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Theorem cdlemk26-3 31400
Description: Part of proof of Lemma K of [Crawley] p. 118. Eliminate the  x requirements from cdlemk25-3 31398. (Contributed by NM, 10-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
cdlemk26-3  |-  ( ( ( ( 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
) ) )  -> 
( ( 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
Allowed substitution hints:    A( e, f, d)    B( e, f, i, j, d)    S( f, i)    H( e, f, d)    K( e, f, d)    .<_ ( f, d)    N( e, d)    Y( e, f, i, j, d)

Proof of Theorem cdlemk26-3
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 simp11l 1068 . . 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
) ) )  ->  K  e.  HL )
2 simp11r 1069 . . 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
) ) )  ->  W  e.  H )
3 cdlemk3.b . . . 4  |-  B  =  ( Base `  K
)
4 cdlemk3.h . . . 4  |-  H  =  ( LHyp `  K
)
5 cdlemk3.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
6 cdlemk3.r . . . 4  |-  R  =  ( ( trL `  K
) `  W )
73, 4, 5, 6cdlemftr3 31059 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. x  e.  T  ( x  =/=  (  _I  |`  B )  /\  ( ( R `  x )  =/=  ( R `  F )  /\  ( R `  x
)  =/=  ( R `
 G )  /\  ( R `  x )  =/=  ( R `  D ) ) ) )
81, 2, 7syl2anc 643 . 2  |-  ( ( ( ( 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
) ) )  ->  E. x  e.  T  ( x  =/=  (  _I  |`  B )  /\  ( ( R `  x )  =/=  ( R `  F )  /\  ( R `  x
)  =/=  ( R `
 G )  /\  ( R `  x )  =/=  ( R `  D ) ) ) )
9 simp111 1086 . . . 4  |-  ( ( ( ( ( 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
) ) )  /\  x  e.  T  /\  ( x  =/=  (  _I  |`  B )  /\  ( ( R `  x )  =/=  ( R `  F )  /\  ( R `  x
)  =/=  ( R `
 G )  /\  ( R `  x )  =/=  ( R `  D ) ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
10 simp112 1087 . . . 4  |-  ( ( ( ( ( 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
) ) )  /\  x  e.  T  /\  ( x  =/=  (  _I  |`  B )  /\  ( ( R `  x )  =/=  ( R `  F )  /\  ( R `  x
)  =/=  ( R `
 G )  /\  ( R `  x )  =/=  ( R `  D ) ) ) )  ->  ( F  e.  T  /\  D  e.  T  /\  N  e.  T ) )
11 simp13l 1072 . . . . . 6  |-  ( ( ( ( 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
) ) )  ->  G  e.  T )
12113ad2ant1 978 . . . . 5  |-  ( ( ( ( ( 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
) ) )  /\  x  e.  T  /\  ( x  =/=  (  _I  |`  B )  /\  ( ( R `  x )  =/=  ( R `  F )  /\  ( R `  x
)  =/=  ( R `
 G )  /\  ( R `  x )  =/=  ( R `  D ) ) ) )  ->  G  e.  T )
13 simp13r 1073 . . . . . 6  |-  ( ( ( ( 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
) ) )  ->  C  e.  T )
14133ad2ant1 978 . . . . 5  |-  ( ( ( ( ( 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
) ) )  /\  x  e.  T  /\  ( x  =/=  (  _I  |`  B )  /\  ( ( R `  x )  =/=  ( R `  F )  /\  ( R `  x
)  =/=  ( R `
 G )  /\  ( R `  x )  =/=  ( R `  D ) ) ) )  ->  C  e.  T )
15 simp2 958 . . . . 5  |-  ( ( ( ( ( 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
) ) )  /\  x  e.  T  /\  ( x  =/=  (  _I  |`  B )  /\  ( ( R `  x )  =/=  ( R `  F )  /\  ( R `  x
)  =/=  ( R `
 G )  /\  ( R `  x )  =/=  ( R `  D ) ) ) )  ->  x  e.  T )
1612, 14, 153jca 1134 . . . 4  |-  ( ( ( ( ( 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
) ) )  /\  x  e.  T  /\  ( x  =/=  (  _I  |`  B )  /\  ( ( R `  x )  =/=  ( R `  F )  /\  ( R `  x
)  =/=  ( R `
 G )  /\  ( R `  x )  =/=  ( R `  D ) ) ) )  ->  ( G  e.  T  /\  C  e.  T  /\  x  e.  T ) )
17 simp121 1089 . . . 4  |-  ( ( ( ( ( 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
) ) )  /\  x  e.  T  /\  ( x  =/=  (  _I  |`  B )  /\  ( ( R `  x )  =/=  ( R `  F )  /\  ( R `  x
)  =/=  ( R `
 G )  /\  ( R `  x )  =/=  ( R `  D ) ) ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
18 simp122 1090 . . . 4  |-  ( ( ( ( ( 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
) ) )  /\  x  e.  T  /\  ( x  =/=  (  _I  |`  B )  /\  ( ( R `  x )  =/=  ( R `  F )  /\  ( R `  x
)  =/=  ( R `
 G )  /\  ( R `  x )  =/=  ( R `  D ) ) ) )  ->  ( ( R `  F )  =  ( R `  N )  /\  F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) ) )
19 simp23l 1078 . . . . . 6  |-  ( ( ( ( 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
) ) )  ->  G  =/=  (  _I  |`  B ) )
20193ad2ant1 978 . . . . 5  |-  ( ( ( ( ( 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
) ) )  /\  x  e.  T  /\  ( x  =/=  (  _I  |`  B )  /\  ( ( R `  x )  =/=  ( R `  F )  /\  ( R `  x
)  =/=  ( R `
 G )  /\  ( R `  x )  =/=  ( R `  D ) ) ) )  ->  G  =/=  (  _I  |`  B ) )
21 simp23r 1079 . . . . . 6  |-  ( ( ( ( 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
) ) )  ->  C  =/=  (  _I  |`  B ) )
22213ad2ant1 978 . . . . 5  |-  ( ( ( ( ( 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
) ) )  /\  x  e.  T  /\  ( x  =/=  (  _I  |`  B )  /\  ( ( R `  x )  =/=  ( R `  F )  /\  ( R `  x
)  =/=  ( R `
 G )  /\  ( R `  x )  =/=  ( R `  D ) ) ) )  ->  C  =/=  (  _I  |`  B ) )
23 simp3l 985 . . . . 5  |-  ( ( ( ( ( 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
) ) )  /\  x  e.  T  /\  ( x  =/=  (  _I  |`  B )  /\  ( ( R `  x )  =/=  ( R `  F )  /\  ( R `  x
)  =/=  ( R `
 G )  /\  ( R `  x )  =/=  ( R `  D ) ) ) )  ->  x  =/=  (  _I  |`  B ) )
2420, 22, 233jca 1134 . . . 4  |-  ( ( ( ( ( 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
) ) )  /\  x  e.  T  /\  ( x  =/=  (  _I  |`  B )  /\  ( ( R `  x )  =/=  ( R `  F )  /\  ( R `  x
)  =/=  ( R `
 G )  /\  ( R `  x )  =/=  ( R `  D ) ) ) )  ->  ( G  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B )  /\  x  =/=  (  _I  |`  B ) ) )
25 simp13l 1072 . . . 4  |-  ( ( ( ( ( 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
) ) )  /\  x  e.  T  /\  ( x  =/=  (  _I  |`  B )  /\  ( ( R `  x )  =/=  ( R `  F )  /\  ( R `  x
)  =/=  ( R `
 G )  /\  ( R `  x )  =/=  ( R `  D ) ) ) )  ->  ( ( R `  G )  =/=  ( R `  C
)  /\  ( R `  C )  =/=  ( R `  F )  /\  ( R `  D
)  =/=  ( R `
 F ) ) )
26 simp13r 1073 . . . 4  |-  ( ( ( ( ( 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
) ) )  /\  x  e.  T  /\  ( x  =/=  (  _I  |`  B )  /\  ( ( R `  x )  =/=  ( R `  F )  /\  ( R `  x
)  =/=  ( R `
 G )  /\  ( R `  x )  =/=  ( R `  D ) ) ) )  ->  ( R `  G )  =/=  ( R `  D )
)
27 simp3r3 1067 . . . . 5  |-  ( ( ( ( ( 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
) ) )  /\  x  e.  T  /\  ( x  =/=  (  _I  |`  B )  /\  ( ( R `  x )  =/=  ( R `  F )  /\  ( R `  x
)  =/=  ( R `
 G )  /\  ( R `  x )  =/=  ( R `  D ) ) ) )  ->  ( R `  x )  =/=  ( R `  D )
)
28 simp3r1 1065 . . . . 5  |-  ( ( ( ( ( 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
) ) )  /\  x  e.  T  /\  ( x  =/=  (  _I  |`  B )  /\  ( ( R `  x )  =/=  ( R `  F )  /\  ( R `  x
)  =/=  ( R `
 G )  /\  ( R `  x )  =/=  ( R `  D ) ) ) )  ->  ( R `  x )  =/=  ( R `  F )
)
29 simp3r2 1066 . . . . . 6  |-  ( ( ( ( ( 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
) ) )  /\  x  e.  T  /\  ( x  =/=  (  _I  |`  B )  /\  ( ( R `  x )  =/=  ( R `  F )  /\  ( R `  x
)  =/=  ( R `
 G )  /\  ( R `  x )  =/=  ( R `  D ) ) ) )  ->  ( R `  x )  =/=  ( R `  G )
)
3029necomd 2658 . . . . 5  |-  ( ( ( ( ( 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
) ) )  /\  x  e.  T  /\  ( x  =/=  (  _I  |`  B )  /\  ( ( R `  x )  =/=  ( R `  F )  /\  ( R `  x
)  =/=  ( R `
 G )  /\  ( R `  x )  =/=  ( R `  D ) ) ) )  ->  ( R `  G )  =/=  ( R `  x )
)
3127, 28, 303jca 1134 . . . 4  |-  ( ( ( ( ( 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
) ) )  /\  x  e.  T  /\  ( x  =/=  (  _I  |`  B )  /\  ( ( R `  x )  =/=  ( R `  F )  /\  ( R `  x
)  =/=  ( R `
 G )  /\  ( R `  x )  =/=  ( R `  D ) ) ) )  ->  ( ( R `  x )  =/=  ( R `  D
)  /\  ( R `  x )  =/=  ( R `  F )  /\  ( R `  G
)  =/=  ( R `
 x ) ) )
32 cdlemk3.l . . . . 5  |-  .<_  =  ( le `  K )
33 cdlemk3.j . . . . 5  |-  .\/  =  ( join `  K )
34 cdlemk3.m . . . . 5  |-  ./\  =  ( meet `  K )
35 cdlemk3.a . . . . 5  |-  A  =  ( Atoms `  K )
36 cdlemk3.s . . . . 5  |-  S  =  ( f  e.  T  |->  ( iota_ i  e.  T
( i `  P
)  =  ( ( P  .\/  ( R `
 f ) ) 
./\  ( ( N `
 P )  .\/  ( R `  ( f  o.  `' F ) ) ) ) ) )
37 cdlemk3.u1 . . . . 5  |-  Y  =  ( d  e.  T ,  e  e.  T  |->  ( iota_ j  e.  T
( j `  P
)  =  ( ( P  .\/  ( R `
 e ) ) 
./\  ( ( ( S `  d ) `
 P )  .\/  ( R `  ( e  o.  `' d ) ) ) ) ) )
383, 32, 33, 34, 35, 4, 5, 6, 36, 37cdlemk25-3 31398 . . . 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
) ) ) )  ->  ( ( D Y G ) `  P )  =  ( ( C Y G ) `  P ) )
399, 10, 16, 17, 18, 24, 25, 26, 31, 38syl333anc 1216 . . 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
) ) )  /\  x  e.  T  /\  ( x  =/=  (  _I  |`  B )  /\  ( ( R `  x )  =/=  ( R `  F )  /\  ( R `  x
)  =/=  ( R `
 G )  /\  ( R `  x )  =/=  ( R `  D ) ) ) )  ->  ( ( D Y G ) `  P )  =  ( ( C Y G ) `  P ) )
4039rexlimdv3a 2800 . 2  |-  ( ( ( ( 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
) ) )  -> 
( E. x  e.  T  ( x  =/=  (  _I  |`  B )  /\  ( ( R `
 x )  =/=  ( R `  F
)  /\  ( R `  x )  =/=  ( R `  G )  /\  ( R `  x
)  =/=  ( R `
 D ) ) )  ->  ( ( D Y G ) `  P )  =  ( ( C Y G ) `  P ) ) )
418, 40mpd 15 1  |-  ( ( ( ( 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
) ) )  -> 
( ( D Y G ) `  P
)  =  ( ( C Y G ) `
 P ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2575   E.wrex 2675   class class class wbr 4180    e. cmpt 4234    _I cid 4461   `'ccnv 4844    |` cres 4847    o. ccom 4849   ` cfv 5421  (class class class)co 6048    e. cmpt2 6050   iota_crio 6509   Basecbs 13432   lecple 13499   joincjn 14364   meetcmee 14365   Atomscatm 29758   HLchlt 29845   LHypclh 30478   LTrncltrn 30595   trLctrl 30652
This theorem is referenced by:  cdlemk27-3  31401
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-iin 4064  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  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 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-undef 6510  df-riota 6516  df-map 6987  df-poset 14366  df-plt 14378  df-lub 14394  df-glb 14395  df-join 14396  df-meet 14397  df-p0 14431  df-p1 14432  df-lat 14438  df-clat 14500  df-oposet 29671  df-ol 29673  df-oml 29674  df-covers 29761  df-ats 29762  df-atl 29793  df-cvlat 29817  df-hlat 29846  df-llines 29992  df-lplanes 29993  df-lvols 29994  df-lines 29995  df-psubsp 29997  df-pmap 29998  df-padd 30290  df-lhyp 30482  df-laut 30483  df-ldil 30598  df-ltrn 30599  df-trl 30653
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