let Y be non empty set ; :: thesis: for G being Subset of (PARTITIONS Y)
for A, B, C, D being a_partition of Y st G = {A,B,C,D} & A <> B & A <> C & A <> D holds
CompF A,G = (B '/\' C) '/\' D
let G be Subset of (PARTITIONS Y); :: thesis: for A, B, C, D being a_partition of Y st G = {A,B,C,D} & A <> B & A <> C & A <> D holds
CompF A,G = (B '/\' C) '/\' D
let A, B, C, D be a_partition of Y; :: thesis: ( G = {A,B,C,D} & A <> B & A <> C & A <> D implies CompF A,G = (B '/\' C) '/\' D )
assume that
A1: G = {A,B,C,D} and
A2: A <> B and
A3: A <> C and
A4: A <> D ; :: thesis: CompF A,G = (B '/\' C) '/\' D
per cases ( B = C or B = D or C = D or ( B <> C & B <> D & C <> D ) ) ;
suppose A5: B = C ; :: thesis: CompF A,G = (B '/\' C) '/\' D
then G = {B,B,A,D} by A1, ENUMSET1:116
.= {B,A,D} by ENUMSET1:71
.= {A,B,D} by ENUMSET1:99 ;
hence CompF A,G = B '/\' D by A2, A4, Th4
.= (B '/\' C) '/\' D by A5, PARTIT1:15 ;
:: thesis: verum
end;
suppose A6: B = D ; :: thesis: CompF A,G = (B '/\' C) '/\' D
then G = {B,B,A,C} by A1, ENUMSET1:112
.= {B,A,C} by ENUMSET1:71
.= {A,B,C} by ENUMSET1:99 ;
hence CompF A,G = B '/\' C by A2, A3, Th4
.= (B '/\' D) '/\' C by A6, PARTIT1:15
.= (B '/\' C) '/\' D by PARTIT1:16 ;
:: thesis: verum
end;
suppose A7: C = D ; :: thesis: CompF A,G = (B '/\' C) '/\' D
then G = {C,C,A,B} by A1, ENUMSET1:118
.= {C,A,B} by ENUMSET1:71
.= {A,B,C} by ENUMSET1:100 ;
hence CompF A,G = B '/\' C by A2, A3, Th4
.= B '/\' (C '/\' D) by A7, PARTIT1:15
.= (B '/\' C) '/\' D by PARTIT1:16 ;
:: thesis: verum
end;
suppose A8: ( B <> C & B <> D & C <> D ) ; :: thesis: CompF A,G = (B '/\' C) '/\' D
G \ {A} = ({A} \/ {B,C,D}) \ {A} by A1, ENUMSET1:44;
then A9: G \ {A} = ({A} \ {A}) \/ ({B,C,D} \ {A}) by XBOOLE_1:42;
A10: not B in {A} by A2, TARSKI:def 1;
A11: ( not C in {A} & not D in {A} ) by A3, A4, TARSKI:def 1;
{B,C,D} \ {A} = ({B} \/ {C,D}) \ {A} by ENUMSET1:42
.= ({B} \ {A}) \/ ({C,D} \ {A}) by XBOOLE_1:42
.= ({B} \ {A}) \/ {C,D} by A11, ZFMISC_1:72
.= {B} \/ {C,D} by A10, ZFMISC_1:67
.= {B,C,D} by ENUMSET1:42 ;
then A12: G \ {A} = {} \/ {B,C,D} by A9, XBOOLE_1:37
.= {B,C,D} ;
A13: (B '/\' C) '/\' D c= '/\' (G \ {A})
proof
let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in (B '/\' C) '/\' D or x in '/\' (G \ {A}) )
assume A14: x in (B '/\' C) '/\' D ; :: thesis: x in '/\' (G \ {A})
then A15: x <> {} by EQREL_1:def 6;
x in (INTERSECTION (B '/\' C),D) \ {{} } by A14, PARTIT1:def 4;
then consider a, d being set such that
A16: a in B '/\' C and
A17: d in D and
A18: x = a /\ d by SETFAM_1:def 5;
a in (INTERSECTION B,C) \ {{} } by A16, PARTIT1:def 4;
then consider b, c being set such that
A19: b in B and
A20: c in C and
A21: a = b /\ c by SETFAM_1:def 5;
set h = ((B .--> b) +* (C .--> c)) +* (D .--> d);
A22: (((B .--> b) +* (C .--> c)) +* (D .--> d)) . D = d by FUNCT_7:96;
A23: (((B .--> b) +* (C .--> c)) +* (D .--> d)) . C = c by A8, Lm2;
A24: rng (((B .--> b) +* (C .--> c)) +* (D .--> d)) = {((((B .--> b) +* (C .--> c)) +* (D .--> d)) . B),((((B .--> b) +* (C .--> c)) +* (D .--> d)) . C),((((B .--> b) +* (C .--> c)) +* (D .--> d)) . D)} by Lm4
.= {((((B .--> b) +* (C .--> c)) +* (D .--> d)) . D),((((B .--> b) +* (C .--> c)) +* (D .--> d)) . B),((((B .--> b) +* (C .--> c)) +* (D .--> d)) . C)} by ENUMSET1:100 ;
A25: (((B .--> b) +* (C .--> c)) +* (D .--> d)) . B = b by A8, Lm3;
rng (((B .--> b) +* (C .--> c)) +* (D .--> d)) c= bool Y
proof
let t be set ; :: according to TARSKI:def 3 :: thesis: ( not t in rng (((B .--> b) +* (C .--> c)) +* (D .--> d)) or t in bool Y )
assume A26: t in rng (((B .--> b) +* (C .--> c)) +* (D .--> d)) ; :: thesis: t in bool Y
now
per cases ( t = (((B .--> b) +* (C .--> c)) +* (D .--> d)) . D or t = (((B .--> b) +* (C .--> c)) +* (D .--> d)) . B or t = (((B .--> b) +* (C .--> c)) +* (D .--> d)) . C ) by A24, A26, ENUMSET1:def 1;
case t = (((B .--> b) +* (C .--> c)) +* (D .--> d)) . D ; :: thesis: t in bool Y
hence t in bool Y by A17, A22; :: thesis: verum
end;
case t = (((B .--> b) +* (C .--> c)) +* (D .--> d)) . B ; :: thesis: t in bool Y
hence t in bool Y by A19, A25; :: thesis: verum
end;
case t = (((B .--> b) +* (C .--> c)) +* (D .--> d)) . C ; :: thesis: t in bool Y
hence t in bool Y by A20, A23; :: thesis: verum
end;
end;
end;
hence t in bool Y ; :: thesis: verum
end;
then reconsider F = rng (((B .--> b) +* (C .--> c)) +* (D .--> d)) as Subset-Family of Y ;
A27: x c= Intersect F
proof
let u be set ; :: according to TARSKI:def 3 :: thesis: ( not u in x or u in Intersect F )
assume A28: u in x ; :: thesis: u in Intersect F
for y being set st y in F holds
u in y
proof
let y be set ; :: thesis: ( y in F implies u in y )
assume A29: y in F ; :: thesis: u in y
now
per cases ( y = (((B .--> b) +* (C .--> c)) +* (D .--> d)) . D or y = (((B .--> b) +* (C .--> c)) +* (D .--> d)) . B or y = (((B .--> b) +* (C .--> c)) +* (D .--> d)) . C ) by A24, A29, ENUMSET1:def 1;
case y = (((B .--> b) +* (C .--> c)) +* (D .--> d)) . D ; :: thesis: u in y
hence u in y by A18, A22, A28, XBOOLE_0:def 4; :: thesis: verum
end;
case A30: y = (((B .--> b) +* (C .--> c)) +* (D .--> d)) . B ; :: thesis: u in y
u in b /\ (c /\ d) by A18, A21, A28, XBOOLE_1:16;
hence u in y by A25, A30, XBOOLE_0:def 4; :: thesis: verum
end;
case A31: y = (((B .--> b) +* (C .--> c)) +* (D .--> d)) . C ; :: thesis: u in y
u in c /\ (b /\ d) by A18, A21, A28, XBOOLE_1:16;
hence u in y by A23, A31, XBOOLE_0:def 4; :: thesis: verum
end;
end;
end;
hence u in y ; :: thesis: verum
end;
then u in meet F by A24, SETFAM_1:def 1;
hence u in Intersect F by A24, SETFAM_1:def 10; :: thesis: verum
end;
A32: for p being set st p in G \ {A} holds
(((B .--> b) +* (C .--> c)) +* (D .--> d)) . p in p
proof
let p be set ; :: thesis: ( p in G \ {A} implies (((B .--> b) +* (C .--> c)) +* (D .--> d)) . p in p )
assume A33: p in G \ {A} ; :: thesis: (((B .--> b) +* (C .--> c)) +* (D .--> d)) . p in p
now
per cases ( p = D or p = B or p = C ) by A12, A33, ENUMSET1:def 1;
case p = D ; :: thesis: (((B .--> b) +* (C .--> c)) +* (D .--> d)) . p in p
hence (((B .--> b) +* (C .--> c)) +* (D .--> d)) . p in p by A17, FUNCT_7:96; :: thesis: verum
end;
case p = B ; :: thesis: (((B .--> b) +* (C .--> c)) +* (D .--> d)) . p in p
hence (((B .--> b) +* (C .--> c)) +* (D .--> d)) . p in p by A8, A19, Lm3; :: thesis: verum
end;
case p = C ; :: thesis: (((B .--> b) +* (C .--> c)) +* (D .--> d)) . p in p
hence (((B .--> b) +* (C .--> c)) +* (D .--> d)) . p in p by A8, A20, Lm2; :: thesis: verum
end;
end;
end;
hence (((B .--> b) +* (C .--> c)) +* (D .--> d)) . p in p ; :: thesis: verum
end;
A34: dom (((B .--> b) +* (C .--> c)) +* (D .--> d)) = {B,C,D} by Lm1;
then D in dom (((B .--> b) +* (C .--> c)) +* (D .--> d)) by ENUMSET1:def 1;
then A35: rng (((B .--> b) +* (C .--> c)) +* (D .--> d)) <> {} by FUNCT_1:12;
Intersect F c= x
proof
let t be set ; :: according to TARSKI:def 3 :: thesis: ( not t in Intersect F or t in x )
assume t in Intersect F ; :: thesis: t in x
then A36: t in meet (rng (((B .--> b) +* (C .--> c)) +* (D .--> d))) by A35, SETFAM_1:def 10;
(((B .--> b) +* (C .--> c)) +* (D .--> d)) . D in rng (((B .--> b) +* (C .--> c)) +* (D .--> d)) by A24, ENUMSET1:def 1;
then A37: t in (((B .--> b) +* (C .--> c)) +* (D .--> d)) . D by A36, SETFAM_1:def 1;
(((B .--> b) +* (C .--> c)) +* (D .--> d)) . C in rng (((B .--> b) +* (C .--> c)) +* (D .--> d)) by A24, ENUMSET1:def 1;
then A38: t in (((B .--> b) +* (C .--> c)) +* (D .--> d)) . C by A36, SETFAM_1:def 1;
(((B .--> b) +* (C .--> c)) +* (D .--> d)) . B in rng (((B .--> b) +* (C .--> c)) +* (D .--> d)) by A24, ENUMSET1:def 1;
then t in (((B .--> b) +* (C .--> c)) +* (D .--> d)) . B by A36, SETFAM_1:def 1;
then t in b /\ c by A25, A23, A38, XBOOLE_0:def 4;
hence t in x by A18, A21, A22, A37, XBOOLE_0:def 4; :: thesis: verum
end;
then x = Intersect F by A27, XBOOLE_0:def 10;
hence x in '/\' (G \ {A}) by A12, A34, A32, A15, BVFUNC_2:def 1; :: thesis: verum
end;
'/\' (G \ {A}) c= (B '/\' C) '/\' D
proof
let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in '/\' (G \ {A}) or x in (B '/\' C) '/\' D )
assume x in '/\' (G \ {A}) ; :: thesis: x in (B '/\' C) '/\' D
then consider h being Function, F being Subset-Family of Y such that
A39: dom h = G \ {A} and
A40: rng h = F and
A41: for d being set st d in G \ {A} holds
h . d in d and
A42: x = Intersect F and
A43: x <> {} by BVFUNC_2:def 1;
D in dom h by A12, A39, ENUMSET1:def 1;
then A44: h . D in rng h by FUNCT_1:def 5;
set m = (h . B) /\ (h . C);
B in dom h by A12, A39, ENUMSET1:def 1;
then A45: h . B in rng h by FUNCT_1:def 5;
C in dom h by A12, A39, ENUMSET1:def 1;
then A46: h . C in rng h by FUNCT_1:def 5;
A47: x c= ((h . B) /\ (h . C)) /\ (h . D)
proof
let m be set ; :: according to TARSKI:def 3 :: thesis: ( not m in x or m in ((h . B) /\ (h . C)) /\ (h . D) )
assume m in x ; :: thesis: m in ((h . B) /\ (h . C)) /\ (h . D)
then A48: m in meet (rng h) by A40, A42, A45, SETFAM_1:def 10;
then ( m in h . B & m in h . C ) by A45, A46, SETFAM_1:def 1;
then A49: m in (h . B) /\ (h . C) by XBOOLE_0:def 4;
m in h . D by A44, A48, SETFAM_1:def 1;
hence m in ((h . B) /\ (h . C)) /\ (h . D) by A49, XBOOLE_0:def 4; :: thesis: verum
end;
then (h . B) /\ (h . C) <> {} by A43;
then A50: not (h . B) /\ (h . C) in {{} } by TARSKI:def 1;
D in G \ {A} by A12, ENUMSET1:def 1;
then A51: h . D in D by A41;
A52: not x in {{} } by A43, TARSKI:def 1;
C in G \ {A} by A12, ENUMSET1:def 1;
then A53: h . C in C by A41;
B in G \ {A} by A12, ENUMSET1:def 1;
then h . B in B by A41;
then (h . B) /\ (h . C) in INTERSECTION B,C by A53, SETFAM_1:def 5;
then (h . B) /\ (h . C) in (INTERSECTION B,C) \ {{} } by A50, XBOOLE_0:def 5;
then A54: (h . B) /\ (h . C) in B '/\' C by PARTIT1:def 4;
((h . B) /\ (h . C)) /\ (h . D) c= x
proof
let m be set ; :: according to TARSKI:def 3 :: thesis: ( not m in ((h . B) /\ (h . C)) /\ (h . D) or m in x )
assume A55: m in ((h . B) /\ (h . C)) /\ (h . D) ; :: thesis: m in x
then A56: m in (h . B) /\ (h . C) by XBOOLE_0:def 4;
A57: rng h c= {(h . B),(h . C),(h . D)}
proof
let u be set ; :: according to TARSKI:def 3 :: thesis: ( not u in rng h or u in {(h . B),(h . C),(h . D)} )
assume u in rng h ; :: thesis: u in {(h . B),(h . C),(h . D)}
then consider x1 being set such that
A58: x1 in dom h and
A59: u = h . x1 by FUNCT_1:def 5;
now
per cases ( x1 = B or x1 = C or x1 = D ) by A12, A39, A58, ENUMSET1:def 1;
case x1 = B ; :: thesis: u in {(h . B),(h . C),(h . D)}
hence u in {(h . B),(h . C),(h . D)} by A59, ENUMSET1:def 1; :: thesis: verum
end;
case x1 = C ; :: thesis: u in {(h . B),(h . C),(h . D)}
hence u in {(h . B),(h . C),(h . D)} by A59, ENUMSET1:def 1; :: thesis: verum
end;
case x1 = D ; :: thesis: u in {(h . B),(h . C),(h . D)}
hence u in {(h . B),(h . C),(h . D)} by A59, ENUMSET1:def 1; :: thesis: verum
end;
end;
end;
hence u in {(h . B),(h . C),(h . D)} ; :: thesis: verum
end;
for y being set st y in rng h holds
m in y
proof
let y be set ; :: thesis: ( y in rng h implies m in y )
assume A60: y in rng h ; :: thesis: m in y
now
per cases ( y = h . B or y = h . C or y = h . D ) by A57, A60, ENUMSET1:def 1;
end;
end;
hence m in y ; :: thesis: verum
end;
then m in meet (rng h) by A45, SETFAM_1:def 1;
hence m in x by A40, A42, A45, SETFAM_1:def 10; :: thesis: verum
end;
then ((h . B) /\ (h . C)) /\ (h . D) = x by A47, XBOOLE_0:def 10;
then x in INTERSECTION (B '/\' C),D by A51, A54, SETFAM_1:def 5;
then x in (INTERSECTION (B '/\' C),D) \ {{} } by A52, XBOOLE_0:def 5;
hence x in (B '/\' C) '/\' D by PARTIT1:def 4; :: thesis: verum
end;
then '/\' (G \ {A}) = (B '/\' C) '/\' D by A13, XBOOLE_0:def 10;
hence CompF A,G = (B '/\' C) '/\' D by BVFUNC_2:def 7; :: thesis: verum
end;
end;