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Theorem cdlemk23-3 29892
Description: Part of proof of Lemma K of [Crawley] p. 118. Eliminate the  ( R `  C )  =/=  ( R `  D ) requirement from cdlemk22-3 29891. (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
cdlemk23-3  |-  ( ( ( ( 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 `
 C ) )  /\  ( ( 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 cdlemk23-3
StepHypRef Expression
1 simp11 990 . . 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 `
 C ) )  /\  ( ( R `
 x )  =/=  ( R `  D
)  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  G
)  =/=  ( R `
 x ) ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
2 simp121 1092 . . 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 `
 C ) )  /\  ( ( R `
 x )  =/=  ( R `  D
)  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  G
)  =/=  ( R `
 x ) ) ) )  ->  F  e.  T )
3 simp122 1093 . . 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 `
 C ) )  /\  ( ( R `
 x )  =/=  ( R `  D
)  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  G
)  =/=  ( R `
 x ) ) ) )  ->  D  e.  T )
4 simp123 1094 . . . 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 `
 C ) )  /\  ( ( R `
 x )  =/=  ( R `  D
)  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  G
)  =/=  ( R `
 x ) ) ) )  ->  N  e.  T )
5 simp131 1095 . . . 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 `
 C ) )  /\  ( ( R `
 x )  =/=  ( R `  D
)  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  G
)  =/=  ( R `
 x ) ) ) )  ->  G  e.  T )
6 simp133 1097 . . . 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 `
 C ) )  /\  ( ( R `
 x )  =/=  ( R `  D
)  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  G
)  =/=  ( R `
 x ) ) ) )  ->  x  e.  T )
74, 5, 63jca 1137 . . 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 `
 C ) )  /\  ( ( R `
 x )  =/=  ( R `  D
)  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  G
)  =/=  ( R `
 x ) ) ) )  ->  ( N  e.  T  /\  G  e.  T  /\  x  e.  T )
)
8 simp21 993 . . 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 `
 C ) )  /\  ( ( R `
 x )  =/=  ( R `  D
)  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  G
)  =/=  ( R `
 x ) ) ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
9 simp221 1101 . . 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 `
 C ) )  /\  ( ( R `
 x )  =/=  ( R `  D
)  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  G
)  =/=  ( R `
 x ) ) ) )  ->  ( R `  F )  =  ( R `  N ) )
10 simp222 1102 . . . 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 `
 C ) )  /\  ( ( R `
 x )  =/=  ( R `  D
)  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  G
)  =/=  ( R `
 x ) ) ) )  ->  F  =/=  (  _I  |`  B ) )
11 simp223 1103 . . . 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 `
 C ) )  /\  ( ( R `
 x )  =/=  ( R `  D
)  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  G
)  =/=  ( R `
 x ) ) ) )  ->  D  =/=  (  _I  |`  B ) )
12 simp231 1104 . . . 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 `
 C ) )  /\  ( ( R `
 x )  =/=  ( R `  D
)  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  G
)  =/=  ( R `
 x ) ) ) )  ->  G  =/=  (  _I  |`  B ) )
1310, 11, 123jca 1137 . . 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 `
 C ) )  /\  ( ( R `
 x )  =/=  ( R `  D
)  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  G
)  =/=  ( R `
 x ) ) ) )  ->  ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) ) )
14 simp233 1106 . . . 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 `
 C ) )  /\  ( ( R `
 x )  =/=  ( R `  D
)  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  G
)  =/=  ( R `
 x ) ) ) )  ->  x  =/=  (  _I  |`  B ) )
15 simp333 1115 . . . 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 `
 C ) )  /\  ( ( R `
 x )  =/=  ( R `  D
)  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  G
)  =/=  ( R `
 x ) ) ) )  ->  ( R `  G )  =/=  ( R `  x
) )
16 simp332 1114 . . . 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 `
 C ) )  /\  ( ( R `
 x )  =/=  ( R `  D
)  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  G
)  =/=  ( R `
 x ) ) ) )  ->  ( R `  x )  =/=  ( R `  F
) )
1714, 15, 163jca 1137 . . 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 `
 C ) )  /\  ( ( R `
 x )  =/=  ( R `  D
)  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  G
)  =/=  ( R `
 x ) ) ) )  ->  (
x  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  x
)  /\  ( R `  x )  =/=  ( R `  F )
) )
18 simp313 1109 . . . 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 `
 C ) )  /\  ( ( R `
 x )  =/=  ( R `  D
)  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  G
)  =/=  ( R `
 x ) ) ) )  ->  ( R `  D )  =/=  ( R `  F
) )
19 simp32l 1085 . . . 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 `
 C ) )  /\  ( ( R `
 x )  =/=  ( R `  D
)  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  G
)  =/=  ( R `
 x ) ) ) )  ->  ( R `  G )  =/=  ( R `  D
) )
20 simp331 1113 . . . 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 `
 C ) )  /\  ( ( R `
 x )  =/=  ( R `  D
)  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  G
)  =/=  ( R `
 x ) ) ) )  ->  ( R `  x )  =/=  ( R `  D
) )
2118, 19, 203jca 1137 . . 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 `
 C ) )  /\  ( ( R `
 x )  =/=  ( R `  D
)  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  G
)  =/=  ( R `
 x ) ) ) )  ->  (
( R `  D
)  =/=  ( R `
 F )  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  x )  =/=  ( R `  D
) ) )
22 cdlemk3.b . . . 4  |-  B  =  ( Base `  K
)
23 cdlemk3.l . . . 4  |-  .<_  =  ( le `  K )
24 cdlemk3.j . . . 4  |-  .\/  =  ( join `  K )
25 cdlemk3.m . . . 4  |-  ./\  =  ( meet `  K )
26 cdlemk3.a . . . 4  |-  A  =  ( Atoms `  K )
27 cdlemk3.h . . . 4  |-  H  =  ( LHyp `  K
)
28 cdlemk3.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
29 cdlemk3.r . . . 4  |-  R  =  ( ( trL `  K
) `  W )
30 cdlemk3.s . . . 4  |-  S  =  ( f  e.  T  |->  ( iota_ i  e.  T
( i `  P
)  =  ( ( P  .\/  ( R `
 f ) ) 
./\  ( ( N `
 P )  .\/  ( R `  ( f  o.  `' F ) ) ) ) ) )
31 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 ) ) ) ) ) )
3222, 23, 24, 25, 26, 27, 28, 29, 30, 31cdlemk22-3 29891 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  x  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( x  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  x )  /\  ( R `  x
)  =/=  ( R `
 F ) )  /\  ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  x
)  =/=  ( R `
 D ) ) ) )  ->  (
( D Y G ) `  P )  =  ( ( x Y G ) `  P ) )
331, 2, 3, 7, 8, 9, 13, 17, 21, 32syl333anc 1219 . 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 `
 C ) )  /\  ( ( R `
 x )  =/=  ( R `  D
)  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  G
)  =/=  ( R `
 x ) ) ) )  ->  (
( D Y G ) `  P )  =  ( ( x Y G ) `  P ) )
34 simp132 1096 . . 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 `
 C ) )  /\  ( ( R `
 x )  =/=  ( R `  D
)  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  G
)  =/=  ( R `
 x ) ) ) )  ->  C  e.  T )
35 simp232 1105 . . . 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 `
 C ) )  /\  ( ( R `
 x )  =/=  ( R `  D
)  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  G
)  =/=  ( R `
 x ) ) ) )  ->  C  =/=  (  _I  |`  B ) )
3610, 35, 123jca 1137 . . 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 `
 C ) )  /\  ( ( R `
 x )  =/=  ( R `  D
)  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  G
)  =/=  ( R `
 x ) ) ) )  ->  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) ) )
37 simp312 1108 . . . 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 `
 C ) )  /\  ( ( R `
 x )  =/=  ( R `  D
)  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  G
)  =/=  ( R `
 x ) ) ) )  ->  ( R `  C )  =/=  ( R `  F
) )
38 simp311 1107 . . . 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 `
 C ) )  /\  ( ( R `
 x )  =/=  ( R `  D
)  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  G
)  =/=  ( R `
 x ) ) ) )  ->  ( R `  G )  =/=  ( R `  C
) )
39 simp32r 1086 . . . 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 `
 C ) )  /\  ( ( R `
 x )  =/=  ( R `  D
)  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  G
)  =/=  ( R `
 x ) ) ) )  ->  ( R `  x )  =/=  ( R `  C
) )
4037, 38, 393jca 1137 . . 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 `
 C ) )  /\  ( ( R `
 x )  =/=  ( R `  D
)  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  G
)  =/=  ( R `
 x ) ) ) )  ->  (
( R `  C
)  =/=  ( R `
 F )  /\  ( R `  G )  =/=  ( R `  C )  /\  ( R `  x )  =/=  ( R `  C
) ) )
4122, 23, 24, 25, 26, 27, 28, 29, 30, 31cdlemk22-3 29891 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  C  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  x  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( x  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  x )  /\  ( R `  x
)  =/=  ( R `
 F ) )  /\  ( ( R `
 C )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  C )  /\  ( R `  x
)  =/=  ( R `
 C ) ) ) )  ->  (
( C Y G ) `  P )  =  ( ( x Y G ) `  P ) )
421, 2, 34, 7, 8, 9, 36, 17, 40, 41syl333anc 1219 . 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 `
 C ) )  /\  ( ( R `
 x )  =/=  ( R `  D
)  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  G
)  =/=  ( R `
 x ) ) ) )  ->  (
( C Y G ) `  P )  =  ( ( x Y G ) `  P ) )
4333, 42eqtr4d 2288 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 `
 C ) )  /\  ( ( R `
 x )  =/=  ( R `  D
)  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  G
)  =/=  ( R `
 x ) ) ) )  ->  (
( D Y G ) `  P )  =  ( ( C Y G ) `  P ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2412   class class class wbr 3920    e. cmpt 3974    _I cid 4197   `'ccnv 4579    |` cres 4582    o. ccom 4584   ` cfv 4592  (class class class)co 5710    e. cmpt2 5712   iota_crio 6181   Basecbs 13022   lecple 13089   joincjn 13922   meetcmee 13923   Atomscatm 28254   HLchlt 28341   LHypclh 28974   LTrncltrn 29091   trLctrl 29148
This theorem is referenced by:  cdlemk24-3  29893
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-iin 3806  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-iota 6143  df-undef 6182  df-riota 6190  df-map 6660  df-poset 13924  df-plt 13936  df-lub 13952  df-glb 13953  df-join 13954  df-meet 13955  df-p0 13989  df-p1 13990  df-lat 13996  df-clat 14058  df-oposet 28167  df-ol 28169  df-oml 28170  df-covers 28257  df-ats 28258  df-atl 28289  df-cvlat 28313  df-hlat 28342  df-llines 28488  df-lplanes 28489  df-lvols 28490  df-lines 28491  df-psubsp 28493  df-pmap 28494  df-padd 28786  df-lhyp 28978  df-laut 28979  df-ldil 29094  df-ltrn 29095  df-trl 29149
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