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Theorem cdlemk52 34937
Description: Part of proof of Lemma K of [Crawley] p. 118. Line 6, p. 120.  G,  I stand for g, h.  X represents tau. (Contributed by NM, 23-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
cdlemk52  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  I )
) )  ->  (
( [_ G  /  g ]_ X  o.  [_ I  /  g ]_ X
) `  P )  =  ( [_ ( G  o.  I )  /  g ]_ X `  P ) )
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 cdlemk52
StepHypRef Expression
1 cdlemk5.b . . . 4  |-  B  =  ( Base `  K
)
2 cdlemk5.l . . . 4  |-  .<_  =  ( le `  K )
3 simp11l 1099 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  I )
) )  ->  K  e.  HL )
4 hllat 33347 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
53, 4syl 16 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  I )
) )  ->  K  e.  Lat )
6 simp11 1018 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  I )
) )  ->  ( K  e.  HL  /\  W  e.  H ) )
7 simp12 1019 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  I )
) )  ->  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) ) )
8 simp13 1020 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  I )
) )  ->  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )
9 simp21 1021 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  I )
) )  ->  N  e.  T )
10 simp22 1022 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  I )
) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
11 simp23 1023 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  I )
) )  ->  ( R `  F )  =  ( R `  N ) )
12 cdlemk5.j . . . . . . . . 9  |-  .\/  =  ( join `  K )
13 cdlemk5.m . . . . . . . . 9  |-  ./\  =  ( meet `  K )
14 cdlemk5.a . . . . . . . . 9  |-  A  =  ( Atoms `  K )
15 cdlemk5.h . . . . . . . . 9  |-  H  =  ( LHyp `  K
)
16 cdlemk5.t . . . . . . . . 9  |-  T  =  ( ( LTrn `  K
) `  W )
17 cdlemk5.r . . . . . . . . 9  |-  R  =  ( ( trL `  K
) `  W )
18 cdlemk5.z . . . . . . . . 9  |-  Z  =  ( ( P  .\/  ( R `  b ) )  ./\  ( ( N `  P )  .\/  ( R `  (
b  o.  `' F
) ) ) )
19 cdlemk5.y . . . . . . . . 9  |-  Y  =  ( ( P  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
g  o.  `' b ) ) ) )
20 cdlemk5.x . . . . . . . . 9  |-  X  =  ( iota_ z  e.  T  A. b  e.  T  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 g ) )  ->  ( z `  P )  =  Y ) )
211, 2, 12, 13, 14, 15, 16, 17, 18, 19, 20cdlemk35s 34920 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  N  e.  T )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) ) )  ->  [_ G  /  g ]_ X  e.  T )
226, 7, 8, 9, 10, 11, 21syl132anc 1237 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  I )
) )  ->  [_ G  /  g ]_ X  e.  T )
23 simp31 1024 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  I )
) )  ->  I  e.  T )
24 simp32 1025 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  I )
) )  ->  I  =/=  (  _I  |`  B ) )
2523, 24jca 532 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  I )
) )  ->  (
I  e.  T  /\  I  =/=  (  _I  |`  B ) ) )
261, 2, 12, 13, 14, 15, 16, 17, 18, 19, 20cdlemk35s 34920 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B ) )  /\  N  e.  T )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) ) )  ->  [_ I  /  g ]_ X  e.  T )
276, 7, 25, 9, 10, 11, 26syl132anc 1237 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  I )
) )  ->  [_ I  /  g ]_ X  e.  T )
2815, 16ltrnco 34702 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  [_ G  / 
g ]_ X  e.  T  /\  [_ I  /  g ]_ X  e.  T
)  ->  ( [_ G  /  g ]_ X  o.  [_ I  /  g ]_ X )  e.  T
)
296, 22, 27, 28syl3anc 1219 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  I )
) )  ->  ( [_ G  /  g ]_ X  o.  [_ I  /  g ]_ X
)  e.  T )
30 simp22l 1107 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  I )
) )  ->  P  e.  A )
312, 14, 15, 16ltrnat 34123 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( [_ G  /  g ]_ X  o.  [_ I  /  g ]_ X )  e.  T  /\  P  e.  A
)  ->  ( ( [_ G  /  g ]_ X  o.  [_ I  /  g ]_ X
) `  P )  e.  A )
326, 29, 30, 31syl3anc 1219 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  I )
) )  ->  (
( [_ G  /  g ]_ X  o.  [_ I  /  g ]_ X
) `  P )  e.  A )
331, 14atbase 33273 . . . . 5  |-  ( ( ( [_ G  / 
g ]_ X  o.  [_ I  /  g ]_ X
) `  P )  e.  A  ->  ( (
[_ G  /  g ]_ X  o.  [_ I  /  g ]_ X
) `  P )  e.  B )
3432, 33syl 16 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  I )
) )  ->  (
( [_ G  /  g ]_ X  o.  [_ I  /  g ]_ X
) `  P )  e.  B )
352, 14, 15, 16ltrnat 34123 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  [_ G  / 
g ]_ X  e.  T  /\  P  e.  A
)  ->  ( [_ G  /  g ]_ X `  P )  e.  A
)
366, 22, 30, 35syl3anc 1219 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  I )
) )  ->  ( [_ G  /  g ]_ X `  P )  e.  A )
371, 14atbase 33273 . . . . . . 7  |-  ( (
[_ G  /  g ]_ X `  P )  e.  A  ->  ( [_ G  /  g ]_ X `  P )  e.  B )
3836, 37syl 16 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  I )
) )  ->  ( [_ G  /  g ]_ X `  P )  e.  B )
391, 15, 16, 17trlcl 34147 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  [_ I  / 
g ]_ X  e.  T
)  ->  ( R `  [_ I  /  g ]_ X )  e.  B
)
406, 27, 39syl2anc 661 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  I )
) )  ->  ( R `  [_ I  / 
g ]_ X )  e.  B )
411, 12latjcl 15341 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( [_ G  /  g ]_ X `  P )  e.  B  /\  ( R `  [_ I  / 
g ]_ X )  e.  B )  ->  (
( [_ G  /  g ]_ X `  P ) 
.\/  ( R `  [_ I  /  g ]_ X ) )  e.  B )
425, 38, 40, 41syl3anc 1219 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  I )
) )  ->  (
( [_ G  /  g ]_ X `  P ) 
.\/  ( R `  [_ I  /  g ]_ X ) )  e.  B )
432, 14, 15, 16ltrnat 34123 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  [_ I  / 
g ]_ X  e.  T  /\  P  e.  A
)  ->  ( [_ I  /  g ]_ X `  P )  e.  A
)
446, 27, 30, 43syl3anc 1219 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  I )
) )  ->  ( [_ I  /  g ]_ X `  P )  e.  A )
451, 14atbase 33273 . . . . . . 7  |-  ( (
[_ I  /  g ]_ X `  P )  e.  A  ->  ( [_ I  /  g ]_ X `  P )  e.  B )
4644, 45syl 16 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  I )
) )  ->  ( [_ I  /  g ]_ X `  P )  e.  B )
471, 15, 16, 17trlcl 34147 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  [_ G  / 
g ]_ X  e.  T
)  ->  ( R `  [_ G  /  g ]_ X )  e.  B
)
486, 22, 47syl2anc 661 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  I )
) )  ->  ( R `  [_ G  / 
g ]_ X )  e.  B )
491, 12latjcl 15341 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( [_ I  /  g ]_ X `  P )  e.  B  /\  ( R `  [_ G  / 
g ]_ X )  e.  B )  ->  (
( [_ I  /  g ]_ X `  P ) 
.\/  ( R `  [_ G  /  g ]_ X ) )  e.  B )
505, 46, 48, 49syl3anc 1219 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  I )
) )  ->  (
( [_ I  /  g ]_ X `  P ) 
.\/  ( R `  [_ G  /  g ]_ X ) )  e.  B )
511, 13latmcl 15342 . . . . 5  |-  ( ( K  e.  Lat  /\  ( ( [_ G  /  g ]_ X `  P )  .\/  ( R `  [_ I  / 
g ]_ X ) )  e.  B  /\  (
( [_ I  /  g ]_ X `  P ) 
.\/  ( R `  [_ G  /  g ]_ X ) )  e.  B )  ->  (
( ( [_ G  /  g ]_ X `  P )  .\/  ( R `  [_ I  / 
g ]_ X ) ) 
./\  ( ( [_ I  /  g ]_ X `  P )  .\/  ( R `  [_ G  / 
g ]_ X ) ) )  e.  B )
525, 42, 50, 51syl3anc 1219 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  I )
) )  ->  (
( ( [_ G  /  g ]_ X `  P )  .\/  ( R `  [_ I  / 
g ]_ X ) ) 
./\  ( ( [_ I  /  g ]_ X `  P )  .\/  ( R `  [_ G  / 
g ]_ X ) ) )  e.  B )
53 simp11r 1100 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  I )
) )  ->  W  e.  H )
541, 14, 15, 16, 17trlnidat 34156 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  I  e.  T  /\  I  =/=  (  _I  |`  B ) )  ->  ( R `  I )  e.  A
)
553, 53, 23, 24, 54syl211anc 1225 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  I )
) )  ->  ( R `  I )  e.  A )
561, 12, 14hlatjcl 33350 . . . . . 6  |-  ( ( K  e.  HL  /\  ( [_ G  /  g ]_ X `  P )  e.  A  /\  ( R `  I )  e.  A )  ->  (
( [_ G  /  g ]_ X `  P ) 
.\/  ( R `  I ) )  e.  B )
573, 36, 55, 56syl3anc 1219 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  I )
) )  ->  (
( [_ G  /  g ]_ X `  P ) 
.\/  ( R `  I ) )  e.  B )
58 simp13l 1103 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  I )
) )  ->  G  e.  T )
59 simp13r 1104 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  I )
) )  ->  G  =/=  (  _I  |`  B ) )
601, 14, 15, 16, 17trlnidat 34156 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T  /\  G  =/=  (  _I  |`  B ) )  ->  ( R `  G )  e.  A
)
613, 53, 58, 59, 60syl211anc 1225 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  I )
) )  ->  ( R `  G )  e.  A )
621, 12, 14hlatjcl 33350 . . . . . 6  |-  ( ( K  e.  HL  /\  ( [_ I  /  g ]_ X `  P )  e.  A  /\  ( R `  G )  e.  A )  ->  (
( [_ I  /  g ]_ X `  P ) 
.\/  ( R `  G ) )  e.  B )
633, 44, 61, 62syl3anc 1219 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  I )
) )  ->  (
( [_ I  /  g ]_ X `  P ) 
.\/  ( R `  G ) )  e.  B )
641, 13latmcl 15342 . . . . 5  |-  ( ( K  e.  Lat  /\  ( ( [_ G  /  g ]_ X `  P )  .\/  ( R `  I )
)  e.  B  /\  ( ( [_ I  /  g ]_ X `  P )  .\/  ( R `  G )
)  e.  B )  ->  ( ( (
[_ G  /  g ]_ X `  P ) 
.\/  ( R `  I ) )  ./\  ( ( [_ I  /  g ]_ X `  P )  .\/  ( R `  G )
) )  e.  B
)
655, 57, 63, 64syl3anc 1219 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  I )
) )  ->  (
( ( [_ G  /  g ]_ X `  P )  .\/  ( R `  I )
)  ./\  ( ( [_ I  /  g ]_ X `  P ) 
.\/  ( R `  G ) ) )  e.  B )
661, 2, 12, 13, 14, 15, 16, 17, 18, 19, 20cdlemk50 34935 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B ) ) )  ->  (
( [_ G  /  g ]_ X  o.  [_ I  /  g ]_ X
) `  P )  .<_  ( ( ( [_ G  /  g ]_ X `  P )  .\/  ( R `  [_ I  / 
g ]_ X ) ) 
./\  ( ( [_ I  /  g ]_ X `  P )  .\/  ( R `  [_ G  / 
g ]_ X ) ) ) )
6725, 66syld3an3 1264 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  I )
) )  ->  (
( [_ G  /  g ]_ X  o.  [_ I  /  g ]_ X
) `  P )  .<_  ( ( ( [_ G  /  g ]_ X `  P )  .\/  ( R `  [_ I  / 
g ]_ X ) ) 
./\  ( ( [_ I  /  g ]_ X `  P )  .\/  ( R `  [_ G  / 
g ]_ X ) ) ) )
681, 2, 12, 13, 14, 15, 16, 17, 18, 19, 20cdlemk51 34936 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B ) ) )  ->  (
( ( [_ G  /  g ]_ X `  P )  .\/  ( R `  [_ I  / 
g ]_ X ) ) 
./\  ( ( [_ I  /  g ]_ X `  P )  .\/  ( R `  [_ G  / 
g ]_ X ) ) )  .<_  ( (
( [_ G  /  g ]_ X `  P ) 
.\/  ( R `  I ) )  ./\  ( ( [_ I  /  g ]_ X `  P )  .\/  ( R `  G )
) ) )
6925, 68syld3an3 1264 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  I )
) )  ->  (
( ( [_ G  /  g ]_ X `  P )  .\/  ( R `  [_ I  / 
g ]_ X ) ) 
./\  ( ( [_ I  /  g ]_ X `  P )  .\/  ( R `  [_ G  / 
g ]_ X ) ) )  .<_  ( (
( [_ G  /  g ]_ X `  P ) 
.\/  ( R `  I ) )  ./\  ( ( [_ I  /  g ]_ X `  P )  .\/  ( R `  G )
) ) )
701, 2, 5, 34, 52, 65, 67, 69lattrd 15348 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  I )
) )  ->  (
( [_ G  /  g ]_ X  o.  [_ I  /  g ]_ X
) `  P )  .<_  ( ( ( [_ G  /  g ]_ X `  P )  .\/  ( R `  I )
)  ./\  ( ( [_ I  /  g ]_ X `  P ) 
.\/  ( R `  G ) ) ) )
711, 2, 12, 13, 14, 15, 16, 17, 18, 19, 20cdlemk47 34932 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  I )
) )  ->  ( [_ ( G  o.  I
)  /  g ]_ X `  P )  =  ( ( (
[_ G  /  g ]_ X `  P ) 
.\/  ( R `  I ) )  ./\  ( ( [_ I  /  g ]_ X `  P )  .\/  ( R `  G )
) ) )
7270, 71breqtrrd 4427 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  I )
) )  ->  (
( [_ G  /  g ]_ X  o.  [_ I  /  g ]_ X
) `  P )  .<_  ( [_ ( G  o.  I )  / 
g ]_ X `  P
) )
73 hlatl 33344 . . . 4  |-  ( K  e.  HL  ->  K  e.  AtLat )
743, 73syl 16 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  I )
) )  ->  K  e.  AtLat )
7515, 16ltrnco 34702 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T  /\  I  e.  T
)  ->  ( G  o.  I )  e.  T
)
766, 58, 23, 75syl3anc 1219 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  I )
) )  ->  ( G  o.  I )  e.  T )
7758, 23jca 532 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  I )
) )  ->  ( G  e.  T  /\  I  e.  T )
)
78 simp33 1026 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  I )
) )  ->  ( R `  G )  =/=  ( R `  I
) )
791, 15, 16, 17trlconid 34708 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( G  e.  T  /\  I  e.  T )  /\  ( R `  G )  =/=  ( R `  I
) )  ->  ( G  o.  I )  =/=  (  _I  |`  B ) )
806, 77, 78, 79syl3anc 1219 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  I )
) )  ->  ( G  o.  I )  =/=  (  _I  |`  B ) )
8176, 80jca 532 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  I )
) )  ->  (
( G  o.  I
)  e.  T  /\  ( G  o.  I
)  =/=  (  _I  |`  B ) ) )
821, 2, 12, 13, 14, 15, 16, 17, 18, 19, 20cdlemk35s 34920 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( ( G  o.  I )  e.  T  /\  ( G  o.  I )  =/=  (  _I  |`  B ) )  /\  N  e.  T )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) ) )  ->  [_ ( G  o.  I )  /  g ]_ X  e.  T )
836, 7, 81, 9, 10, 11, 82syl132anc 1237 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  I )
) )  ->  [_ ( G  o.  I )  /  g ]_ X  e.  T )
842, 14, 15, 16ltrnat 34123 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  [_ ( G  o.  I )  / 
g ]_ X  e.  T  /\  P  e.  A
)  ->  ( [_ ( G  o.  I
)  /  g ]_ X `  P )  e.  A )
856, 83, 30, 84syl3anc 1219 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  I )
) )  ->  ( [_ ( G  o.  I
)  /  g ]_ X `  P )  e.  A )
862, 14atcmp 33295 . . 3  |-  ( ( K  e.  AtLat  /\  (
( [_ G  /  g ]_ X  o.  [_ I  /  g ]_ X
) `  P )  e.  A  /\  ( [_ ( G  o.  I
)  /  g ]_ X `  P )  e.  A )  ->  (
( ( [_ G  /  g ]_ X  o.  [_ I  /  g ]_ X ) `  P
)  .<_  ( [_ ( G  o.  I )  /  g ]_ X `  P )  <->  ( ( [_ G  /  g ]_ X  o.  [_ I  /  g ]_ X
) `  P )  =  ( [_ ( G  o.  I )  /  g ]_ X `  P ) ) )
8774, 32, 85, 86syl3anc 1219 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  I )
) )  ->  (
( ( [_ G  /  g ]_ X  o.  [_ I  /  g ]_ X ) `  P
)  .<_  ( [_ ( G  o.  I )  /  g ]_ X `  P )  <->  ( ( [_ G  /  g ]_ X  o.  [_ I  /  g ]_ X
) `  P )  =  ( [_ ( G  o.  I )  /  g ]_ X `  P ) ) )
8872, 87mpbid 210 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  I )
) )  ->  (
( [_ G  /  g ]_ X  o.  [_ I  /  g ]_ X
) `  P )  =  ( [_ ( G  o.  I )  /  g ]_ X `  P ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2648   A.wral 2799   [_csb 3396   class class class wbr 4401    _I cid 4740   `'ccnv 4948    |` cres 4951    o. ccom 4953   ` cfv 5527   iota_crio 6161  (class class class)co 6201   Basecbs 14293   lecple 14365   joincjn 15234   meetcmee 15235   Latclat 15335   Atomscatm 33247   AtLatcal 33248   HLchlt 33334   LHypclh 33967   LTrncltrn 34084   trLctrl 34141
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-riotaBAD 32943
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-iin 4283  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-1st 6688  df-2nd 6689  df-undef 6903  df-map 7327  df-poset 15236  df-plt 15248  df-lub 15264  df-glb 15265  df-join 15266  df-meet 15267  df-p0 15329  df-p1 15330  df-lat 15336  df-clat 15398  df-oposet 33160  df-ol 33162  df-oml 33163  df-covers 33250  df-ats 33251  df-atl 33282  df-cvlat 33306  df-hlat 33335  df-llines 33481  df-lplanes 33482  df-lvols 33483  df-lines 33484  df-psubsp 33486  df-pmap 33487  df-padd 33779  df-lhyp 33971  df-laut 33972  df-ldil 34087  df-ltrn 34088  df-trl 34142
This theorem is referenced by:  cdlemk53a  34938
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