0} by RLVECT_2:def 6; then ex u st v = u & l.u <> 0; hence contradiction by A1,A5,RLVECT_1:24; end; hence thesis by A4,TARSKI:def 1; end; hence thesis; end; theorem {0.V} is linearly-dependent by Th9; theorem Th11: {v1,v2} is linearly-independent implies v1 <> 0.V & v2 <> 0.V proof assume A1: {v1,v2} is linearly-independent; v1 in {v1,v2} & v2 in {v1,v2} by TARSKI:def 2; hence thesis by A1,Th7; end; theorem {v,0.V} is linearly-dependent & {0.V,v} is linearly-dependent by Th11; theorem Th13: v1 <> v2 & {v1,v2} is linearly-independent iff v2 <> 0.V & for a holds v1 <> a * v2 proof thus v1 <> v2 & {v1,v2} is linearly-independent implies v2 <> 0.V & for a holds v1 <> a * v2 proof assume that A1: v1 <> v2 and A2: {v1,v2} is linearly-independent; thus v2 <> 0.V by A2,Th11; let a; assume A3: v1 = a * v2; deffunc F(Element of V)=0; consider f such that A4: f.v1 = - 1 & f.v2 = a and A5: for v being Element of V st v <> v1 & v <> v2 holds f.v = F(v) from LambdaSep2(A1); reconsider f as Element of Funcs(the carrier of V, REAL) by FUNCT_2:11; now let v; assume not v in {v1,v2}; then v <> v1 & v <> v2 by TARSKI:def 2; hence f.v = 0 by A5; end; then reconsider f as Linear_Combination of V by RLVECT_2:def 5; Carrier(f) c= {v1,v2} proof let x; assume x in Carrier(f); then x in {u : f.u <> 0} by RLVECT_2:def 6; then A6: ex u st x = u & f.u <> 0; assume not x in {v1,v2}; then x <> v1 & x <> v2 by TARSKI:def 2; hence thesis by A5,A6; end; then reconsider f as Linear_Combination of {v1,v2} by RLVECT_2:def 8; A7: now assume not v1 in Carrier(f); then not v1 in {u : f.u <> 0} by RLVECT_2:def 6; hence contradiction by A4; end; set w = a * v2; Sum(f) = (- 1) * w + w by A1,A3,A4,RLVECT_2:51 .= (- w) + w by RLVECT_1:29 .= - (w - w) by RLVECT_1:47 .= - 0.V by RLVECT_1:28 .= 0.V by RLVECT_1:25; hence thesis by A2,A7,Def1; end; assume A8: v2 <> 0.V; assume A9: for a holds v1 <> a * v2; then A10: v1 <> 1 * v2 & 1 * v2 = v2 by RLVECT_1:def 9; hence v1 <> v2; let l be Linear_Combination of {v1,v2}; assume that A11: Sum(l) = 0.V and A12: Carrier(l) <> {}; consider x being Element of Carrier(l); x in Carrier(l) by A12; then x in {u : l.u <> 0} by RLVECT_2:def 6; then A13: ex u st x = u & l.u <> 0; Carrier(l) c= {v1,v2} by RLVECT_2:def 8; then A14: x in {v1,v2} by A12,TARSKI:def 3; A15: 0.V = l.v1 * v1 + l.v2 * v2 by A10,A11,RLVECT_2:51; now per cases by A13,A14,TARSKI:def 2; suppose A16: l.v1 <> 0; 0.V = (l.v1)" * (l.v1 * v1 + l.v2 * v2) by A15,RLVECT_1:23 .= (l.v1)" * (l.v1 * v1) + (l.v1)" * (l.v2 * v2) by RLVECT_1:def 9 .= (l.v1)" * l.v1 * v1 + (l.v1)" * (l.v2 * v2) by RLVECT_1:def 9 .= (l.v1)" * l.v1 * v1 + (l.v1)" * l.v2 * v2 by RLVECT_1:def 9 .= 1 * v1 + (l.v1)" * l.v2 * v2 by A16,XCMPLX_0:def 7 .= v1 + (l.v1)" * l.v2 * v2 by RLVECT_1:def 9; then v1 = - ((l.v1)" * l.v2 * v2) by RLVECT_1:19 .= (- 1) * ((l.v1)" * l.v2 * v2) by RLVECT_1:29 .= ((- 1) * ((l.v1)" * l.v2)) * v2 by RLVECT_1:def 9; hence thesis by A9; suppose A17: l.v2 <> 0 & l.v1 = 0; 0.V = (l.v2)" * (l.v1 * v1 + l.v2 * v2) by A15,RLVECT_1:23 .= (l.v2)" * (l.v1 * v1) + (l.v2)" * (l.v2 * v2) by RLVECT_1:def 9 .= (l.v2)" * l.v1 * v1 + (l.v2)" * (l.v2 * v2) by RLVECT_1:def 9 .= (l.v2)" * l.v1 * v1 + (l.v2)" * l.v2 * v2 by RLVECT_1:def 9 .= (l.v2)" * l.v1 * v1 + 1 * v2 by A17,XCMPLX_0:def 7 .= 0 * v1 + v2 by A17,RLVECT_1:def 9 .= 0.V + v2 by RLVECT_1:23 .= v2 by RLVECT_1:10; hence thesis by A8; end; hence thesis; end; theorem v1 <> v2 & {v1,v2} is linearly-independent iff for a,b st a * v1 + b * v2 = 0.V holds a = 0 & b = 0 proof thus v1 <> v2 & {v1,v2} is linearly-independent implies for a,b st a * v1 + b * v2 = 0.V holds a = 0 & b = 0 proof assume that A1: v1 <> v2 and A2: {v1,v2} is linearly-independent; let a,b; assume that A3: a * v1 + b * v2 = 0.V and A4: a <> 0 or b <> 0; now per cases by A4; suppose A5: a <> 0; 0.V = a" * (a * v1 + b * v2) by A3,RLVECT_1:23 .= a" * (a * v1) + a" * (b * v2) by RLVECT_1:def 9 .= (a" * a) * v1 + a" * (b * v2) by RLVECT_1:def 9 .= (a" * a) * v1 + (a" * b) * v2 by RLVECT_1:def 9 .= 1 * v1 + (a" * b) * v2 by A5,XCMPLX_0:def 7 .= v1 + (a" * b) * v2 by RLVECT_1:def 9; then v1 = - ((a" * b) * v2) by RLVECT_1:19 .= (- 1) * ((a" * b) * v2) by RLVECT_1:29 .= (- 1) * (a" * b) * v2 by RLVECT_1:def 9; hence thesis by A1,A2,Th13; suppose A6: b <> 0; 0.V = b" * (a * v1 + b * v2) by A3,RLVECT_1:23 .= b" * (a * v1) + b" * (b * v2) by RLVECT_1:def 9 .= (b" * a) * v1 + b" * (b * v2) by RLVECT_1:def 9 .= (b" * a) * v1 + (b" * b) * v2 by RLVECT_1:def 9 .= (b" * a) * v1 + 1* v2 by A6,XCMPLX_0:def 7 .= (b" * a) * v1 + v2 by RLVECT_1:def 9; then v2 = - ((b" * a) * v1) by RLVECT_1:def 10 .= (- 1) * ((b" * a) * v1) by RLVECT_1:29 .= (- 1) * (b" * a) * v1 by RLVECT_1:def 9; hence thesis by A1,A2,Th13; end; hence thesis; end; assume A7: for a,b st a * v1 + b * v2 = 0.V holds a = 0 & b = 0; A8: now assume A9: v2 = 0.V; 0.V = 0.V + 0.V by RLVECT_1:10 .= 0 * v1 + 0.V by RLVECT_1:23 .= 0 * v1 + 1 * v2 by A9,RLVECT_1:23; hence contradiction by A7; end; now let a; assume v1 = a * v2; then v1 = 0.V + a * v2 by RLVECT_1:10; then 0.V = v1 - a * v2 by RLSUB_2:78 .= v1 + (- a * v2) by RLVECT_1:def 11 .= v1 + a * (- v2) by RLVECT_1:39 .= v1 + ((- a) * v2) by RLVECT_1:38 .= 1 * v1 + (- a) * v2 by RLVECT_1:def 9; hence contradiction by A7; end; hence thesis by A8,Th13; end; definition let V; let A; func Lin(A) -> strict Subspace of V means :Def2: the carrier of it = {Sum(l) : not contradiction}; existence proof set A1 = {Sum(l) : not contradiction}; A1 c= the carrier of V proof let x; assume x in A1; then ex l st x = Sum(l); hence thesis; end; then reconsider A1 as Subset of V; reconsider l = ZeroLC(V) as Linear_Combination of A by RLVECT_2:34; Sum(l) = 0.V by RLVECT_2:48; then A1: 0.V in A1; A1 is lineary-closed proof thus for v,u st v in A1 & u in A1 holds v + u in A1 proof let v,u; assume that A2: v in A1 and A3: u in A1; consider l1 being Linear_Combination of A such that A4: v = Sum(l1) by A2; consider l2 being Linear_Combination of A such that A5: u = Sum(l2) by A3; reconsider f = l1 + l2 as Linear_Combination of A by RLVECT_2:59; v + u = Sum(f) by A4,A5,Th1; hence thesis; end; let a,v; assume v in A1; then consider l such that A6: v = Sum(l); reconsider f = a * l as Linear_Combination of A by RLVECT_2:67; a * v = Sum(f) by A6,Th2; hence thesis; end; hence thesis by A1,RLSUB_1:43; end; uniqueness by RLSUB_1:38; end; canceled 2; theorem Th17: x in Lin(A) iff ex l st x = Sum(l) proof thus x in Lin(A) implies ex l st x = Sum(l) proof assume x in Lin(A); then x in the carrier of Lin(A) by RLVECT_1:def 1; then x in {Sum(l) : not contradiction} by Def2; hence thesis; end; given k being Linear_Combination of A such that A1: x = Sum(k); x in {Sum(l): not contradiction} by A1; then x in the carrier of Lin(A) by Def2; hence thesis by RLVECT_1:def 1; end; theorem Th18: x in A implies x in Lin(A) proof assume A1: x in A; then reconsider v = x as VECTOR of V; deffunc F(Element of V)=0; consider f being Function of the carrier of V, REAL such that A2: f.v = 1 and A3: for u st u <> v holds f.u = F(u) from LambdaSep1; reconsider f as Element of Funcs(the carrier of V, REAL) by FUNCT_2:11; ex T st for u st not u in T holds f.u = 0 proof take T = {v}; let u; assume not u in T; then u <> v by TARSKI:def 1; hence thesis by A3; end; then reconsider f as Linear_Combination of V by RLVECT_2:def 5; A4: Carrier(f) c= {v} proof let x; assume x in Carrier(f); then x in {u : f.u <> 0} by RLVECT_2:def 6; then consider u such that A5: x = u and A6: f.u <> 0; u = v by A3,A6; hence thesis by A5,TARSKI:def 1; end; then reconsider f as Linear_Combination of {v} by RLVECT_2:def 8; A7: Sum(f) = 1 * v by A2,RLVECT_2:50 .= v by RLVECT_1:def 9; {v} c= A by A1,ZFMISC_1:37; then Carrier(f) c= A by A4,XBOOLE_1:1; then reconsider f as Linear_Combination of A by RLVECT_2:def 8; Sum(f) = v by A7; hence thesis by Th17; end; Lm2: x in (0).V iff x = 0.V proof thus x in (0).V implies x = 0.V proof assume x in (0).V; then x in the carrier of (0).V by RLVECT_1:def 1; then x in {0.V} by RLSUB_1:def 3; hence thesis by TARSKI:def 1; end; thus thesis by RLSUB_1:25; end; reserve l0 for Linear_Combination of {}(the carrier of V); theorem Lin({}(the carrier of V)) = (0).V proof set A = Lin({}(the carrier of V)); now let v; thus v in A implies v in (0).V proof assume v in A; then v in the carrier of A & the carrier of A = {Sum (l0) : not contradiction} by Def2,RLVECT_1:def 1; then ex l0 st v = Sum(l0); then v = 0.V by RLVECT_2:49; hence thesis by Lm2; end; assume v in (0).V; then v = 0.V by Lm2; hence v in A by RLSUB_1:25; end; hence thesis by RLSUB_1:39; end; theorem Lin(A) = (0).V implies A = {} or A = {0.V} proof assume that A1: Lin(A) = (0).V and A2: A <> {}; thus A c= {0.V} proof let x; assume x in A; then x in Lin(A) by Th18; then x = 0.V by A1,Lm2; hence thesis by TARSKI:def 1; end; let x; assume x in {0.V}; then A3: x = 0.V by TARSKI:def 1; consider y being Element of A; A4: y in A by A2; y in Lin(A) by A2,Th18; hence thesis by A1,A3,A4,Lm2; end; theorem Th21: for W being strict Subspace of V holds A = the carrier of W implies Lin(A) = W proof let W be strict Subspace of V; assume A1: A = the carrier of W; now let v; thus v in Lin(A) implies v in W proof assume v in Lin(A); then A2: ex l st v = Sum(l) by Th17; A is lineary-closed & A <> {} by A1,RLSUB_1:42; then v in the carrier of W by A1,A2,RLVECT_2:47; hence thesis by RLVECT_1:def 1; end; v in W iff v in the carrier of W by RLVECT_1:def 1; hence v in W implies v in Lin(A) by A1,Th18; end; hence thesis by RLSUB_1:39; end; theorem for V being strict RealLinearSpace,A being Subset of V holds A = the carrier of V implies Lin(A) = V proof let V be strict RealLinearSpace,A be Subset of V; assume A = the carrier of V; then A = the carrier of (Omega).V by RLSUB_1:def 4; hence Lin(A) = (Omega).V by Th21 .= V by RLSUB_1:def 4; end; Lm3: W1 is Subspace of W3 implies W1 /\ W2 is Subspace of W3 proof assume A1: W1 is Subspace of W3; W1 /\ W2 is Subspace of W1 by RLSUB_2:20; hence thesis by A1,RLSUB_1:35; end; Lm4: W1 is Subspace of W2 & W1 is Subspace of W3 implies W1 is Subspace of W2 /\ W3 proof assume A1: W1 is Subspace of W2 & W1 is Subspace of W3; now let v; assume v in W1; then v in W2 & v in W3 by A1,RLSUB_1:16; hence v in W2 /\ W3 by RLSUB_2:7; end; hence thesis by RLSUB_1:37; end; Lm5: W1 is Subspace of W2 implies W1 is Subspace of W2 + W3 proof assume A1: W1 is Subspace of W2; W2 is Subspace of W2 + W3 by RLSUB_2:11; hence thesis by A1,RLSUB_1:35; end; Lm6: W1 is Subspace of W3 & W2 is Subspace of W3 implies W1 + W2 is Subspace of W3 proof assume A1: W1 is Subspace of W3 & W2 is Subspace of W3; now let v; assume v in W1 + W2; then consider v1,v2 such that A2: v1 in W1 & v2 in W2 and A3: v = v1 + v2 by RLSUB_2:5; v1 in W3 & v2 in W3 by A1,A2,RLSUB_1:16; hence v in W3 by A3,RLSUB_1:28; end; hence thesis by RLSUB_1:37; end; theorem Th23: A c= B implies Lin(A) is Subspace of Lin(B) proof assume A1: A c= B; now let v; assume v in Lin(A); then consider l such that A2: v = Sum(l) by Th17; reconsider l as Linear_Combination of B by A1,RLVECT_2:33; Sum(l) = v by A2; hence v in Lin(B) by Th17; end; hence thesis by RLSUB_1:37; end; theorem for V being strict RealLinearSpace,A,B being Subset of V holds Lin(A) = V & A c= B implies Lin(B) = V proof let V be strict RealLinearSpace,A,B be Subset of V; assume Lin(A) = V & A c= B; then V is Subspace of Lin(B) by Th23; hence thesis by RLSUB_1:34; end; theorem Lin(A \/ B) = Lin(A) + Lin(B) proof A c= A \/ B & B c= A \/ B by XBOOLE_1:7; then Lin(A) is Subspace of Lin(A \/ B) & Lin(B) is Subspace of Lin(A \/ B) by Th23; then A1: Lin(A) + Lin(B) is Subspace of Lin(A \/ B) by Lm6; now let v; assume v in Lin(A \/ B); then consider l being Linear_Combination of A \/ B such that A2: v = Sum(l) by Th17; set C = Carrier(l) /\ A; set D = Carrier(l) \ A; defpred P[set] means $1 in C; deffunc F(set)=l.$1; deffunc G(set)= 0; A3: for x st x in the carrier of V holds (P[x] implies F(x) in REAL) & (not P[x] implies G(x) in REAL) proof now let x; assume x in the carrier of V; then reconsider v = x as VECTOR of V; f.v in REAL; hence x in C implies l.x in REAL; assume not x in C; thus 0 in REAL; end; hence thesis; end; consider f being Function of the carrier of V, REAL such that A4: for x st x in the carrier of V holds (P[x] implies f.x = F(x)) & (not P[x] implies f.x = G(x)) from Lambda1C(A3); reconsider f as Element of Funcs(the carrier of V, REAL) by FUNCT_2:11; defpred D[set] means $1 in D; A5: for x st x in the carrier of V holds (D[x] implies F(x) in REAL) & (not D[x] implies G(x) in REAL) proof now let x; assume x in the carrier of V; then reconsider v = x as VECTOR of V; g.v in REAL; hence x in D implies l.x in REAL; assume not x in D; thus 0 in REAL; end; hence thesis; end; consider g being Function of the carrier of V, REAL such that A6: for x st x in the carrier of V holds (D[x] implies g.x = F(x)) & (not D[x] implies g.x = G(x)) from Lambda1C(A5); reconsider g as Element of Funcs(the carrier of V, REAL) by FUNCT_2:11; reconsider C as finite Subset of V; for u holds not u in C implies f.u = 0 by A4; then reconsider f as Linear_Combination of V by RLVECT_2:def 5; A7: Carrier(f) c= C proof let x; assume x in Carrier(f); then x in {u : f.u <> 0} by RLVECT_2:def 6; then A8: ex u st x = u & f.u <> 0; assume not x in C; hence thesis by A4,A8; end; C c= A by XBOOLE_1:17; then Carrier(f) c= A by A7,XBOOLE_1:1; then reconsider f as Linear_Combination of A by RLVECT_2:def 8; reconsider D as finite Subset of V; for u holds not u in D implies g.u = 0 by A6; then reconsider g as Linear_Combination of V by RLVECT_2:def 5; A9: Carrier(g) c= D proof let x; assume x in Carrier(g); then x in {u : g.u <> 0} by RLVECT_2:def 6; then A10: ex u st x = u & g.u <> 0; assume not x in D; hence thesis by A6,A10; end; D c= B proof let x; assume x in D; then x in Carrier(l) & not x in A & Carrier(l) c= A \/ B by RLVECT_2:def 8,XBOOLE_0:def 4; hence thesis by XBOOLE_0:def 2; end; then Carrier(g) c= B by A9,XBOOLE_1:1; then reconsider g as Linear_Combination of B by RLVECT_2:def 8; l = f + g proof let v; now per cases; suppose A11: v in C; A12: now assume v in D; then not v in A by XBOOLE_0:def 4; hence contradiction by A11,XBOOLE_0:def 3; end; thus (f + g).v = f.v + g.v by RLVECT_2:def 12 .= l.v + g.v by A4,A11 .= l.v + 0 by A6,A12 .= l.v; suppose A13: not v in C; now per cases; suppose A14: v in Carrier(l); A15: now assume not v in D; then not v in Carrier(l) or v in A by XBOOLE_0:def 4; hence contradiction by A13,A14,XBOOLE_0:def 3; end; thus (f + g). v = f.v + g.v by RLVECT_2:def 12 .= 0 + g.v by A4,A13 .= l.v by A6,A15; suppose A16: not v in Carrier(l); then A17: not v in C & not v in D by XBOOLE_0:def 3,def 4; thus (f + g).v = f.v + g.v by RLVECT_2:def 12 .= 0 + g.v by A4,A17 .= 0 + 0 by A6,A17 .= l.v by A16,RLVECT_2:28; end; hence (f + g).v = l.v; end; hence thesis; end; then A18: v = Sum(f) + Sum(g) by A2,Th1; Sum(f) in Lin(A) & Sum(g) in Lin(B) by Th17; hence v in Lin(A) + Lin(B) by A18,RLSUB_2:5; end; then Lin(A \/ B) is Subspace of Lin(A) + Lin(B) by RLSUB_1:37; hence thesis by A1,RLSUB_1:34; end; theorem Lin(A /\ B) is Subspace of Lin(A) /\ Lin(B) proof A /\ B c= A & A /\ B c= B by XBOOLE_1:17; then Lin(A /\ B) is Subspace of Lin(A) & Lin(A /\ B) is Subspace of Lin(B) by Th23; hence thesis by Lm4; end; Lm7: not {} in M implies dom CF = M & rng CF c= union M proof consider x being Element of M; assume A1: not {} in M; x in M; then union M <> {} by A1,ORDERS_1:91; hence dom CF = M & rng CF c= union M by FUNCT_2:def 1,RELSET_1:12; end; Lm8: for Z being finite set holds card Z = 0 & Z <> {} & (for X,Y st X in Z & Y in Z holds X c= Y or Y c= X) implies union Z in Z by CARD_2:59; Lm9: now let n; assume A1: for Z being finite set holds card Z = n & Z <> {} & (for X,Y st X in Z & Y in Z holds X c= Y or Y c= X) implies union Z in Z; let Z be finite set; assume that A2: card Z = n + 1 and A3: Z <> {} and A4: for X,Y st X in Z & Y in Z holds X c= Y or Y c= X; consider y being Element of Z; now per cases; suppose n = 0; then consider x such that A5: Z = {x} by A2,CARD_2:60; y = x & union Z = x by A5,TARSKI:def 1,ZFMISC_1:31; hence union Z in Z by A3; suppose A6: n <> 0; set Y = Z \ {y}; A7: Y c= Z by XBOOLE_1:36; reconsider Y as finite set; {y} c= Z by A3,ZFMISC_1:37; then A8: card Y = (n + 1) - card{y} by A2,CARD_2:63 .= n + 1 - 1 by CARD_1:79 .= n by XCMPLX_1:26; for a,b being set st a in Y & b in Y holds a c= b or b c= a by A4,A7; then A9: union Y in Y by A1,A6,A8,CARD_1:47; then A10: union Y in Z by A7; A11: y c= union Y or union Y c= y by A4,A7,A9; A12: y in Z by A3; Z = (Z \ {y}) \/ {y} proof thus Z c= (Z \ {y}) \/ {y} proof let x; assume x in Z; then x in Z \ {y} or x in {y} by XBOOLE_0:def 4; hence thesis by XBOOLE_0:def 2; end; let x; assume x in (Z \ {y}) \/ {y}; then x in (Z \ {y}) or x in {y} by XBOOLE_0:def 2; then (x in Z & not x in {y}) or x in {y} by XBOOLE_0:def 4; then x in Z \/ {y} & {y} c= Z by A3,XBOOLE_0:def 2,ZFMISC_1:37; hence thesis by XBOOLE_1:12; end; then union Z = union Y \/ union {y} by ZFMISC_1:96 .= union Y \/ y by ZFMISC_1:31; hence union Z in Z by A10,A11,A12,XBOOLE_1:12; end; hence union Z in Z; end; Lm10: for Z being finite set holds Z <> {} & (for X,Y st X in Z & Y in Z holds X c= Y or Y c= X) implies union Z in Z proof let Z be finite set; A1: card Z = card Z; defpred P[Nat] means for Z being finite set st card Z = $1 & Z <> {} & (for X,Y st X in Z & Y in Z holds X c= Y or Y c= X) holds union Z in Z; A2: P[0] by Lm8; A3: for k be Nat st P[k] holds P[k+1] by Lm9; for n holds P[n] from Ind(A2,A3); hence thesis by A1; end; theorem Th27: A is linearly-independent implies ex B st A c= B & B is linearly-independent & Lin(B) = the RLSStruct of V proof assume A1: A is linearly-independent; defpred P[set] means (ex B being Subset of V st B = $1 & A c= B & B is linearly-independent); consider Q being set such that A2: for Z holds Z in Q iff Z in bool(the carrier of V) & P[Z] from Separation; A3: Q <> {} by A1,A2; now let Z; assume that A4: Z <> {} and A5: Z c= Q and A6: Z is c=-linear; set W = union Z; W c= the carrier of V proof let x; assume x in W; then consider X such that A7: x in X and A8: X in Z by TARSKI:def 4; X in bool(the carrier of V) by A2,A5,A8; hence thesis by A7; end; then reconsider W as Subset of V; consider x being Element of Z; x in Q by A4,A5,TARSKI:def 3; then A9: ex B being Subset of V st B = x & A c= B & B is linearly-independent by A2; x c= W by A4,ZFMISC_1:92; then A10: A c= W by A9,XBOOLE_1:1; W is linearly-independent proof let l be Linear_Combination of W; assume that A11: Sum(l) = 0.V and A12: Carrier(l) <> {}; deffunc Q(set)={C where C is Subset of V: $1 in C & C in Z}; consider f being Function such that A13: dom f = Carrier(l) and A14: for x st x in Carrier(l) holds f.x = Q(x) from Lambda; reconsider M = rng f as non empty set by A12,A13,RELAT_1:65; consider F being Choice_Function of M; A15: now assume {} in M; then consider x such that A16: x in dom f and A17: f.x = {} by FUNCT_1:def 5; Carrier(l) c= W by RLVECT_2:def 8; then consider X such that A18: x in X and A19: X in Z by A13, A16,TARSKI:def 4; reconsider X as Subset of V by A2,A5,A19; X in {C where C is Subset of V: x in C & C in Z} by A18,A19; hence contradiction by A13,A14,A16,A17; end; set S = rng F; A20: dom F = M & M <> {} by A15,Lm7; then A21: S <> {} by RELAT_1:65; A22: now let X; assume X in S; then consider x such that A23: x in dom F and A24: F.x = X by FUNCT_1:def 5; consider y such that A25: y in dom f and A26: f.y = x by A20,A23,FUNCT_1:def 5; A27: X in x by A15,A20,A23,A24,ORDERS_1:def 1; x = Q(y) by A13,A14,A25,A26; then ex C being Subset of V st C = X & y in C & C in Z by A27; hence X in Z; end; A28: now let X,Y; assume X in S & Y in S; then X in Z & Y in Z by A22; then X,Y are_c=-comparable by A6,ORDINAL1:def 9; hence X c= Y or Y c= X by XBOOLE_0:def 9; end; dom F is finite by A13,A20,FINSET_1:26; then S is finite by FINSET_1:26; then union S in S by A21,A28,Lm10; then union S in Z by A22; then consider B being Subset of V such that A29: B = union S and A c= B and A30: B is linearly-independent by A2,A5; Carrier(l) c= union S proof let x; assume A31: x in Carrier(l); then A32: f.x in M by A13,FUNCT_1:def 5; set X = f.x; A33: F.X in X by A15,A32,ORDERS_1:def 1; f.x = {C where C is Subset of V: x in C & C in Z} by A14,A31; then A34: ex C being Subset of V st F.X = C & x in C & C in Z by A33; F.X in S by A20,A32,FUNCT_1:def 5; hence x in union S by A34,TARSKI:def 4; end; then l is Linear_Combination of B by A29,RLVECT_2:def 8; hence thesis by A11,A12,A30,Def1; end; hence union Z in Q by A2,A10; end; then consider X such that A35: X in Q and A36: for Z st Z in Q & Z <> X holds not X c= Z by A3,ORDERS_2:79; consider B being Subset of V such that A37: B = X and A38: A c= B and A39: B is linearly-independent by A2,A35; take B; thus A c= B & B is linearly-independent by A38,A39; A40: (Omega).V = the RLSStruct of V by RLSUB_1:def 4; assume Lin(B) <> the RLSStruct of V; then consider v being VECTOR of V such that A41: not(v in Lin(B) iff v in the RLSStruct of V) by A40,RLSUB_1:39; A42: B \/ {v} is linearly-independent proof let l be Linear_Combination of B \/ {v}; assume A43: Sum(l) = 0.V; now per cases; suppose A44: v in Carrier(l); deffunc L(VECTOR of V)=l.$1; consider f being Function of the carrier of V, REAL such that A45: f.v = 0 and A46: for u being VECTOR of V st u <> v holds f.u = L(u) from LambdaSep1; reconsider f as Element of Funcs(the carrier of V, REAL) by FUNCT_2:11; now let u be VECTOR of V; assume not u in Carrier(l) \ {v}; then not u in Carrier(l) or u in {v} by XBOOLE_0:def 4; then (l.u = 0 & u <> v) or u = v by RLVECT_2:28,TARSKI:def 1 ; hence f.u = 0 by A45,A46; end; then reconsider f as Linear_Combination of V by RLVECT_2:def 5; Carrier(f) c= B proof let x; assume x in Carrier(f); then x in {u where u is VECTOR of V: f.u <> 0} by RLVECT_2:def 6; then consider u being VECTOR of V such that A47: u = x and A48: f.u <> 0; f.u = l.u by A45,A46,A48; then u in {v1 where v1 is VECTOR of V: l.v1 <> 0} by A48; then u in Carrier(l) & Carrier(l) c= B \/ {v} by RLVECT_2:def 6,def 8 ; then u in B or u in {v} by XBOOLE_0:def 2; hence thesis by A45,A47,A48,TARSKI:def 1; end; then reconsider f as Linear_Combination of B by RLVECT_2:def 8; deffunc G(VECTOR of V)=0; consider g being Function of the carrier of V, REAL such that A49: g.v = - l.v and A50: for u being VECTOR of V st u <> v holds g.u = G(u) from LambdaSep1; reconsider g as Element of Funcs(the carrier of V, REAL) by FUNCT_2:11; now let u be VECTOR of V; assume not u in {v}; then u <> v by TARSKI:def 1; hence g.u = 0 by A50; end; then reconsider g as Linear_Combination of V by RLVECT_2:def 5; Carrier(g) c= {v} proof let x; assume x in Carrier(g); then x in {u where u is VECTOR of V: g.u <> 0} by RLVECT_2:def 6; then ex u being VECTOR of V st x = u & g.u <> 0; then x = v by A50; hence thesis by TARSKI:def 1; end; then reconsider g as Linear_Combination of {v} by RLVECT_2:def 8; A51: f - g = l proof let u be VECTOR of V; now per cases; suppose A52: v = u; thus (f - g).u = f.u - g.u by RLVECT_2:79 .= 0 + (- (- l.v)) by A45,A49,A52,XCMPLX_0:def 8 .= l.u by A52; suppose A53: v <> u; thus (f - g).u = f.u - g.u by RLVECT_2:79 .= l.u - g.u by A46,A53 .= l.u - 0 by A50,A53 .= l.u; end; hence thesis; end; A54: Sum(g) = (- l.v) * v by A49,RLVECT_2:50; 0.V = Sum(f) - Sum(g) by A43,A51,Th4; then Sum(f) = 0.V + Sum(g) by RLSUB_2:78 .= (- l.v) * v by A54,RLVECT_1:10; then A55: (- l.v) * v in Lin(B) by Th17; l.v <> 0 by A44,RLVECT_2:28; then A56: - l.v <> 0 by XCMPLX_1:135; (- l.v)" * ((- l.v) * v) = (- l.v)" * (- l.v) * v by RLVECT_1: def 9 .= 1 * v by A56,XCMPLX_0:def 7 .= v by RLVECT_1:def 9; hence thesis by A41,A55,RLSUB_1:29,RLVECT_1:3; suppose A57: not v in Carrier(l); Carrier(l) c= B proof let x; assume A58: x in Carrier(l); Carrier(l) c= B \/ {v} by RLVECT_2:def 8; then x in B or x in {v} by A58,XBOOLE_0:def 2; hence thesis by A57,A58,TARSKI:def 1; end; then l is Linear_Combination of B by RLVECT_2:def 8; hence thesis by A39,A43,Def1; end; hence thesis; end; v in {v} by TARSKI:def 1; then A59: v in B \/ {v} & not v in B by A41,Th18,RLVECT_1:3,XBOOLE_0:def 2; A60: B c= B \/ {v} by XBOOLE_1:7; then A c= B \/ {v} by A38,XBOOLE_1:1; then B \/ {v} in Q by A2,A42; hence contradiction by A36,A37,A59,A60; end; theorem Th28: Lin(A) = V implies ex B st B c= A & B is linearly-independent & Lin(B) = V proof assume A1: Lin(A) = V; defpred P[set] means (ex B st B = $1 & B c= A & B is linearly-independent); consider Q being set such that A2: for Z holds Z in Q iff Z in bool(the carrier of V) & P[Z] from Separation; {}(the carrier of V) in bool(the carrier of V) & {}(the carrier of V) c= A & {}(the carrier of V) is linearly-independent by Th8,XBOOLE_1:2; then A3: Q <> {} by A2; now let Z; assume that Z <> {} and A4: Z c= Q and A5: Z is c=-linear; set W = union Z; W c= the carrier of V proof let x; assume x in W; then consider X such that A6: x in X and A7: X in Z by TARSKI:def 4; X in bool(the carrier of V) by A2,A4,A7; hence thesis by A6; end; then reconsider W as Subset of V; A8: W c= A proof let x; assume x in W; then consider X such that A9: x in X and A10: X in Z by TARSKI: def 4; ex B st B = X & B c= A & B is linearly-independent by A2,A4, A10 ; hence thesis by A9; end; W is linearly-independent proof let l be Linear_Combination of W; assume that A11: Sum(l) = 0.V and A12: Carrier(l) <> {}; deffunc F(set)= {C : $1 in C & C in Z}; consider f being Function such that A13: dom f = Carrier(l) and A14: for x st x in Carrier(l) holds f.x = F(x) from Lambda; reconsider M = rng f as non empty set by A12,A13,RELAT_1:65; consider F being Choice_Function of M; A15: now assume {} in M; then consider x such that A16: x in dom f and A17: f.x = {} by FUNCT_1:def 5 ; Carrier(l) c= W by RLVECT_2:def 8; then consider X such that A18: x in X and A19: X in Z by A13, A16,TARSKI:def 4; reconsider X as Subset of V by A2,A4,A19; X in {C : x in C & C in Z} by A18,A19; hence contradiction by A13,A14,A16,A17; end; set S = rng F; A20: dom F = M & M <> {} by A15,Lm7; then A21: S <> {} by RELAT_1:65; A22: now let X; assume X in S; then consider x such that A23: x in dom F and A24: F.x = X by FUNCT_1:def 5; consider y such that A25: y in dom f and A26: f.y = x by A20,A23,FUNCT_1: def 5 ; A27: X in x by A15,A20,A23,A24,ORDERS_1:def 1; x = {C : y in C & C in Z} by A13,A14,A25,A26; then ex C st C = X & y in C & C in Z by A27; hence X in Z; end; A28: now let X,Y; assume X in S & Y in S; then X in Z & Y in Z by A22; then X,Y are_c=-comparable by A5,ORDINAL1:def 9; hence X c= Y or Y c= X by XBOOLE_0:def 9; end; dom F is finite by A13,A20,FINSET_1:26; then S is finite by FINSET_1:26; then union S in S by A21,A28,Lm10; then union S in Z by A22; then consider B such that A29: B = union S and B c= A and A30: B is linearly-independent by A2,A4 ; Carrier(l) c= union S proof let x; assume A31: x in Carrier(l); then A32: f.x in M by A13,FUNCT_1:def 5; set X = f.x; A33: F.X in X by A15,A32,ORDERS_1:def 1; f.x = {C : x in C & C in Z} by A14,A31; then A34: ex C st F.X = C & x in C & C in Z by A33 ; F.X in S by A20,A32,FUNCT_1:def 5; hence x in union S by A34,TARSKI:def 4; end; then l is Linear_Combination of B by A29,RLVECT_2:def 8; hence thesis by A11,A12,A30,Def1; end; hence union Z in Q by A2,A8; end; then consider X such that A35: X in Q and A36: for Z st Z in Q & Z <> X holds not X c= Z by A3,ORDERS_2:79; consider B such that A37: B = X and A38: B c= A and A39: B is linearly-independent by A2,A35; take B; thus B c= A & B is linearly-independent by A38,A39; assume A40: Lin(B) <> V; now assume A41: for v st v in A holds v in Lin(B); now let v; assume v in Lin(A); then consider l such that A42: v = Sum(l) by Th17; A43: Carrier(l) c= the carrier of Lin(B) proof let x; assume A44: x in Carrier(l); then reconsider a = x as VECTOR of V; Carrier(l) c= A by RLVECT_2:def 8; then a in Lin(B) by A41,A44; hence thesis by RLVECT_1:def 1; end; reconsider F = the carrier of Lin(B) as Subset of V by RLSUB_1:def 2 ; reconsider l as Linear_Combination of F by A43,RLVECT_2:def 8; Sum(l) = v by A42; then v in Lin(F) by Th17; hence v in Lin(B) by Th21; end; then Lin(A) is Subspace of Lin(B) by RLSUB_1:37; hence contradiction by A1,A40,RLSUB_1:34; end; then consider v such that A45: v in A and A46: not v in Lin(B); A47: B \/ {v} is linearly-independent proof let l be Linear_Combination of B \/ {v}; assume A48: Sum(l) = 0.V; now per cases; suppose A49: v in Carrier(l); deffunc F(VECTOR of V)=l.$1; consider f such that A50: f.v = 0 and A51: for u st u <> v holds f.u = F(u) from LambdaSep1; reconsider f as Element of Funcs(the carrier of V, REAL) by FUNCT_2:11; now let u; assume not u in Carrier(l) \ {v}; then not u in Carrier(l) or u in {v} by XBOOLE_0:def 4; then (l.u = 0 & u <> v) or u = v by RLVECT_2:28,TARSKI:def 1 ; hence f.u = 0 by A50,A51; end; then reconsider f as Linear_Combination of V by RLVECT_2:def 5; Carrier(f) c= B proof let x; assume x in Carrier(f); then x in {u : f.u <> 0} by RLVECT_2:def 6; then consider u such that A52: u = x and A53: f.u <> 0; f.u = l.u by A50,A51,A53; then u in {v1 : l.v1 <> 0} by A53; then u in Carrier(l) & Carrier(l) c= B \/ {v} by RLVECT_2:def 6,def 8 ; then u in B or u in {v} by XBOOLE_0:def 2; hence thesis by A50,A52,A53,TARSKI:def 1; end; then reconsider f as Linear_Combination of B by RLVECT_2:def 8; deffunc G(VECTOR of V)= 0; consider g such that A54: g.v = - l.v and A55: for u st u <> v holds g.u = G(u) from LambdaSep1; reconsider g as Element of Funcs(the carrier of V, REAL) by FUNCT_2:11; now let u; assume not u in {v}; then u <> v by TARSKI:def 1; hence g.u = 0 by A55; end; then reconsider g as Linear_Combination of V by RLVECT_2:def 5; Carrier(g) c= {v} proof let x; assume x in Carrier(g); then x in {u : g.u <> 0} by RLVECT_2:def 6; then ex u st x = u & g.u <> 0; then x = v by A55; hence thesis by TARSKI:def 1; end; then reconsider g as Linear_Combination of {v} by RLVECT_2:def 8; A56: f - g = l proof let u; now per cases; suppose A57: v = u; thus (f - g).u = f.u - g.u by RLVECT_2:79 .= 0 + (- (- l.v)) by A50,A54,A57,XCMPLX_0:def 8 .= l.u by A57; suppose A58: v <> u; thus (f - g).u = f.u - g.u by RLVECT_2:79 .= l.u - g.u by A51,A58 .= l.u - 0 by A55,A58 .= l.u; end; hence thesis; end; A59: Sum(g) = (- l.v) * v by A54,RLVECT_2:50; 0.V = Sum(f) - Sum(g) by A48,A56,Th4; then Sum(f) = 0.V + Sum(g) by RLSUB_2:78 .= (- l.v) * v by A59,RLVECT_1:10; then A60: (- l.v) * v in Lin(B) by Th17; l.v <> 0 by A49,RLVECT_2:28; then A61: - l.v <> 0 by XCMPLX_1:135; (- l.v)" * ((- l.v) * v) = (- l.v)" * (- l.v) * v by RLVECT_1: def 9 .= 1 * v by A61,XCMPLX_0:def 7 .= v by RLVECT_1:def 9; hence thesis by A46,A60,RLSUB_1:29; suppose A62: not v in Carrier(l); Carrier(l) c= B proof let x; assume A63: x in Carrier(l); Carrier(l) c= B \/ {v} by RLVECT_2:def 8; then x in B or x in {v} by A63,XBOOLE_0:def 2; hence thesis by A62,A63,TARSKI:def 1; end; then l is Linear_Combination of B by RLVECT_2:def 8; hence thesis by A39,A48,Def1; end; hence thesis; end; v in {v} by TARSKI:def 1; then A64: v in B \/ {v} & not v in B by A46,Th18,XBOOLE_0:def 2; A65: B c= B \/ {v} by XBOOLE_1:7; {v} c= A by A45,ZFMISC_1:37; then B \/ {v} c= A by A38,XBOOLE_1:8; then B \/ {v} in Q by A2,A47; hence contradiction by A36,A37,A64,A65; end; definition let V be RealLinearSpace; mode Basis of V -> Subset of V means :Def3: it is linearly-independent & Lin(it) = the RLSStruct of V; existence proof {}(the carrier of V) is linearly-independent by Th8; then ex B being Subset of V st {}(the carrier of V) c= B & B is linearly-independent & Lin(B) = the RLSStruct of V by Th27; hence thesis; end; end; reserve I for Basis of V; canceled 3; theorem for V being strict RealLinearSpace,A being Subset of V holds A is linearly-independent implies ex I being Basis of V st A c= I proof let V be strict RealLinearSpace,A be Subset of V; assume A is linearly-independent; then consider B being Subset of V such that A1: A c= B and A2: B is linearly-independent & Lin(B) = V by Th27; reconsider B as Basis of V by A2,Def3; take B; thus thesis by A1; end; theorem Lin(A) = V implies ex I st I c= A proof assume Lin(A) = V; then consider B such that A1: B c= A and A2: B is linearly-independent & Lin(B) = V by Th28; reconsider B as Basis of V by A2,Def3; take B; thus thesis by A1; end; :: :: Auxiliary theorems. :: theorem Z <> {} & Z is finite & (for X,Y st X in Z & Y in Z holds X c= Y or Y c= X) implies union Z in Z by Lm10; theorem not {} in M implies dom CF = M & rng CF c= union M by Lm7; theorem x in (0).V iff x = 0.V by Lm2; theorem W1 is Subspace of W3 implies W1 /\ W2 is Subspace of W3 by Lm3; theorem W1 is Subspace of W2 & W1 is Subspace of W3 implies W1 is Subspace of W2 /\ W3 by Lm4; theorem W1 is Subspace of W3 & W2 is Subspace of W3 implies W1 + W2 is Subspace of W3 by Lm6; theorem W1 is Subspace of W2 implies W1 is Subspace of W2 + W3 by Lm5; theorem f (#) (F ^ G) = (f (#) F) ^ (f (#) G) by Lm1;