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Theorem cdlemk53 31205
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 980 . . . . . 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 1094 . . . . . . . 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 1095 . . . . . . . 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 518 . . . . . . 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 990 . . . . . . 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 1096 . . . . . . 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 991 . . . . . . 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 981 . . . . . . 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 31186 . . . . . . 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 1201 . . . . . 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 30372 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  [_ G  / 
g ]_ X  e.  T
)  ->  [_ G  / 
g ]_ X : B -1-1-onto-> B
)
231, 21, 22syl2anc 642 . . . . 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 5578 . . . . 5  |-  ( [_ G  /  g ]_ X : B -1-1-onto-> B  ->  [_ G  / 
g ]_ X : B --> B )
25 fcoi1 5521 . . . . 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 451 . . 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 1007 . . . . 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 1133 . . . . . 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 451 . . . . 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 1036 . . . . 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 447 . . . . 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 31184 . . . . 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 1187 . . . 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 4949 . . 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 4949 . . . . 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 30372 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T
)  ->  G : B
-1-1-onto-> B )
381, 5, 37syl2anc 642 . . . . . . 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 5578 . . . . . . 7  |-  ( G : B -1-1-onto-> B  ->  G : B
--> B )
40 fcoi1 5521 . . . . . . 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 451 . . . . 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 2398 . . . 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 3173 . . 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 2409 . 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 959 . . 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 960 . . 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 1011 . . 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 447 . . 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 1012 . . 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 31204 . . 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 1195 . 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 2607 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 set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 935    = wceq 1647    e. wcel 1715    =/= wne 2529   A.wral 2628   [_csb 3167   class class class wbr 4125    _I cid 4407   `'ccnv 4791    |` cres 4794    o. ccom 4796   -->wf 5354   -1-1-onto->wf1o 5357   ` cfv 5358  (class class class)co 5981   iota_crio 6439   Basecbs 13356   lecple 13423   joincjn 14288   meetcmee 14289   Atomscatm 29512   HLchlt 29599   LHypclh 30232   LTrncltrn 30349   trLctrl 30406
This theorem is referenced by:  cdlemk54  31206  cdlemk55  31209
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-iin 4010  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-undef 6440  df-riota 6446  df-map 6917  df-poset 14290  df-plt 14302  df-lub 14318  df-glb 14319  df-join 14320  df-meet 14321  df-p0 14355  df-p1 14356  df-lat 14362  df-clat 14424  df-oposet 29425  df-ol 29427  df-oml 29428  df-covers 29515  df-ats 29516  df-atl 29547  df-cvlat 29571  df-hlat 29600  df-llines 29746  df-lplanes 29747  df-lvols 29748  df-lines 29749  df-psubsp 29751  df-pmap 29752  df-padd 30044  df-lhyp 30236  df-laut 30237  df-ldil 30352  df-ltrn 30353  df-trl 30407
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