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Theorem cdlemk53 36826
Description: Part of proof of Lemma K of [Crawley] p. 118. Line 7, p. 120.  G,  I stand for g, h.  X represents tau. (Contributed by NM, 26-Jul-2013.)
Hypotheses
Ref Expression
cdlemk5.b  |-  B  =  ( Base `  K
)
cdlemk5.l  |-  .<_  =  ( le `  K )
cdlemk5.j  |-  .\/  =  ( join `  K )
cdlemk5.m  |-  ./\  =  ( meet `  K )
cdlemk5.a  |-  A  =  ( Atoms `  K )
cdlemk5.h  |-  H  =  ( LHyp `  K
)
cdlemk5.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemk5.r  |-  R  =  ( ( trL `  K
) `  W )
cdlemk5.z  |-  Z  =  ( ( P  .\/  ( R `  b ) )  ./\  ( ( N `  P )  .\/  ( R `  (
b  o.  `' F
) ) ) )
cdlemk5.y  |-  Y  =  ( ( P  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
g  o.  `' b ) ) ) )
cdlemk5.x  |-  X  =  ( iota_ z  e.  T  A. b  e.  T  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 g ) )  ->  ( z `  P )  =  Y ) )
Assertion
Ref Expression
cdlemk53  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( I  e.  T  /\  ( R `  G
)  =/=  ( R `
 I ) ) )  ->  [_ ( G  o.  I )  / 
g ]_ X  =  (
[_ G  /  g ]_ X  o.  [_ I  /  g ]_ X
) )
Distinct variable groups:    ./\ , g    .\/ , g    B, g    P, g    R, g    T, g    g, Z    g, b, G, z    ./\ , b, z    .<_ , b    z,
g,  .<_    .\/ , b, z    A, b, g, z    B, b, z    F, b, g, z   
z, G    H, b,
g, z    K, b,
g, z    N, b,
g, z    P, b,
z    R, b, z    T, b, z    W, b, g, z    z, Y    G, b    I, b, g, z
Allowed substitution hints:    X( z, g, b)    Y( g, b)    Z( z, b)

Proof of Theorem cdlemk53
StepHypRef Expression
1 simp1l 1020 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( I  e.  T  /\  ( R `  G
)  =/=  ( R `
 I ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
2 simp211 1134 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( I  e.  T  /\  ( R `  G
)  =/=  ( R `
 I ) ) )  ->  F  e.  T )
3 simp212 1135 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( I  e.  T  /\  ( R `  G
)  =/=  ( R `
 I ) ) )  ->  F  =/=  (  _I  |`  B ) )
42, 3jca 532 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( I  e.  T  /\  ( R `  G
)  =/=  ( R `
 I ) ) )  ->  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) ) )
5 simp22 1030 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( I  e.  T  /\  ( R `  G
)  =/=  ( R `
 I ) ) )  ->  G  e.  T )
6 simp213 1136 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( I  e.  T  /\  ( R `  G
)  =/=  ( R `
 I ) ) )  ->  N  e.  T )
7 simp23 1031 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( I  e.  T  /\  ( R `  G
)  =/=  ( R `
 I ) ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
8 simp1r 1021 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( I  e.  T  /\  ( R `  G
)  =/=  ( R `
 I ) ) )  ->  ( R `  F )  =  ( R `  N ) )
9 cdlemk5.b . . . . . . . 8  |-  B  =  ( Base `  K
)
10 cdlemk5.l . . . . . . . 8  |-  .<_  =  ( le `  K )
11 cdlemk5.j . . . . . . . 8  |-  .\/  =  ( join `  K )
12 cdlemk5.m . . . . . . . 8  |-  ./\  =  ( meet `  K )
13 cdlemk5.a . . . . . . . 8  |-  A  =  ( Atoms `  K )
14 cdlemk5.h . . . . . . . 8  |-  H  =  ( LHyp `  K
)
15 cdlemk5.t . . . . . . . 8  |-  T  =  ( ( LTrn `  K
) `  W )
16 cdlemk5.r . . . . . . . 8  |-  R  =  ( ( trL `  K
) `  W )
17 cdlemk5.z . . . . . . . 8  |-  Z  =  ( ( P  .\/  ( R `  b ) )  ./\  ( ( N `  P )  .\/  ( R `  (
b  o.  `' F
) ) ) )
18 cdlemk5.y . . . . . . . 8  |-  Y  =  ( ( P  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
g  o.  `' b ) ) ) )
19 cdlemk5.x . . . . . . . 8  |-  X  =  ( iota_ z  e.  T  A. b  e.  T  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 g ) )  ->  ( z `  P )  =  Y ) )
209, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19cdlemk35s-id 36807 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  G  e.  T  /\  N  e.  T )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) ) )  ->  [_ G  /  g ]_ X  e.  T )
211, 4, 5, 6, 7, 8, 20syl132anc 1246 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( I  e.  T  /\  ( R `  G
)  =/=  ( R `
 I ) ) )  ->  [_ G  / 
g ]_ X  e.  T
)
229, 14, 15ltrn1o 35991 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  [_ G  / 
g ]_ X  e.  T
)  ->  [_ G  / 
g ]_ X : B -1-1-onto-> B
)
231, 21, 22syl2anc 661 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( I  e.  T  /\  ( R `  G
)  =/=  ( R `
 I ) ) )  ->  [_ G  / 
g ]_ X : B -1-1-onto-> B
)
24 f1of 5822 . . . . 5  |-  ( [_ G  /  g ]_ X : B -1-1-onto-> B  ->  [_ G  / 
g ]_ X : B --> B )
25 fcoi1 5765 . . . . 5  |-  ( [_ G  /  g ]_ X : B --> B  ->  ( [_ G  /  g ]_ X  o.  (  _I  |`  B ) )  =  [_ G  / 
g ]_ X )
2623, 24, 253syl 20 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( I  e.  T  /\  ( R `  G
)  =/=  ( R `
 I ) ) )  ->  ( [_ G  /  g ]_ X  o.  (  _I  |`  B ) )  =  [_ G  /  g ]_ X
)
2726adantr 465 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( I  e.  T  /\  ( R `  G
)  =/=  ( R `
 I ) ) )  /\  I  =  (  _I  |`  B ) )  ->  ( [_ G  /  g ]_ X  o.  (  _I  |`  B ) )  =  [_ G  /  g ]_ X
)
28 simpl1l 1047 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( I  e.  T  /\  ( R `  G
)  =/=  ( R `
 I ) ) )  /\  I  =  (  _I  |`  B ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
292, 6, 83jca 1176 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( I  e.  T  /\  ( R `  G
)  =/=  ( R `
 I ) ) )  ->  ( F  e.  T  /\  N  e.  T  /\  ( R `
 F )  =  ( R `  N
) ) )
3029adantr 465 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( I  e.  T  /\  ( R `  G
)  =/=  ( R `
 I ) ) )  /\  I  =  (  _I  |`  B ) )  ->  ( F  e.  T  /\  N  e.  T  /\  ( R `
 F )  =  ( R `  N
) ) )
31 simpl23 1076 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( I  e.  T  /\  ( R `  G
)  =/=  ( R `
 I ) ) )  /\  I  =  (  _I  |`  B ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
32 simpr 461 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( I  e.  T  /\  ( R `  G
)  =/=  ( R `
 I ) ) )  /\  I  =  (  _I  |`  B ) )  ->  I  =  (  _I  |`  B ) )
339, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19cdlemkid 36805 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  N  e.  T  /\  ( R `
 F )  =  ( R `  N
) )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  I  =  (  _I  |`  B )
) )  ->  [_ I  /  g ]_ X  =  (  _I  |`  B ) )
3428, 30, 31, 32, 33syl112anc 1232 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( I  e.  T  /\  ( R `  G
)  =/=  ( R `
 I ) ) )  /\  I  =  (  _I  |`  B ) )  ->  [_ I  / 
g ]_ X  =  (  _I  |`  B )
)
3534coeq2d 5175 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( I  e.  T  /\  ( R `  G
)  =/=  ( R `
 I ) ) )  /\  I  =  (  _I  |`  B ) )  ->  ( [_ G  /  g ]_ X  o.  [_ I  /  g ]_ X )  =  (
[_ G  /  g ]_ X  o.  (  _I  |`  B ) ) )
3632coeq2d 5175 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( I  e.  T  /\  ( R `  G
)  =/=  ( R `
 I ) ) )  /\  I  =  (  _I  |`  B ) )  ->  ( G  o.  I )  =  ( G  o.  (  _I  |`  B ) ) )
379, 14, 15ltrn1o 35991 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T
)  ->  G : B
-1-1-onto-> B )
381, 5, 37syl2anc 661 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( I  e.  T  /\  ( R `  G
)  =/=  ( R `
 I ) ) )  ->  G : B
-1-1-onto-> B )
39 f1of 5822 . . . . . . 7  |-  ( G : B -1-1-onto-> B  ->  G : B
--> B )
40 fcoi1 5765 . . . . . . 7  |-  ( G : B --> B  -> 
( G  o.  (  _I  |`  B ) )  =  G )
4138, 39, 403syl 20 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( I  e.  T  /\  ( R `  G
)  =/=  ( R `
 I ) ) )  ->  ( G  o.  (  _I  |`  B ) )  =  G )
4241adantr 465 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( I  e.  T  /\  ( R `  G
)  =/=  ( R `
 I ) ) )  /\  I  =  (  _I  |`  B ) )  ->  ( G  o.  (  _I  |`  B ) )  =  G )
4336, 42eqtrd 2498 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( I  e.  T  /\  ( R `  G
)  =/=  ( R `
 I ) ) )  /\  I  =  (  _I  |`  B ) )  ->  ( G  o.  I )  =  G )
4443csbeq1d 3437 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( I  e.  T  /\  ( R `  G
)  =/=  ( R `
 I ) ) )  /\  I  =  (  _I  |`  B ) )  ->  [_ ( G  o.  I )  / 
g ]_ X  =  [_ G  /  g ]_ X
)
4527, 35, 443eqtr4rd 2509 . 2  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( I  e.  T  /\  ( R `  G
)  =/=  ( R `
 I ) ) )  /\  I  =  (  _I  |`  B ) )  ->  [_ ( G  o.  I )  / 
g ]_ X  =  (
[_ G  /  g ]_ X  o.  [_ I  /  g ]_ X
) )
46 simpl1 999 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( I  e.  T  /\  ( R `  G
)  =/=  ( R `
 I ) ) )  /\  I  =/=  (  _I  |`  B ) )  ->  ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) ) )
47 simpl2 1000 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( I  e.  T  /\  ( R `  G
)  =/=  ( R `
 I ) ) )  /\  I  =/=  (  _I  |`  B ) )  ->  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )
48 simpl3l 1051 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( I  e.  T  /\  ( R `  G
)  =/=  ( R `
 I ) ) )  /\  I  =/=  (  _I  |`  B ) )  ->  I  e.  T )
49 simpr 461 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( I  e.  T  /\  ( R `  G
)  =/=  ( R `
 I ) ) )  /\  I  =/=  (  _I  |`  B ) )  ->  I  =/=  (  _I  |`  B ) )
50 simpl3r 1052 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( I  e.  T  /\  ( R `  G
)  =/=  ( R `
 I ) ) )  /\  I  =/=  (  _I  |`  B ) )  ->  ( R `  G )  =/=  ( R `  I )
)
519, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19cdlemk53b 36825 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  I ) ) )  ->  [_ ( G  o.  I )  /  g ]_ X  =  ( [_ G  /  g ]_ X  o.  [_ I  /  g ]_ X
) )
5246, 47, 48, 49, 50, 51syl113anc 1240 . 2  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( I  e.  T  /\  ( R `  G
)  =/=  ( R `
 I ) ) )  /\  I  =/=  (  _I  |`  B ) )  ->  [_ ( G  o.  I )  / 
g ]_ X  =  (
[_ G  /  g ]_ X  o.  [_ I  /  g ]_ X
) )
5345, 52pm2.61dane 2775 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( I  e.  T  /\  ( R `  G
)  =/=  ( R `
 I ) ) )  ->  [_ ( G  o.  I )  / 
g ]_ X  =  (
[_ G  /  g ]_ X  o.  [_ I  /  g ]_ X
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819    =/= wne 2652   A.wral 2807   [_csb 3430   class class class wbr 4456    _I cid 4799   `'ccnv 5007    |` cres 5010    o. ccom 5012   -->wf 5590   -1-1-onto->wf1o 5593   ` cfv 5594   iota_crio 6257  (class class class)co 6296   Basecbs 14644   lecple 14719   joincjn 15700   meetcmee 15701   Atomscatm 35131   HLchlt 35218   LHypclh 35851   LTrncltrn 35968   trLctrl 36026
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-riotaBAD 34827
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-undef 7020  df-map 7440  df-preset 15684  df-poset 15702  df-plt 15715  df-lub 15731  df-glb 15732  df-join 15733  df-meet 15734  df-p0 15796  df-p1 15797  df-lat 15803  df-clat 15865  df-oposet 35044  df-ol 35046  df-oml 35047  df-covers 35134  df-ats 35135  df-atl 35166  df-cvlat 35190  df-hlat 35219  df-llines 35365  df-lplanes 35366  df-lvols 35367  df-lines 35368  df-psubsp 35370  df-pmap 35371  df-padd 35663  df-lhyp 35855  df-laut 35856  df-ldil 35971  df-ltrn 35972  df-trl 36027
This theorem is referenced by:  cdlemk54  36827  cdlemk55  36830
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