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Theorem cdlemk54 31594
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 5379 . . 3  |-  ( ( G  o.  I )  o.  j )  =  ( G  o.  (
I  o.  j ) )
2 csbeq1 3246 . . 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 8 . 2  |-  [_ (
( G  o.  I
)  o.  j )  /  g ]_ X  =  [_ ( G  o.  ( I  o.  j
) )  /  g ]_ X
4 simp1 957 . . 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 990 . . 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 981 . . . 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 991 . . . 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 1080 . . . 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 31355 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T  /\  I  e.  T
)  ->  ( G  o.  I )  e.  T
)
126, 7, 8, 11syl3anc 1184 . . 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 992 . . 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 994 . . 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 1112 . . . 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 2681 . . 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 31593 . . 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 1202 . 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 958 . . . 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 31355 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  I  e.  T  /\  j  e.  T
)  ->  ( I  o.  j )  e.  T
)
306, 8, 14, 29syl3anc 1184 . . . 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 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
) )  /\  j  e.  T  /\  (
j  =/=  (  _I  |`  B )  /\  ( R `  j )  =/=  ( R `  G
)  /\  ( R `  j )  =/=  ( R `  ( G  o.  I ) ) ) ) )  ->  ( R `  G )  =  ( R `  I ) )
32 simp332 1111 . . . . . . . 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 2620 . . . . . . 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 2681 . . . . . 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 1110 . . . . . 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 31364 . . . . . 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 1193 . . . . 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 2616 . . . 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 31593 . . . 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 1188 . . 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 31593 . . . . . 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 1202 . . . . 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 5026 . . . 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 5379 . . . 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 2485 . . 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 2467 . 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 2491 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 set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   [_csb 3243   class class class wbr 4204    _I cid 4485   `'ccnv 4868    |` cres 4871    o. ccom 4873   ` cfv 5445  (class class class)co 6072   iota_crio 6533   Basecbs 13457   lecple 13524   joincjn 14389   meetcmee 14390   Atomscatm 29900   HLchlt 29987   LHypclh 30620   LTrncltrn 30737   trLctrl 30794
This theorem is referenced by:  cdlemk55a  31595
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-1st 6340  df-2nd 6341  df-undef 6534  df-riota 6540  df-map 7011  df-poset 14391  df-plt 14403  df-lub 14419  df-glb 14420  df-join 14421  df-meet 14422  df-p0 14456  df-p1 14457  df-lat 14463  df-clat 14525  df-oposet 29813  df-ol 29815  df-oml 29816  df-covers 29903  df-ats 29904  df-atl 29935  df-cvlat 29959  df-hlat 29988  df-llines 30134  df-lplanes 30135  df-lvols 30136  df-lines 30137  df-psubsp 30139  df-pmap 30140  df-padd 30432  df-lhyp 30624  df-laut 30625  df-ldil 30740  df-ltrn 30741  df-trl 30795
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