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Theorem cdlemk23-3 34381
Description: Part of proof of Lemma K of [Crawley] p. 118. Eliminate the  ( R `  C )  =/=  ( R `  D ) requirement from cdlemk22-3 34380. (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 1035 . . 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 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  e.  T )
3 simp122 1138 . . 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 1139 . . . 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 1140 . . . 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 1142 . . . 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 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 `
 C ) )  /\  ( ( R `
 x )  =/=  ( R `  D
)  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  G
)  =/=  ( R `
 x ) ) ) )  ->  ( N  e.  T  /\  G  e.  T  /\  x  e.  T )
)
8 simp21 1038 . . 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 1146 . . 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 1147 . . . 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 1148 . . . 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 1149 . . . 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 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 `
 C ) )  /\  ( ( R `
 x )  =/=  ( R `  D
)  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  G
)  =/=  ( R `
 x ) ) ) )  ->  ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) ) )
14 simp233 1151 . . . 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 1160 . . . 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 1159 . . . 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 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 `
 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 1154 . . . 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 1130 . . . 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 1158 . . . 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 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 `
 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 34380 . . 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 1296 . 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 1141 . . 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 1150 . . . 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 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 `
 C ) )  /\  ( ( R `
 x )  =/=  ( R `  D
)  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  G
)  =/=  ( R `
 x ) ) ) )  ->  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) ) )
37 simp312 1153 . . . 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 1152 . . . 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 1131 . . . 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 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 `
 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 34380 . . 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 1296 . 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 2460 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872    =/= wne 2594   class class class wbr 4361    |-> cmpt 4420    _I cid 4701   `'ccnv 4790    |` cres 4793    o. ccom 4795   ` cfv 5539   iota_crio 6205  (class class class)co 6244    |-> cmpt2 6246   Basecbs 15059   lecple 15135   joincjn 16127   meetcmee 16128   Atomscatm 32741   HLchlt 32828   LHypclh 33461   LTrncltrn 33578   trLctrl 33636
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2058  ax-ext 2403  ax-rep 4474  ax-sep 4484  ax-nul 4493  ax-pow 4540  ax-pr 4598  ax-un 6536  ax-riotaBAD 32437
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 1658  df-nf 1662  df-sb 1791  df-eu 2275  df-mo 2276  df-clab 2410  df-cleq 2416  df-clel 2419  df-nfc 2553  df-ne 2596  df-nel 2597  df-ral 2714  df-rex 2715  df-reu 2716  df-rmo 2717  df-rab 2718  df-v 3019  df-sbc 3238  df-csb 3334  df-dif 3377  df-un 3379  df-in 3381  df-ss 3388  df-nul 3700  df-if 3850  df-pw 3921  df-sn 3937  df-pr 3939  df-op 3943  df-uni 4158  df-iun 4239  df-iin 4240  df-br 4362  df-opab 4421  df-mpt 4422  df-id 4706  df-xp 4797  df-rel 4798  df-cnv 4799  df-co 4800  df-dm 4801  df-rn 4802  df-res 4803  df-ima 4804  df-iota 5503  df-fun 5541  df-fn 5542  df-f 5543  df-f1 5544  df-fo 5545  df-f1o 5546  df-fv 5547  df-riota 6206  df-ov 6247  df-oprab 6248  df-mpt2 6249  df-1st 6746  df-2nd 6747  df-undef 6970  df-map 7424  df-preset 16111  df-poset 16129  df-plt 16142  df-lub 16158  df-glb 16159  df-join 16160  df-meet 16161  df-p0 16223  df-p1 16224  df-lat 16230  df-clat 16292  df-oposet 32654  df-ol 32656  df-oml 32657  df-covers 32744  df-ats 32745  df-atl 32776  df-cvlat 32800  df-hlat 32829  df-llines 32975  df-lplanes 32976  df-lvols 32977  df-lines 32978  df-psubsp 32980  df-pmap 32981  df-padd 33273  df-lhyp 33465  df-laut 33466  df-ldil 33581  df-ltrn 33582  df-trl 33637
This theorem is referenced by:  cdlemk24-3  34382
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