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Theorem cdlemk25-3 34209
Description: Part of proof of Lemma K of [Crawley] p. 118. Eliminate the  ( R `  C )  =  ( R `  D ) requirement from cdlemk24-3 34208. (Contributed by NM, 7-Jul-2013.)
Hypotheses
Ref Expression
cdlemk3.b  |-  B  =  ( Base `  K
)
cdlemk3.l  |-  .<_  =  ( le `  K )
cdlemk3.j  |-  .\/  =  ( join `  K )
cdlemk3.m  |-  ./\  =  ( meet `  K )
cdlemk3.a  |-  A  =  ( Atoms `  K )
cdlemk3.h  |-  H  =  ( LHyp `  K
)
cdlemk3.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemk3.r  |-  R  =  ( ( trL `  K
) `  W )
cdlemk3.s  |-  S  =  ( f  e.  T  |->  ( iota_ i  e.  T  ( i `  P
)  =  ( ( P  .\/  ( R `
 f ) ) 
./\  ( ( N `
 P )  .\/  ( R `  ( f  o.  `' F ) ) ) ) ) )
cdlemk3.u1  |-  Y  =  ( d  e.  T ,  e  e.  T  |->  ( iota_ j  e.  T  ( j `  P
)  =  ( ( P  .\/  ( R `
 e ) ) 
./\  ( ( ( S `  d ) `
 P )  .\/  ( R `  ( e  o.  `' d ) ) ) ) ) )
Assertion
Ref Expression
cdlemk25-3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T )  /\  ( G  e.  T  /\  C  e.  T  /\  x  e.  T
) )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( ( R `
 F )  =  ( R `  N
)  /\  F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( G  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B )  /\  x  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  G )  =/=  ( R `  C )  /\  ( R `  C
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  F ) )  /\  ( R `  G )  =/=  ( R `  D )  /\  (
( R `  x
)  =/=  ( R `
 D )  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  G )  =/=  ( R `  x
) ) ) )  ->  ( ( D Y G ) `  P )  =  ( ( C Y G ) `  P ) )
Distinct variable groups:    e, d,
f, i,  ./\    .<_ , i    .\/ , d, e, f, i    A, i    j, d, D, e, f, i    f, F, i    G, d, e, j   
i, H    i, K    f, N, i    P, d, e, f, i    R, d, e, f, i    T, d, e, f, i    W, d, e, f, i    ./\ , j    .<_ , j    .\/ , j    A, j    j, F    j, H    j, K    j, N    P, j    R, j    S, d, e, j    T, j    j, W    F, d, e    .<_ , e    C, d, e, f, i, j   
f, G, i    x, d, e, f, i, j
Allowed substitution hints:    A( x, e, f, d)    B( x, e, f, i, j, d)    C( x)    D( x)    P( x)    R( x)    S( x, f, i)    T( x)    F( x)    G( x)    H( x, e, f, d)    .\/ ( x)    K( x, e, f, d)    .<_ ( x, f, d)    ./\ ( x)    N( x, e, d)    W( x)    Y( x, e, f, i, j, d)

Proof of Theorem cdlemk25-3
StepHypRef Expression
1 simpl1 1008 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T )  /\  ( G  e.  T  /\  C  e.  T  /\  x  e.  T
) )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( ( R `
 F )  =  ( R `  N
)  /\  F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( G  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B )  /\  x  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  G )  =/=  ( R `  C )  /\  ( R `  C
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  F ) )  /\  ( R `  G )  =/=  ( R `  D )  /\  (
( R `  x
)  =/=  ( R `
 D )  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  G )  =/=  ( R `  x
) ) ) )  /\  ( R `  C )  =  ( R `  D ) )  ->  ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T )  /\  ( G  e.  T  /\  C  e.  T  /\  x  e.  T
) ) )
2 simpl2 1009 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T )  /\  ( G  e.  T  /\  C  e.  T  /\  x  e.  T
) )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( ( R `
 F )  =  ( R `  N
)  /\  F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( G  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B )  /\  x  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  G )  =/=  ( R `  C )  /\  ( R `  C
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  F ) )  /\  ( R `  G )  =/=  ( R `  D )  /\  (
( R `  x
)  =/=  ( R `
 D )  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  G )  =/=  ( R `  x
) ) ) )  /\  ( R `  C )  =  ( R `  D ) )  ->  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( ( R `
 F )  =  ( R `  N
)  /\  F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( G  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B )  /\  x  =/=  (  _I  |`  B ) ) ) )
3 simpl31 1086 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T )  /\  ( G  e.  T  /\  C  e.  T  /\  x  e.  T
) )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( ( R `
 F )  =  ( R `  N
)  /\  F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( G  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B )  /\  x  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  G )  =/=  ( R `  C )  /\  ( R `  C
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  F ) )  /\  ( R `  G )  =/=  ( R `  D )  /\  (
( R `  x
)  =/=  ( R `
 D )  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  G )  =/=  ( R `  x
) ) ) )  /\  ( R `  C )  =  ( R `  D ) )  ->  ( ( R `  G )  =/=  ( R `  C
)  /\  ( R `  C )  =/=  ( R `  F )  /\  ( R `  D
)  =/=  ( R `
 F ) ) )
4 simpl32 1087 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T )  /\  ( G  e.  T  /\  C  e.  T  /\  x  e.  T
) )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( ( R `
 F )  =  ( R `  N
)  /\  F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( G  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B )  /\  x  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  G )  =/=  ( R `  C )  /\  ( R `  C
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  F ) )  /\  ( R `  G )  =/=  ( R `  D )  /\  (
( R `  x
)  =/=  ( R `
 D )  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  G )  =/=  ( R `  x
) ) ) )  /\  ( R `  C )  =  ( R `  D ) )  ->  ( R `  G )  =/=  ( R `  D )
)
5 simpr 462 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T )  /\  ( G  e.  T  /\  C  e.  T  /\  x  e.  T
) )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( ( R `
 F )  =  ( R `  N
)  /\  F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( G  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B )  /\  x  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  G )  =/=  ( R `  C )  /\  ( R `  C
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  F ) )  /\  ( R `  G )  =/=  ( R `  D )  /\  (
( R `  x
)  =/=  ( R `
 D )  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  G )  =/=  ( R `  x
) ) ) )  /\  ( R `  C )  =  ( R `  D ) )  ->  ( R `  C )  =  ( R `  D ) )
64, 5jca 534 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T )  /\  ( G  e.  T  /\  C  e.  T  /\  x  e.  T
) )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( ( R `
 F )  =  ( R `  N
)  /\  F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( G  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B )  /\  x  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  G )  =/=  ( R `  C )  /\  ( R `  C
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  F ) )  /\  ( R `  G )  =/=  ( R `  D )  /\  (
( R `  x
)  =/=  ( R `
 D )  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  G )  =/=  ( R `  x
) ) ) )  /\  ( R `  C )  =  ( R `  D ) )  ->  ( ( R `  G )  =/=  ( R `  D
)  /\  ( R `  C )  =  ( R `  D ) ) )
7 simpl33 1088 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T )  /\  ( G  e.  T  /\  C  e.  T  /\  x  e.  T
) )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( ( R `
 F )  =  ( R `  N
)  /\  F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( G  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B )  /\  x  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  G )  =/=  ( R `  C )  /\  ( R `  C
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  F ) )  /\  ( R `  G )  =/=  ( R `  D )  /\  (
( R `  x
)  =/=  ( R `
 D )  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  G )  =/=  ( R `  x
) ) ) )  /\  ( R `  C )  =  ( R `  D ) )  ->  ( ( R `  x )  =/=  ( R `  D
)  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  G
)  =/=  ( R `
 x ) ) )
8 cdlemk3.b . . . 4  |-  B  =  ( Base `  K
)
9 cdlemk3.l . . . 4  |-  .<_  =  ( le `  K )
10 cdlemk3.j . . . 4  |-  .\/  =  ( join `  K )
11 cdlemk3.m . . . 4  |-  ./\  =  ( meet `  K )
12 cdlemk3.a . . . 4  |-  A  =  ( Atoms `  K )
13 cdlemk3.h . . . 4  |-  H  =  ( LHyp `  K
)
14 cdlemk3.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
15 cdlemk3.r . . . 4  |-  R  =  ( ( trL `  K
) `  W )
16 cdlemk3.s . . . 4  |-  S  =  ( f  e.  T  |->  ( iota_ i  e.  T  ( i `  P
)  =  ( ( P  .\/  ( R `
 f ) ) 
./\  ( ( N `
 P )  .\/  ( R `  ( f  o.  `' F ) ) ) ) ) )
17 cdlemk3.u1 . . . 4  |-  Y  =  ( d  e.  T ,  e  e.  T  |->  ( iota_ j  e.  T  ( j `  P
)  =  ( ( P  .\/  ( R `
 e ) ) 
./\  ( ( ( S `  d ) `
 P )  .\/  ( R `  ( e  o.  `' d ) ) ) ) ) )
188, 9, 10, 11, 12, 13, 14, 15, 16, 17cdlemk24-3 34208 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T )  /\  ( G  e.  T  /\  C  e.  T  /\  x  e.  T
) )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( ( R `
 F )  =  ( R `  N
)  /\  F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( G  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B )  /\  x  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  G )  =/=  ( R `  C )  /\  ( R `  C
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  F ) )  /\  ( ( R `  G )  =/=  ( R `  D )  /\  ( R `  C
)  =  ( R `
 D ) )  /\  ( ( R `
 x )  =/=  ( R `  D
)  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  G
)  =/=  ( R `
 x ) ) ) )  ->  (
( D Y G ) `  P )  =  ( ( C Y G ) `  P ) )
191, 2, 3, 6, 7, 18syl113anc 1276 . 2  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T )  /\  ( G  e.  T  /\  C  e.  T  /\  x  e.  T
) )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( ( R `
 F )  =  ( R `  N
)  /\  F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( G  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B )  /\  x  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  G )  =/=  ( R `  C )  /\  ( R `  C
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  F ) )  /\  ( R `  G )  =/=  ( R `  D )  /\  (
( R `  x
)  =/=  ( R `
 D )  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  G )  =/=  ( R `  x
) ) ) )  /\  ( R `  C )  =  ( R `  D ) )  ->  ( ( D Y G ) `  P )  =  ( ( C Y G ) `  P ) )
20 simp11 1035 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T )  /\  ( G  e.  T  /\  C  e.  T  /\  x  e.  T
) )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( ( R `
 F )  =  ( R `  N
)  /\  F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( G  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B )  /\  x  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  G )  =/=  ( R `  C )  /\  ( R `  C
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  F ) )  /\  ( R `  G )  =/=  ( R `  D )  /\  (
( R `  x
)  =/=  ( R `
 D )  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  G )  =/=  ( R `  x
) ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
21 simp121 1137 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T )  /\  ( G  e.  T  /\  C  e.  T  /\  x  e.  T
) )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( ( R `
 F )  =  ( R `  N
)  /\  F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( G  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B )  /\  x  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  G )  =/=  ( R `  C )  /\  ( R `  C
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  F ) )  /\  ( R `  G )  =/=  ( R `  D )  /\  (
( R `  x
)  =/=  ( R `
 D )  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  G )  =/=  ( R `  x
) ) ) )  ->  F  e.  T
)
22 simp122 1138 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T )  /\  ( G  e.  T  /\  C  e.  T  /\  x  e.  T
) )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( ( R `
 F )  =  ( R `  N
)  /\  F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( G  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B )  /\  x  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  G )  =/=  ( R `  C )  /\  ( R `  C
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  F ) )  /\  ( R `  G )  =/=  ( R `  D )  /\  (
( R `  x
)  =/=  ( R `
 D )  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  G )  =/=  ( R `  x
) ) ) )  ->  D  e.  T
)
2320, 21, 223jca 1185 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T )  /\  ( G  e.  T  /\  C  e.  T  /\  x  e.  T
) )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( ( R `
 F )  =  ( R `  N
)  /\  F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( G  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B )  /\  x  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  G )  =/=  ( R `  C )  /\  ( R `  C
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  F ) )  /\  ( R `  G )  =/=  ( R `  D )  /\  (
( R `  x
)  =/=  ( R `
 D )  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  G )  =/=  ( R `  x
) ) ) )  ->  ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T ) )
2423adantr 466 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T )  /\  ( G  e.  T  /\  C  e.  T  /\  x  e.  T
) )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( ( R `
 F )  =  ( R `  N
)  /\  F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( G  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B )  /\  x  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  G )  =/=  ( R `  C )  /\  ( R `  C
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  F ) )  /\  ( R `  G )  =/=  ( R `  D )  /\  (
( R `  x
)  =/=  ( R `
 D )  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  G )  =/=  ( R `  x
) ) ) )  /\  ( R `  C )  =/=  ( R `  D )
)  ->  ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T ) )
25 simp123 1139 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T )  /\  ( G  e.  T  /\  C  e.  T  /\  x  e.  T
) )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( ( R `
 F )  =  ( R `  N
)  /\  F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( G  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B )  /\  x  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  G )  =/=  ( R `  C )  /\  ( R `  C
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  F ) )  /\  ( R `  G )  =/=  ( R `  D )  /\  (
( R `  x
)  =/=  ( R `
 D )  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  G )  =/=  ( R `  x
) ) ) )  ->  N  e.  T
)
26 simp131 1140 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T )  /\  ( G  e.  T  /\  C  e.  T  /\  x  e.  T
) )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( ( R `
 F )  =  ( R `  N
)  /\  F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( G  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B )  /\  x  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  G )  =/=  ( R `  C )  /\  ( R `  C
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  F ) )  /\  ( R `  G )  =/=  ( R `  D )  /\  (
( R `  x
)  =/=  ( R `
 D )  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  G )  =/=  ( R `  x
) ) ) )  ->  G  e.  T
)
27 simp132 1141 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T )  /\  ( G  e.  T  /\  C  e.  T  /\  x  e.  T
) )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( ( R `
 F )  =  ( R `  N
)  /\  F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( G  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B )  /\  x  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  G )  =/=  ( R `  C )  /\  ( R `  C
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  F ) )  /\  ( R `  G )  =/=  ( R `  D )  /\  (
( R `  x
)  =/=  ( R `
 D )  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  G )  =/=  ( R `  x
) ) ) )  ->  C  e.  T
)
2825, 26, 273jca 1185 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T )  /\  ( G  e.  T  /\  C  e.  T  /\  x  e.  T
) )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( ( R `
 F )  =  ( R `  N
)  /\  F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( G  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B )  /\  x  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  G )  =/=  ( R `  C )  /\  ( R `  C
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  F ) )  /\  ( R `  G )  =/=  ( R `  D )  /\  (
( R `  x
)  =/=  ( R `
 D )  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  G )  =/=  ( R `  x
) ) ) )  ->  ( N  e.  T  /\  G  e.  T  /\  C  e.  T ) )
29 simp21 1038 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T )  /\  ( G  e.  T  /\  C  e.  T  /\  x  e.  T
) )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( ( R `
 F )  =  ( R `  N
)  /\  F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( G  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B )  /\  x  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  G )  =/=  ( R `  C )  /\  ( R `  C
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  F ) )  /\  ( R `  G )  =/=  ( R `  D )  /\  (
( R `  x
)  =/=  ( R `
 D )  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  G )  =/=  ( R `  x
) ) ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
30 simp221 1146 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T )  /\  ( G  e.  T  /\  C  e.  T  /\  x  e.  T
) )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( ( R `
 F )  =  ( R `  N
)  /\  F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( G  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B )  /\  x  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  G )  =/=  ( R `  C )  /\  ( R `  C
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  F ) )  /\  ( R `  G )  =/=  ( R `  D )  /\  (
( R `  x
)  =/=  ( R `
 D )  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  G )  =/=  ( R `  x
) ) ) )  ->  ( R `  F )  =  ( R `  N ) )
3128, 29, 303jca 1185 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T )  /\  ( G  e.  T  /\  C  e.  T  /\  x  e.  T
) )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( ( R `
 F )  =  ( R `  N
)  /\  F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( G  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B )  /\  x  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  G )  =/=  ( R `  C )  /\  ( R `  C
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  F ) )  /\  ( R `  G )  =/=  ( R `  D )  /\  (
( R `  x
)  =/=  ( R `
 D )  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  G )  =/=  ( R `  x
) ) ) )  ->  ( ( N  e.  T  /\  G  e.  T  /\  C  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) ) )
3231adantr 466 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T )  /\  ( G  e.  T  /\  C  e.  T  /\  x  e.  T
) )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( ( R `
 F )  =  ( R `  N
)  /\  F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( G  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B )  /\  x  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  G )  =/=  ( R `  C )  /\  ( R `  C
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  F ) )  /\  ( R `  G )  =/=  ( R `  D )  /\  (
( R `  x
)  =/=  ( R `
 D )  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  G )  =/=  ( R `  x
) ) ) )  /\  ( R `  C )  =/=  ( R `  D )
)  ->  ( ( N  e.  T  /\  G  e.  T  /\  C  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) ) )
33 simp222 1147 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T )  /\  ( G  e.  T  /\  C  e.  T  /\  x  e.  T
) )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( ( R `
 F )  =  ( R `  N
)  /\  F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( G  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B )  /\  x  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  G )  =/=  ( R `  C )  /\  ( R `  C
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  F ) )  /\  ( R `  G )  =/=  ( R `  D )  /\  (
( R `  x
)  =/=  ( R `
 D )  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  G )  =/=  ( R `  x
) ) ) )  ->  F  =/=  (  _I  |`  B ) )
34 simp223 1148 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T )  /\  ( G  e.  T  /\  C  e.  T  /\  x  e.  T
) )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( ( R `
 F )  =  ( R `  N
)  /\  F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( G  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B )  /\  x  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  G )  =/=  ( R `  C )  /\  ( R `  C
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  F ) )  /\  ( R `  G )  =/=  ( R `  D )  /\  (
( R `  x
)  =/=  ( R `
 D )  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  G )  =/=  ( R `  x
) ) ) )  ->  D  =/=  (  _I  |`  B ) )
35 simp231 1149 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T )  /\  ( G  e.  T  /\  C  e.  T  /\  x  e.  T
) )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( ( R `
 F )  =  ( R `  N
)  /\  F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( G  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B )  /\  x  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  G )  =/=  ( R `  C )  /\  ( R `  C
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  F ) )  /\  ( R `  G )  =/=  ( R `  D )  /\  (
( R `  x
)  =/=  ( R `
 D )  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  G )  =/=  ( R `  x
) ) ) )  ->  G  =/=  (  _I  |`  B ) )
3633, 34, 353jca 1185 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T )  /\  ( G  e.  T  /\  C  e.  T  /\  x  e.  T
) )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( ( R `
 F )  =  ( R `  N
)  /\  F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( G  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B )  /\  x  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  G )  =/=  ( R `  C )  /\  ( R `  C
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  F ) )  /\  ( R `  G )  =/=  ( R `  D )  /\  (
( R `  x
)  =/=  ( R `
 D )  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  G )  =/=  ( R `  x
) ) ) )  ->  ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) ) )
3736adantr 466 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T )  /\  ( G  e.  T  /\  C  e.  T  /\  x  e.  T
) )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( ( R `
 F )  =  ( R `  N
)  /\  F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( G  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B )  /\  x  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  G )  =/=  ( R `  C )  /\  ( R `  C
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  F ) )  /\  ( R `  G )  =/=  ( R `  D )  /\  (
( R `  x
)  =/=  ( R `
 D )  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  G )  =/=  ( R `  x
) ) ) )  /\  ( R `  C )  =/=  ( R `  D )
)  ->  ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) ) )
38 simp232 1150 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T )  /\  ( G  e.  T  /\  C  e.  T  /\  x  e.  T
) )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( ( R `
 F )  =  ( R `  N
)  /\  F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( G  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B )  /\  x  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  G )  =/=  ( R `  C )  /\  ( R `  C
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  F ) )  /\  ( R `  G )  =/=  ( R `  D )  /\  (
( R `  x
)  =/=  ( R `
 D )  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  G )  =/=  ( R `  x
) ) ) )  ->  C  =/=  (  _I  |`  B ) )
39 simp311 1152 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T )  /\  ( G  e.  T  /\  C  e.  T  /\  x  e.  T
) )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( ( R `
 F )  =  ( R `  N
)  /\  F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( G  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B )  /\  x  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  G )  =/=  ( R `  C )  /\  ( R `  C
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  F ) )  /\  ( R `  G )  =/=  ( R `  D )  /\  (
( R `  x
)  =/=  ( R `
 D )  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  G )  =/=  ( R `  x
) ) ) )  ->  ( R `  G )  =/=  ( R `  C )
)
40 simp312 1153 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T )  /\  ( G  e.  T  /\  C  e.  T  /\  x  e.  T
) )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( ( R `
 F )  =  ( R `  N
)  /\  F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( G  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B )  /\  x  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  G )  =/=  ( R `  C )  /\  ( R `  C
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  F ) )  /\  ( R `  G )  =/=  ( R `  D )  /\  (
( R `  x
)  =/=  ( R `
 D )  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  G )  =/=  ( R `  x
) ) ) )  ->  ( R `  C )  =/=  ( R `  F )
)
4138, 39, 403jca 1185 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T )  /\  ( G  e.  T  /\  C  e.  T  /\  x  e.  T
) )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( ( R `
 F )  =  ( R `  N
)  /\  F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( G  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B )  /\  x  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  G )  =/=  ( R `  C )  /\  ( R `  C
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  F ) )  /\  ( R `  G )  =/=  ( R `  D )  /\  (
( R `  x
)  =/=  ( R `
 D )  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  G )  =/=  ( R `  x
) ) ) )  ->  ( C  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  C )  /\  ( R `  C
)  =/=  ( R `
 F ) ) )
4241adantr 466 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T )  /\  ( G  e.  T  /\  C  e.  T  /\  x  e.  T
) )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( ( R `
 F )  =  ( R `  N
)  /\  F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( G  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B )  /\  x  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  G )  =/=  ( R `  C )  /\  ( R `  C
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  F ) )  /\  ( R `  G )  =/=  ( R `  D )  /\  (
( R `  x
)  =/=  ( R `
 D )  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  G )  =/=  ( R `  x
) ) ) )  /\  ( R `  C )  =/=  ( R `  D )
)  ->  ( C  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  C )  /\  ( R `  C
)  =/=  ( R `
 F ) ) )
43 simp313 1154 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T )  /\  ( G  e.  T  /\  C  e.  T  /\  x  e.  T
) )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( ( R `
 F )  =  ( R `  N
)  /\  F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( G  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B )  /\  x  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  G )  =/=  ( R `  C )  /\  ( R `  C
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  F ) )  /\  ( R `  G )  =/=  ( R `  D )  /\  (
( R `  x
)  =/=  ( R `
 D )  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  G )  =/=  ( R `  x
) ) ) )  ->  ( R `  D )  =/=  ( R `  F )
)
4443adantr 466 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T )  /\  ( G  e.  T  /\  C  e.  T  /\  x  e.  T
) )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( ( R `
 F )  =  ( R `  N
)  /\  F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( G  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B )  /\  x  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  G )  =/=  ( R `  C )  /\  ( R `  C
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  F ) )  /\  ( R `  G )  =/=  ( R `  D )  /\  (
( R `  x
)  =/=  ( R `
 D )  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  G )  =/=  ( R `  x
) ) ) )  /\  ( R `  C )  =/=  ( R `  D )
)  ->  ( R `  D )  =/=  ( R `  F )
)
45 simpl32 1087 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T )  /\  ( G  e.  T  /\  C  e.  T  /\  x  e.  T
) )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( ( R `
 F )  =  ( R `  N
)  /\  F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( G  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B )  /\  x  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  G )  =/=  ( R `  C )  /\  ( R `  C
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  F ) )  /\  ( R `  G )  =/=  ( R `  D )  /\  (
( R `  x
)  =/=  ( R `
 D )  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  G )  =/=  ( R `  x
) ) ) )  /\  ( R `  C )  =/=  ( R `  D )
)  ->  ( R `  G )  =/=  ( R `  D )
)
46 simpr 462 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T )  /\  ( G  e.  T  /\  C  e.  T  /\  x  e.  T
) )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( ( R `
 F )  =  ( R `  N
)  /\  F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( G  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B )  /\  x  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  G )  =/=  ( R `  C )  /\  ( R `  C
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  F ) )  /\  ( R `  G )  =/=  ( R `  D )  /\  (
( R `  x
)  =/=  ( R `
 D )  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  G )  =/=  ( R `  x
) ) ) )  /\  ( R `  C )  =/=  ( R `  D )
)  ->  ( R `  C )  =/=  ( R `  D )
)
4744, 45, 463jca 1185 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T )  /\  ( G  e.  T  /\  C  e.  T  /\  x  e.  T
) )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( ( R `
 F )  =  ( R `  N
)  /\  F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( G  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B )  /\  x  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  G )  =/=  ( R `  C )  /\  ( R `  C
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  F ) )  /\  ( R `  G )  =/=  ( R `  D )  /\  (
( R `  x
)  =/=  ( R `
 D )  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  G )  =/=  ( R `  x
) ) ) )  /\  ( R `  C )  =/=  ( R `  D )
)  ->  ( ( R `  D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  C
)  =/=  ( R `
 D ) ) )
488, 9, 10, 11, 12, 13, 14, 15, 16, 17cdlemk22-3 34206 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  C  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( C  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  C )  /\  ( R `  C
)  =/=  ( R `
 F ) )  /\  ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  C
)  =/=  ( R `
 D ) ) ) )  ->  (
( D Y G ) `  P )  =  ( ( C Y G ) `  P ) )
4924, 32, 37, 42, 47, 48syl113anc 1276 . 2  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T )  /\  ( G  e.  T  /\  C  e.  T  /\  x  e.  T
) )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( ( R `
 F )  =  ( R `  N
)  /\  F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( G  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B )  /\  x  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  G )  =/=  ( R `  C )  /\  ( R `  C
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  F ) )  /\  ( R `  G )  =/=  ( R `  D )  /\  (
( R `  x
)  =/=  ( R `
 D )  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  G )  =/=  ( R `  x
) ) ) )  /\  ( R `  C )  =/=  ( R `  D )
)  ->  ( ( D Y G ) `  P )  =  ( ( C Y G ) `  P ) )
5019, 49pm2.61dane 2740 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T )  /\  ( G  e.  T  /\  C  e.  T  /\  x  e.  T
) )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( ( R `
 F )  =  ( R `  N
)  /\  F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( G  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B )  /\  x  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  G )  =/=  ( R `  C )  /\  ( R `  C
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  F ) )  /\  ( R `  G )  =/=  ( R `  D )  /\  (
( R `  x
)  =/=  ( R `
 D )  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  G )  =/=  ( R `  x
) ) ) )  ->  ( ( D Y G ) `  P )  =  ( ( C Y G ) `  P ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1867    =/= wne 2616   class class class wbr 4417    |-> cmpt 4475    _I cid 4755   `'ccnv 4844    |` cres 4847    o. ccom 4849   ` cfv 5592   iota_crio 6257  (class class class)co 6296    |-> cmpt2 6298   Basecbs 15073   lecple 15149   joincjn 16133   meetcmee 16134   Atomscatm 32567   HLchlt 32654   LHypclh 33287   LTrncltrn 33404   trLctrl 33462
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 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-riotaBAD 32263
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-iun 4295  df-iin 4296  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6798  df-2nd 6799  df-undef 7019  df-map 7473  df-preset 16117  df-poset 16135  df-plt 16148  df-lub 16164  df-glb 16165  df-join 16166  df-meet 16167  df-p0 16229  df-p1 16230  df-lat 16236  df-clat 16298  df-oposet 32480  df-ol 32482  df-oml 32483  df-covers 32570  df-ats 32571  df-atl 32602  df-cvlat 32626  df-hlat 32655  df-llines 32801  df-lplanes 32802  df-lvols 32803  df-lines 32804  df-psubsp 32806  df-pmap 32807  df-padd 33099  df-lhyp 33291  df-laut 33292  df-ldil 33407  df-ltrn 33408  df-trl 33463
This theorem is referenced by:  cdlemk26-3  34211
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