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Theorem cdlemk53 34595
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 1054 . . . . . 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 1168 . . . . . . . 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 1169 . . . . . . . 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 541 . . . . . . 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 1064 . . . . . . 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 1170 . . . . . . 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 1065 . . . . . . 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 1055 . . . . . . 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 34576 . . . . . . 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 1310 . . . . . 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 33760 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  [_ G  / 
g ]_ X  e.  T
)  ->  [_ G  / 
g ]_ X : B -1-1-onto-> B
)
231, 21, 22syl2anc 673 . . . . 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 5828 . . . . 5  |-  ( [_ G  /  g ]_ X : B -1-1-onto-> B  ->  [_ G  / 
g ]_ X : B --> B )
25 fcoi1 5769 . . . . 5  |-  ( [_ G  /  g ]_ X : B --> B  ->  ( [_ G  /  g ]_ X  o.  (  _I  |`  B ) )  =  [_ G  / 
g ]_ X )
2623, 24, 253syl 18 . . . 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 472 . . 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 1081 . . . . 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 1210 . . . . . 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 472 . . . . 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 1110 . . . . 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 468 . . . . 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 34574 . . . . 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 1296 . . . 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 5002 . . 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 5002 . . . . 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 33760 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T
)  ->  G : B
-1-1-onto-> B )
381, 5, 37syl2anc 673 . . . . . . 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 5828 . . . . . . 7  |-  ( G : B -1-1-onto-> B  ->  G : B
--> B )
40 fcoi1 5769 . . . . . . 7  |-  ( G : B --> B  -> 
( G  o.  (  _I  |`  B ) )  =  G )
4138, 39, 403syl 18 . . . . . 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 472 . . . . 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 2505 . . . 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 3356 . . 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 2516 . 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 1033 . . 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 1034 . . 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 1085 . . 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 468 . . 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 1086 . . 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 34594 . . 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 1304 . 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 2730 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 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    =/= wne 2641   A.wral 2756   [_csb 3349   class class class wbr 4395    _I cid 4749   `'ccnv 4838    |` cres 4841    o. ccom 4843   -->wf 5585   -1-1-onto->wf1o 5588   ` cfv 5589   iota_crio 6269  (class class class)co 6308   Basecbs 15199   lecple 15275   joincjn 16267   meetcmee 16268   Atomscatm 32900   HLchlt 32987   LHypclh 33620   LTrncltrn 33737   trLctrl 33795
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-riotaBAD 32589
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-1st 6812  df-2nd 6813  df-undef 7038  df-map 7492  df-preset 16251  df-poset 16269  df-plt 16282  df-lub 16298  df-glb 16299  df-join 16300  df-meet 16301  df-p0 16363  df-p1 16364  df-lat 16370  df-clat 16432  df-oposet 32813  df-ol 32815  df-oml 32816  df-covers 32903  df-ats 32904  df-atl 32935  df-cvlat 32959  df-hlat 32988  df-llines 33134  df-lplanes 33135  df-lvols 33136  df-lines 33137  df-psubsp 33139  df-pmap 33140  df-padd 33432  df-lhyp 33624  df-laut 33625  df-ldil 33740  df-ltrn 33741  df-trl 33796
This theorem is referenced by:  cdlemk54  34596  cdlemk55  34599
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