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Theorem cdlemk54 34234
Description: Part of proof of Lemma K of [Crawley] p. 118. Line 10, 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
cdlemk54  |-  ( ( ( ( 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
) )  /\  j  e.  T  /\  (
j  =/=  (  _I  |`  B )  /\  ( R `  j )  =/=  ( R `  G
)  /\  ( R `  j )  =/=  ( R `  ( G  o.  I ) ) ) ) )  ->  ( [_ ( G  o.  I
)  /  g ]_ X  o.  [_ j  / 
g ]_ X )  =  ( ( [_ G  /  g ]_ X  o.  [_ I  /  g ]_ X )  o.  [_ j  /  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   
j, b, g, z
Allowed substitution hints:    A( j)    B( j)    P( j)    R( j)    T( j)    F( j)    G( j)    H( j)    I( j)    .\/ ( j)    K( j)    .<_ ( j)    ./\ ( j)    N( j)    W( j)    X( z, g, j, b)    Y( g, j, b)    Z( z, j, b)

Proof of Theorem cdlemk54
StepHypRef Expression
1 coass 5374 . . 3  |-  ( ( G  o.  I )  o.  j )  =  ( G  o.  (
I  o.  j ) )
2 csbeq1 3404 . . 3  |-  ( ( ( G  o.  I
)  o.  j )  =  ( G  o.  ( I  o.  j
) )  ->  [_ (
( G  o.  I
)  o.  j )  /  g ]_ X  =  [_ ( G  o.  ( I  o.  j
) )  /  g ]_ X )
31, 2ax-mp 5 . 2  |-  [_ (
( G  o.  I
)  o.  j )  /  g ]_ X  =  [_ ( G  o.  ( I  o.  j
) )  /  g ]_ X
4 simp1 1005 . . 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
) )  /\  j  e.  T  /\  (
j  =/=  (  _I  |`  B )  /\  ( R `  j )  =/=  ( R `  G
)  /\  ( R `  j )  =/=  ( R `  ( G  o.  I ) ) ) ) )  ->  (
( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) ) )
5 simp21 1038 . . 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
) )  /\  j  e.  T  /\  (
j  =/=  (  _I  |`  B )  /\  ( R `  j )  =/=  ( R `  G
)  /\  ( R `  j )  =/=  ( R `  ( G  o.  I ) ) ) ) )  ->  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
) )
6 simp1l 1029 . . . 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
) )  /\  j  e.  T  /\  (
j  =/=  (  _I  |`  B )  /\  ( R `  j )  =/=  ( R `  G
)  /\  ( R `  j )  =/=  ( R `  ( G  o.  I ) ) ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
7 simp22 1039 . . . 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
) )  /\  j  e.  T  /\  (
j  =/=  (  _I  |`  B )  /\  ( R `  j )  =/=  ( R `  G
)  /\  ( R `  j )  =/=  ( R `  ( G  o.  I ) ) ) ) )  ->  G  e.  T )
8 simp31l 1128 . . . 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
) )  /\  j  e.  T  /\  (
j  =/=  (  _I  |`  B )  /\  ( R `  j )  =/=  ( R `  G
)  /\  ( R `  j )  =/=  ( R `  ( G  o.  I ) ) ) ) )  ->  I  e.  T )
9 cdlemk5.h . . . . 5  |-  H  =  ( LHyp `  K
)
10 cdlemk5.t . . . . 5  |-  T  =  ( ( LTrn `  K
) `  W )
119, 10ltrnco 33995 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T  /\  I  e.  T
)  ->  ( G  o.  I )  e.  T
)
126, 7, 8, 11syl3anc 1264 . . 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
) )  /\  j  e.  T  /\  (
j  =/=  (  _I  |`  B )  /\  ( R `  j )  =/=  ( R `  G
)  /\  ( R `  j )  =/=  ( R `  ( G  o.  I ) ) ) ) )  ->  ( G  o.  I )  e.  T )
13 simp23 1040 . . 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
) )  /\  j  e.  T  /\  (
j  =/=  (  _I  |`  B )  /\  ( R `  j )  =/=  ( R `  G
)  /\  ( R `  j )  =/=  ( R `  ( G  o.  I ) ) ) ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
14 simp32 1042 . . 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
) )  /\  j  e.  T  /\  (
j  =/=  (  _I  |`  B )  /\  ( R `  j )  =/=  ( R `  G
)  /\  ( R `  j )  =/=  ( R `  ( G  o.  I ) ) ) ) )  ->  j  e.  T )
15 simp333 1160 . . . 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
) )  /\  j  e.  T  /\  (
j  =/=  (  _I  |`  B )  /\  ( R `  j )  =/=  ( R `  G
)  /\  ( R `  j )  =/=  ( R `  ( G  o.  I ) ) ) ) )  ->  ( R `  j )  =/=  ( R `  ( G  o.  I )
) )
1615necomd 2702 . . 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
) )  /\  j  e.  T  /\  (
j  =/=  (  _I  |`  B )  /\  ( R `  j )  =/=  ( R `  G
)  /\  ( R `  j )  =/=  ( R `  ( G  o.  I ) ) ) ) )  ->  ( R `  ( G  o.  I ) )  =/=  ( R `  j
) )
17 cdlemk5.b . . . 4  |-  B  =  ( Base `  K
)
18 cdlemk5.l . . . 4  |-  .<_  =  ( le `  K )
19 cdlemk5.j . . . 4  |-  .\/  =  ( join `  K )
20 cdlemk5.m . . . 4  |-  ./\  =  ( meet `  K )
21 cdlemk5.a . . . 4  |-  A  =  ( Atoms `  K )
22 cdlemk5.r . . . 4  |-  R  =  ( ( trL `  K
) `  W )
23 cdlemk5.z . . . 4  |-  Z  =  ( ( P  .\/  ( R `  b ) )  ./\  ( ( N `  P )  .\/  ( R `  (
b  o.  `' F
) ) ) )
24 cdlemk5.y . . . 4  |-  Y  =  ( ( P  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
g  o.  `' b ) ) ) )
25 cdlemk5.x . . . 4  |-  X  =  ( iota_ z  e.  T  A. b  e.  T  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 g ) )  ->  ( z `  P )  =  Y ) )
2617, 18, 19, 20, 21, 9, 10, 22, 23, 24, 25cdlemk53 34233 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  ( G  o.  I )  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( j  e.  T  /\  ( R `  ( G  o.  I ) )  =/=  ( R `  j
) ) )  ->  [_ ( ( G  o.  I )  o.  j
)  /  g ]_ X  =  ( [_ ( G  o.  I
)  /  g ]_ X  o.  [_ j  / 
g ]_ X ) )
274, 5, 12, 13, 14, 16, 26syl132anc 1282 . 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
) )  /\  j  e.  T  /\  (
j  =/=  (  _I  |`  B )  /\  ( R `  j )  =/=  ( R `  G
)  /\  ( R `  j )  =/=  ( R `  ( G  o.  I ) ) ) ) )  ->  [_ (
( G  o.  I
)  o.  j )  /  g ]_ X  =  ( [_ ( G  o.  I )  /  g ]_ X  o.  [_ j  /  g ]_ X ) )
28 simp2 1006 . . . 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
) )  /\  j  e.  T  /\  (
j  =/=  (  _I  |`  B )  /\  ( R `  j )  =/=  ( R `  G
)  /\  ( R `  j )  =/=  ( R `  ( G  o.  I ) ) ) ) )  ->  (
( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T )  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )
299, 10ltrnco 33995 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  I  e.  T  /\  j  e.  T
)  ->  ( I  o.  j )  e.  T
)
306, 8, 14, 29syl3anc 1264 . . . 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
) )  /\  j  e.  T  /\  (
j  =/=  (  _I  |`  B )  /\  ( R `  j )  =/=  ( R `  G
)  /\  ( R `  j )  =/=  ( R `  ( G  o.  I ) ) ) ) )  ->  (
I  o.  j )  e.  T )
31 simp31r 1129 . . . . 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
) )  /\  j  e.  T  /\  (
j  =/=  (  _I  |`  B )  /\  ( R `  j )  =/=  ( R `  G
)  /\  ( R `  j )  =/=  ( R `  ( G  o.  I ) ) ) ) )  ->  ( R `  G )  =  ( R `  I ) )
32 simp332 1159 . . . . . . . 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
) )  /\  j  e.  T  /\  (
j  =/=  (  _I  |`  B )  /\  ( R `  j )  =/=  ( R `  G
)  /\  ( R `  j )  =/=  ( R `  ( G  o.  I ) ) ) ) )  ->  ( R `  j )  =/=  ( R `  G
) )
3332, 31neeqtrd 2726 . . . . . . 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
) )  /\  j  e.  T  /\  (
j  =/=  (  _I  |`  B )  /\  ( R `  j )  =/=  ( R `  G
)  /\  ( R `  j )  =/=  ( R `  ( G  o.  I ) ) ) ) )  ->  ( R `  j )  =/=  ( R `  I
) )
3433necomd 2702 . . . . . 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
) )  /\  j  e.  T  /\  (
j  =/=  (  _I  |`  B )  /\  ( R `  j )  =/=  ( R `  G
)  /\  ( R `  j )  =/=  ( R `  ( G  o.  I ) ) ) ) )  ->  ( R `  I )  =/=  ( R `  j
) )
35 simp331 1158 . . . . . 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
) )  /\  j  e.  T  /\  (
j  =/=  (  _I  |`  B )  /\  ( R `  j )  =/=  ( R `  G
)  /\  ( R `  j )  =/=  ( R `  ( G  o.  I ) ) ) ) )  ->  j  =/=  (  _I  |`  B ) )
3617, 9, 10, 22trlcone 34004 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( I  e.  T  /\  j  e.  T )  /\  (
( R `  I
)  =/=  ( R `
 j )  /\  j  =/=  (  _I  |`  B ) ) )  ->  ( R `  I )  =/=  ( R `  (
I  o.  j ) ) )
376, 8, 14, 34, 35, 36syl122anc 1273 . . . . 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
) )  /\  j  e.  T  /\  (
j  =/=  (  _I  |`  B )  /\  ( R `  j )  =/=  ( R `  G
)  /\  ( R `  j )  =/=  ( R `  ( G  o.  I ) ) ) ) )  ->  ( R `  I )  =/=  ( R `  (
I  o.  j ) ) )
3831, 37eqnetrd 2724 . . . 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
) )  /\  j  e.  T  /\  (
j  =/=  (  _I  |`  B )  /\  ( R `  j )  =/=  ( R `  G
)  /\  ( R `  j )  =/=  ( R `  ( G  o.  I ) ) ) ) )  ->  ( R `  G )  =/=  ( R `  (
I  o.  j ) ) )
3917, 18, 19, 20, 21, 9, 10, 22, 23, 24, 25cdlemk53 34233 . . . 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  o.  j )  e.  T  /\  ( R `  G
)  =/=  ( R `
 ( I  o.  j ) ) ) )  ->  [_ ( G  o.  ( I  o.  j ) )  / 
g ]_ X  =  (
[_ G  /  g ]_ X  o.  [_ (
I  o.  j )  /  g ]_ X
) )
404, 28, 30, 38, 39syl112anc 1268 . . 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
) )  /\  j  e.  T  /\  (
j  =/=  (  _I  |`  B )  /\  ( R `  j )  =/=  ( R `  G
)  /\  ( R `  j )  =/=  ( R `  ( G  o.  I ) ) ) ) )  ->  [_ ( G  o.  ( I  o.  j ) )  / 
g ]_ X  =  (
[_ G  /  g ]_ X  o.  [_ (
I  o.  j )  /  g ]_ X
) )
4117, 18, 19, 20, 21, 9, 10, 22, 23, 24, 25cdlemk53 34233 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  I  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( j  e.  T  /\  ( R `  I
)  =/=  ( R `
 j ) ) )  ->  [_ ( I  o.  j )  / 
g ]_ X  =  (
[_ I  /  g ]_ X  o.  [_ j  /  g ]_ X
) )
424, 5, 8, 13, 14, 34, 41syl132anc 1282 . . . . 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
) )  /\  j  e.  T  /\  (
j  =/=  (  _I  |`  B )  /\  ( R `  j )  =/=  ( R `  G
)  /\  ( R `  j )  =/=  ( R `  ( G  o.  I ) ) ) ) )  ->  [_ (
I  o.  j )  /  g ]_ X  =  ( [_ I  /  g ]_ X  o.  [_ j  /  g ]_ X ) )
4342coeq2d 5017 . . . 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
) )  /\  j  e.  T  /\  (
j  =/=  (  _I  |`  B )  /\  ( R `  j )  =/=  ( R `  G
)  /\  ( R `  j )  =/=  ( R `  ( G  o.  I ) ) ) ) )  ->  ( [_ G  /  g ]_ X  o.  [_ (
I  o.  j )  /  g ]_ X
)  =  ( [_ G  /  g ]_ X  o.  ( [_ I  / 
g ]_ X  o.  [_ j  /  g ]_ X
) ) )
44 coass 5374 . . . 4  |-  ( (
[_ G  /  g ]_ X  o.  [_ I  /  g ]_ X
)  o.  [_ j  /  g ]_ X
)  =  ( [_ G  /  g ]_ X  o.  ( [_ I  / 
g ]_ X  o.  [_ j  /  g ]_ X
) )
4543, 44syl6eqr 2488 . . 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
) )  /\  j  e.  T  /\  (
j  =/=  (  _I  |`  B )  /\  ( R `  j )  =/=  ( R `  G
)  /\  ( R `  j )  =/=  ( R `  ( G  o.  I ) ) ) ) )  ->  ( [_ G  /  g ]_ X  o.  [_ (
I  o.  j )  /  g ]_ X
)  =  ( (
[_ G  /  g ]_ X  o.  [_ I  /  g ]_ X
)  o.  [_ j  /  g ]_ X
) )
4640, 45eqtrd 2470 . 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
) )  /\  j  e.  T  /\  (
j  =/=  (  _I  |`  B )  /\  ( R `  j )  =/=  ( R `  G
)  /\  ( R `  j )  =/=  ( R `  ( G  o.  I ) ) ) ) )  ->  [_ ( G  o.  ( I  o.  j ) )  / 
g ]_ X  =  ( ( [_ G  / 
g ]_ X  o.  [_ I  /  g ]_ X
)  o.  [_ j  /  g ]_ X
) )
473, 27, 463eqtr3a 2494 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
) )  /\  j  e.  T  /\  (
j  =/=  (  _I  |`  B )  /\  ( R `  j )  =/=  ( R `  G
)  /\  ( R `  j )  =/=  ( R `  ( G  o.  I ) ) ) ) )  ->  ( [_ ( G  o.  I
)  /  g ]_ X  o.  [_ j  / 
g ]_ X )  =  ( ( [_ G  /  g ]_ X  o.  [_ I  /  g ]_ X )  o.  [_ j  /  g ]_ X
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870    =/= wne 2625   A.wral 2782   [_csb 3401   class class class wbr 4426    _I cid 4764   `'ccnv 4853    |` cres 4856    o. ccom 4858   ` cfv 5601   iota_crio 6266  (class class class)co 6305   Basecbs 15084   lecple 15159   joincjn 16140   meetcmee 16141   Atomscatm 32538   HLchlt 32625   LHypclh 33258   LTrncltrn 33375   trLctrl 33433
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-riotaBAD 32234
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-iin 4305  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-1st 6807  df-2nd 6808  df-undef 7028  df-map 7482  df-preset 16124  df-poset 16142  df-plt 16155  df-lub 16171  df-glb 16172  df-join 16173  df-meet 16174  df-p0 16236  df-p1 16237  df-lat 16243  df-clat 16305  df-oposet 32451  df-ol 32453  df-oml 32454  df-covers 32541  df-ats 32542  df-atl 32573  df-cvlat 32597  df-hlat 32626  df-llines 32772  df-lplanes 32773  df-lvols 32774  df-lines 32775  df-psubsp 32777  df-pmap 32778  df-padd 33070  df-lhyp 33262  df-laut 33263  df-ldil 33378  df-ltrn 33379  df-trl 33434
This theorem is referenced by:  cdlemk55a  34235
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