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Theorem cdlemk27-3 33906
Description: Part of proof of Lemma K of [Crawley] p. 118. Eliminate the  P from the conclusion of cdlemk25-3 33903. (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
cdlemk27-3  |-  ( ( ( ( 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 )  =  ( C Y G ) )
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 cdlemk27-3
StepHypRef Expression
1 simp11 1027 . 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
) ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
2 simp221 1138 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T )  /\  ( G  e.  T  /\  C  e.  T
) )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( ( R `
 F )  =  ( R `  N
)  /\  F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( G  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) ) )  /\  ( ( ( R `  G
)  =/=  ( R `
 C )  /\  ( R `  C )  =/=  ( R `  F )  /\  ( R `  D )  =/=  ( R `  F
) )  /\  ( R `  G )  =/=  ( R `  D
) ) )  -> 
( R `  F
)  =  ( R `
 N ) )
3 simp13l 1112 . . 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
) ) )  ->  G  e.  T )
4 simp12 1028 . . 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
) ) )  -> 
( F  e.  T  /\  D  e.  T  /\  N  e.  T
) )
5 simp3l3 1104 . . . 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
) ) )  -> 
( R `  D
)  =/=  ( R `
 F ) )
6 simp3r 1026 . . . . 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
) ) )  -> 
( R `  G
)  =/=  ( R `
 D ) )
76necomd 2674 . . . 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
) ) )  -> 
( R `  D
)  =/=  ( R `
 G ) )
85, 7jca 530 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T )  /\  ( G  e.  T  /\  C  e.  T
) )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( ( R `
 F )  =  ( R `  N
)  /\  F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( G  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) ) )  /\  ( ( ( R `  G
)  =/=  ( R `
 C )  /\  ( R `  C )  =/=  ( R `  F )  /\  ( R `  D )  =/=  ( R `  F
) )  /\  ( R `  G )  =/=  ( R `  D
) ) )  -> 
( ( R `  D )  =/=  ( R `  F )  /\  ( R `  D
)  =/=  ( R `
 G ) ) )
9 simp222 1139 . . . 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
) ) )  ->  F  =/=  (  _I  |`  B ) )
10 simp23l 1118 . . . 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
) ) )  ->  G  =/=  (  _I  |`  B ) )
11 simp223 1140 . . . 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
) ) )  ->  D  =/=  (  _I  |`  B ) )
129, 10, 113jca 1177 . . 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
) ) )  -> 
( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) ) )
13 simp21 1030 . . 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
) ) )  -> 
( P  e.  A  /\  -.  P  .<_  W ) )
14 cdlemk3.b . . . 4  |-  B  =  ( Base `  K
)
15 cdlemk3.l . . . 4  |-  .<_  =  ( le `  K )
16 cdlemk3.j . . . 4  |-  .\/  =  ( join `  K )
17 cdlemk3.m . . . 4  |-  ./\  =  ( meet `  K )
18 cdlemk3.a . . . 4  |-  A  =  ( Atoms `  K )
19 cdlemk3.h . . . 4  |-  H  =  ( LHyp `  K
)
20 cdlemk3.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
21 cdlemk3.r . . . 4  |-  R  =  ( ( trL `  K
) `  W )
22 cdlemk3.s . . . 4  |-  S  =  ( f  e.  T  |->  ( iota_ i  e.  T  ( i `  P
)  =  ( ( P  .\/  ( R `
 f ) ) 
./\  ( ( N `
 P )  .\/  ( R `  ( f  o.  `' F ) ) ) ) ) )
23 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 ) ) ) ) ) )
2414, 15, 16, 17, 18, 19, 20, 21, 22, 23cdlemkuel-3 33897 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N )  /\  G  e.  T )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T )  /\  ( ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  D )  =/=  ( R `  G )
)  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( D Y G )  e.  T
)
251, 2, 3, 4, 8, 12, 13, 24syl313anc 1254 . 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
) ) )  -> 
( D Y G )  e.  T )
26 simp121 1129 . . 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
) ) )  ->  F  e.  T )
27 simp13r 1113 . . 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
) ) )  ->  C  e.  T )
28 simp123 1131 . . 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
) ) )  ->  N  e.  T )
29 simp3l2 1103 . . . 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
) ) )  -> 
( R `  C
)  =/=  ( R `
 F ) )
30 simp3l1 1102 . . . . 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
) ) )  -> 
( R `  G
)  =/=  ( R `
 C ) )
3130necomd 2674 . . . 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
) ) )  -> 
( R `  C
)  =/=  ( R `
 G ) )
3229, 31jca 530 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T )  /\  ( G  e.  T  /\  C  e.  T
) )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( ( R `
 F )  =  ( R `  N
)  /\  F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( G  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) ) )  /\  ( ( ( R `  G
)  =/=  ( R `
 C )  /\  ( R `  C )  =/=  ( R `  F )  /\  ( R `  D )  =/=  ( R `  F
) )  /\  ( R `  G )  =/=  ( R `  D
) ) )  -> 
( ( R `  C )  =/=  ( R `  F )  /\  ( R `  C
)  =/=  ( R `
 G ) ) )
33 simp23r 1119 . . . 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
) ) )  ->  C  =/=  (  _I  |`  B ) )
349, 10, 333jca 1177 . . 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
) ) )  -> 
( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) ) )
3514, 15, 16, 17, 18, 19, 20, 21, 22, 23cdlemkuel-3 33897 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N )  /\  G  e.  T )  /\  ( F  e.  T  /\  C  e.  T  /\  N  e.  T )  /\  ( ( ( R `
 C )  =/=  ( R `  F
)  /\  ( R `  C )  =/=  ( R `  G )
)  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( C Y G )  e.  T
)
361, 2, 3, 26, 27, 28, 32, 34, 13, 35syl333anc 1262 . 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
) ) )  -> 
( C Y G )  e.  T )
3714, 15, 16, 17, 18, 19, 20, 21, 22, 23cdlemk26-3 33905 . 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
) ) )  -> 
( ( D Y G ) `  P
)  =  ( ( C Y G ) `
 P ) )
3815, 18, 19, 20cdlemd 33205 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( D Y G )  e.  T  /\  ( C Y G )  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( ( D Y G ) `  P )  =  ( ( C Y G ) `  P ) )  ->  ( D Y G )  =  ( C Y G ) )
391, 25, 36, 13, 37, 38syl311anc 1244 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 )  =  ( C Y G ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842    =/= wne 2598   class class class wbr 4394    |-> cmpt 4452    _I cid 4732   `'ccnv 4821    |` cres 4824    o. ccom 4826   ` cfv 5568   iota_crio 6238  (class class class)co 6277    |-> cmpt2 6279   Basecbs 14839   lecple 14914   joincjn 15895   meetcmee 15896   Atomscatm 32261   HLchlt 32348   LHypclh 32981   LTrncltrn 33098   trLctrl 33156
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-riotaBAD 31957
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-iin 4273  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6783  df-2nd 6784  df-undef 7004  df-map 7458  df-preset 15879  df-poset 15897  df-plt 15910  df-lub 15926  df-glb 15927  df-join 15928  df-meet 15929  df-p0 15991  df-p1 15992  df-lat 15998  df-clat 16060  df-oposet 32174  df-ol 32176  df-oml 32177  df-covers 32264  df-ats 32265  df-atl 32296  df-cvlat 32320  df-hlat 32349  df-llines 32495  df-lplanes 32496  df-lvols 32497  df-lines 32498  df-psubsp 32500  df-pmap 32501  df-padd 32793  df-lhyp 32985  df-laut 32986  df-ldil 33101  df-ltrn 33102  df-trl 33157
This theorem is referenced by:  cdlemk28-3  33907
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