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Theorem cdlemk23-3 34513
Description: Part of proof of Lemma K of [Crawley] p. 118. Eliminate the  ( R `  C )  =/=  ( R `  D ) requirement from cdlemk22-3 34512. (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 1044 . . 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 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 ) ) ) )  ->  F  e.  T )
3 simp122 1147 . . 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 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 ) ) ) )  ->  N  e.  T )
5 simp131 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  e.  T )
6 simp133 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  e.  T )
74, 5, 63jca 1194 . . 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 1047 . . 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 1155 . . 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 1156 . . . 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 1157 . . . 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 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 ) ) ) )  ->  G  =/=  (  _I  |`  B ) )
1310, 11, 123jca 1194 . . 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 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 ) ) ) )  ->  x  =/=  (  _I  |`  B ) )
15 simp333 1169 . . . 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 1168 . . . 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 1194 . . 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 1163 . . . 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 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 ) ) ) )  ->  ( R `  G )  =/=  ( R `  D
) )
20 simp331 1167 . . . 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 1194 . . 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 34512 . . 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 1308 . 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 1150 . . 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 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 ) ) ) )  ->  C  =/=  (  _I  |`  B ) )
3610, 35, 123jca 1194 . . 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 1162 . . . 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 1161 . . . 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 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 ) ) ) )  ->  ( R `  x )  =/=  ( R `  C
) )
4037, 38, 393jca 1194 . . 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 34512 . . 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 1308 . 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 2498 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 375    /\ w3a 991    = wceq 1454    e. wcel 1897    =/= wne 2632   class class class wbr 4415    |-> cmpt 4474    _I cid 4762   `'ccnv 4851    |` cres 4854    o. ccom 4856   ` cfv 5600   iota_crio 6275  (class class class)co 6314    |-> cmpt2 6316   Basecbs 15169   lecple 15245   joincjn 16237   meetcmee 16238   Atomscatm 32873   HLchlt 32960   LHypclh 33593   LTrncltrn 33710   trLctrl 33768
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-rep 4528  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609  ax-riotaBAD 32569
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-nel 2635  df-ral 2753  df-rex 2754  df-reu 2755  df-rmo 2756  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-nul 3743  df-if 3893  df-pw 3964  df-sn 3980  df-pr 3982  df-op 3986  df-uni 4212  df-iun 4293  df-iin 4294  df-br 4416  df-opab 4475  df-mpt 4476  df-id 4767  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-riota 6276  df-ov 6317  df-oprab 6318  df-mpt2 6319  df-1st 6819  df-2nd 6820  df-undef 7045  df-map 7499  df-preset 16221  df-poset 16239  df-plt 16252  df-lub 16268  df-glb 16269  df-join 16270  df-meet 16271  df-p0 16333  df-p1 16334  df-lat 16340  df-clat 16402  df-oposet 32786  df-ol 32788  df-oml 32789  df-covers 32876  df-ats 32877  df-atl 32908  df-cvlat 32932  df-hlat 32961  df-llines 33107  df-lplanes 33108  df-lvols 33109  df-lines 33110  df-psubsp 33112  df-pmap 33113  df-padd 33405  df-lhyp 33597  df-laut 33598  df-ldil 33713  df-ltrn 33714  df-trl 33769
This theorem is referenced by:  cdlemk24-3  34514
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