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Theorem cdlemk22 34429
Description: Part of proof of Lemma K of [Crawley] p. 118. Lines 26-27, p. 119 for i=1 and j=2. (Contributed by NM, 5-Jul-2013.)
Hypotheses
Ref Expression
cdlemk2.b  |-  B  =  ( Base `  K
)
cdlemk2.l  |-  .<_  =  ( le `  K )
cdlemk2.j  |-  .\/  =  ( join `  K )
cdlemk2.m  |-  ./\  =  ( meet `  K )
cdlemk2.a  |-  A  =  ( Atoms `  K )
cdlemk2.h  |-  H  =  ( LHyp `  K
)
cdlemk2.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemk2.r  |-  R  =  ( ( trL `  K
) `  W )
cdlemk2.s  |-  S  =  ( f  e.  T  |->  ( iota_ i  e.  T  ( i `  P
)  =  ( ( P  .\/  ( R `
 f ) ) 
./\  ( ( N `
 P )  .\/  ( R `  ( f  o.  `' F ) ) ) ) ) )
cdlemk2.q  |-  Q  =  ( S `  C
)
cdlemk2.v  |-  V  =  ( d  e.  T  |->  ( iota_ k  e.  T  ( k `  P
)  =  ( ( P  .\/  ( R `
 d ) ) 
./\  ( ( Q `
 P )  .\/  ( R `  ( d  o.  `' C ) ) ) ) ) )
cdlemk2a.o  |-  O  =  ( S `  D
)
cdlemk2.u  |-  U  =  ( e  e.  T  |->  ( iota_ j  e.  T  ( j `  P
)  =  ( ( P  .\/  ( R `
 e ) ) 
./\  ( ( O `
 P )  .\/  ( R `  ( e  o.  `' D ) ) ) ) ) )
Assertion
Ref Expression
cdlemk22  |-  ( ( ( ( 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 `
 D ) ) ) )  ->  (
( U `  G
) `  P )  =  ( ( V `
 G ) `  P ) )
Distinct variable groups:    f, i,  ./\    .<_ , i    .\/ , f, i    A, i    C, f, i    f, F, i    i, H    i, K    f, N, i    P, f, i    R, f, i    T, f, i    f, W, i    ./\ , d    .\/ , d    C, d, k    G, d, k    Q, d    P, d    R, d    T, d    W, d    ./\ , k    .<_ , k    .\/ , k    A, k    C, k    k, F   
k, H    k, K    k, N    Q, k    P, k    R, k    T, k    k, W    F, d    i, G, f, k    D, k   
i, d, D, f   
e, j,  ./\    .<_ , e, j    .\/ , e, j    A, j    C, e, j    D, e, j    e, F, j   
e, G, j    j, H    j, K    j, N    e, O, j    P, e, j    R, e, j    T, e, j    e, W, j, f, i
Allowed substitution hints:    A( e, f, d)    B( e, f, i, j, k, d)    Q( e, f, i, j)    S( e, f, i, j, k, d)    U( e, f, i, j, k, d)    H( e, f, d)    K( e, f, d)    .<_ ( f, d)    N( e, d)    O( f, i, k, d)    V( e, f, i, j, k, d)

Proof of Theorem cdlemk22
StepHypRef Expression
1 simp11 1035 . . . 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 `
 D ) ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
2 simp212 1144 . . . 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 `
 D ) ) ) )  ->  G  e.  T )
3 simp22 1039 . . . 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 `
 D ) ) ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
4 cdlemk2.l . . . . 5  |-  .<_  =  ( le `  K )
5 cdlemk2.j . . . . 5  |-  .\/  =  ( join `  K )
6 cdlemk2.a . . . . 5  |-  A  =  ( Atoms `  K )
7 cdlemk2.h . . . . 5  |-  H  =  ( LHyp `  K
)
8 cdlemk2.t . . . . 5  |-  T  =  ( ( LTrn `  K
) `  W )
9 cdlemk2.r . . . . 5  |-  R  =  ( ( trL `  K
) `  W )
104, 5, 6, 7, 8, 9trljat1 33701 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( P  .\/  ( R `  G
) )  =  ( P  .\/  ( G `
 P ) ) )
111, 2, 3, 10syl3anc 1264 . . 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 `
 D ) ) ) )  ->  ( P  .\/  ( R `  G ) )  =  ( P  .\/  ( G `  P )
) )
12 simp1 1005 . . . . . 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 )  /\  ( R `  C
)  =/=  ( R `
 D ) ) ) )  ->  (
( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T
) )
13 simp211 1143 . . . . . . 7  |-  ( ( ( ( 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 `
 D ) ) ) )  ->  N  e.  T )
14 simp213 1145 . . . . . . 7  |-  ( ( ( ( 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 `
 D ) ) ) )  ->  C  e.  T )
1513, 14jca 534 . . . . . 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 )  /\  ( R `  C
)  =/=  ( R `
 D ) ) ) )  ->  ( N  e.  T  /\  C  e.  T )
)
16 simp23 1040 . . . . . 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 )  /\  ( R `  C
)  =/=  ( R `
 D ) ) ) )  ->  ( R `  F )  =  ( R `  N ) )
17 simp311 1152 . . . . . . 7  |-  ( ( ( ( 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 `
 D ) ) ) )  ->  F  =/=  (  _I  |`  B ) )
18 simp312 1153 . . . . . . 7  |-  ( ( ( ( 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 `
 D ) ) ) )  ->  D  =/=  (  _I  |`  B ) )
19 simp321 1155 . . . . . . 7  |-  ( ( ( ( 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 `
 D ) ) ) )  ->  C  =/=  (  _I  |`  B ) )
2017, 18, 193jca 1185 . . . . . 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 )  /\  ( R `  C
)  =/=  ( R `
 D ) ) ) )  ->  ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) ) )
21 simp331 1158 . . . . . . 7  |-  ( ( ( ( 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 `
 D ) ) ) )  ->  ( R `  D )  =/=  ( R `  F
) )
22 simp323 1157 . . . . . . 7  |-  ( ( ( ( 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 `
 D ) ) ) )  ->  ( R `  C )  =/=  ( R `  F
) )
23 simp333 1160 . . . . . . 7  |-  ( ( ( ( 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 `
 D ) ) ) )  ->  ( R `  C )  =/=  ( R `  D
) )
2421, 22, 233jca 1185 . . . . . 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 )  /\  ( R `  C
)  =/=  ( R `
 D ) ) ) )  ->  (
( R `  D
)  =/=  ( R `
 F )  /\  ( R `  C )  =/=  ( R `  F )  /\  ( R `  C )  =/=  ( R `  D
) ) )
25 cdlemk2.b . . . . . . 7  |-  B  =  ( Base `  K
)
26 cdlemk2.m . . . . . . 7  |-  ./\  =  ( meet `  K )
27 cdlemk2.s . . . . . . 7  |-  S  =  ( f  e.  T  |->  ( iota_ i  e.  T  ( i `  P
)  =  ( ( P  .\/  ( R `
 f ) ) 
./\  ( ( N `
 P )  .\/  ( R `  ( f  o.  `' F ) ) ) ) ) )
28 cdlemk2a.o . . . . . . 7  |-  O  =  ( S `  D
)
29 cdlemk2.u . . . . . . 7  |-  U  =  ( e  e.  T  |->  ( iota_ j  e.  T  ( j `  P
)  =  ( ( P  .\/  ( R `
 e ) ) 
./\  ( ( O `
 P )  .\/  ( R `  ( e  o.  `' D ) ) ) ) ) )
30 cdlemk2.q . . . . . . 7  |-  Q  =  ( S `  C
)
3125, 4, 5, 26, 6, 7, 8, 9, 27, 28, 29, 30cdlemk20 34410 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  C  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( ( R `  D )  =/=  ( R `  F )  /\  ( R `  C )  =/=  ( R `  F
)  /\  ( R `  C )  =/=  ( R `  D )
) ) )  -> 
( ( U `  C ) `  P
)  =  ( Q `
 P ) )
3212, 15, 3, 16, 20, 24, 31syl132anc 1282 . . . . 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 `  C
)  =/=  ( R `
 D ) ) ) )  ->  (
( U `  C
) `  P )  =  ( Q `  P ) )
3332eqcomd 2430 . . . 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 `
 D ) ) ) )  ->  ( Q `  P )  =  ( ( U `
 C ) `  P ) )
347, 8, 9trlcocnv 34256 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T  /\  C  e.  T
)  ->  ( R `  ( G  o.  `' C ) )  =  ( R `  ( C  o.  `' G
) ) )
351, 2, 14, 34syl3anc 1264 . . . 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 `
 D ) ) ) )  ->  ( R `  ( G  o.  `' C ) )  =  ( R `  ( C  o.  `' G
) ) )
3633, 35oveq12d 6323 . . 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 `
 D ) ) ) )  ->  (
( Q `  P
)  .\/  ( R `  ( G  o.  `' C ) ) )  =  ( ( ( U `  C ) `
 P )  .\/  ( R `  ( C  o.  `' G ) ) ) )
3711, 36oveq12d 6323 . 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 )  /\  ( R `  C
)  =/=  ( R `
 D ) ) ) )  ->  (
( P  .\/  ( R `  G )
)  ./\  ( ( Q `  P )  .\/  ( R `  ( G  o.  `' C
) ) ) )  =  ( ( P 
.\/  ( G `  P ) )  ./\  ( ( ( U `
 C ) `  P )  .\/  ( R `  ( C  o.  `' G ) ) ) ) )
38 simp12 1036 . . 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 `
 D ) ) ) )  ->  F  e.  T )
39 simp322 1156 . . . . 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 `  C
)  =/=  ( R `
 D ) ) ) )  ->  ( R `  G )  =/=  ( R `  C
) )
4039necomd 2691 . . . 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 `
 D ) ) ) )  ->  ( R `  C )  =/=  ( R `  G
) )
4122, 40jca 534 . . 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 `
 D ) ) ) )  ->  (
( R `  C
)  =/=  ( R `
 F )  /\  ( R `  C )  =/=  ( R `  G ) ) )
42 simp313 1154 . . . 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 `
 D ) ) ) )  ->  G  =/=  (  _I  |`  B ) )
4317, 42, 193jca 1185 . . 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 `
 D ) ) ) )  ->  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) ) )
44 cdlemk2.v . . . 4  |-  V  =  ( d  e.  T  |->  ( iota_ k  e.  T  ( k `  P
)  =  ( ( P  .\/  ( R `
 d ) ) 
./\  ( ( Q `
 P )  .\/  ( R `  ( d  o.  `' C ) ) ) ) ) )
4525, 4, 5, 26, 6, 7, 8, 9, 27, 30, 44cdlemkuv2-2 34421 . . 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 ) ) )  ->  ( ( V `  G ) `  P )  =  ( ( P  .\/  ( R `  G )
)  ./\  ( ( Q `  P )  .\/  ( R `  ( G  o.  `' C
) ) ) ) )
461, 16, 2, 38, 14, 13, 41, 43, 3, 45syl333anc 1296 . 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 )  /\  ( R `  C
)  =/=  ( R `
 D ) ) ) )  ->  (
( V `  G
) `  P )  =  ( ( P 
.\/  ( R `  G ) )  ./\  ( ( Q `  P )  .\/  ( R `  ( G  o.  `' C ) ) ) ) )
47 simp31 1041 . . . 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 `
 D ) ) ) )  ->  ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) ) )
4819, 39jca 534 . . . 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 `
 D ) ) ) )  ->  ( C  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  C )
) )
49 simp33 1043 . . . 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 `
 D ) ) ) )  ->  (
( R `  D
)  =/=  ( R `
 F )  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  C )  =/=  ( R `  D
) ) )
5047, 48, 493jca 1185 . . 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 `
 D ) ) ) )  ->  (
( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( C  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  C )
)  /\  ( ( R `  D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  C
)  =/=  ( R `
 D ) ) ) )
5125, 4, 5, 26, 6, 7, 8, 9, 27, 28, 29cdlemk12u 34408 . . 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 `  D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  C
)  =/=  ( R `
 D ) ) ) )  ->  (
( U `  G
) `  P )  =  ( ( P 
.\/  ( G `  P ) )  ./\  ( ( ( U `
 C ) `  P )  .\/  ( R `  ( C  o.  `' G ) ) ) ) )
5250, 51syld3an3 1309 . 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 )  /\  ( R `  C
)  =/=  ( R `
 D ) ) ) )  ->  (
( U `  G
) `  P )  =  ( ( P 
.\/  ( G `  P ) )  ./\  ( ( ( U `
 C ) `  P )  .\/  ( R `  ( C  o.  `' G ) ) ) ) )
5337, 46, 523eqtr4rd 2474 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 )  /\  ( R `  C
)  =/=  ( R `
 D ) ) ) )  ->  (
( U `  G
) `  P )  =  ( ( V `
 G ) `  P ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872    =/= wne 2614   class class class wbr 4423    |-> cmpt 4482    _I cid 4763   `'ccnv 4852    |` cres 4855    o. ccom 4857   ` cfv 5601   iota_crio 6266  (class class class)co 6305   Basecbs 15120   lecple 15196   joincjn 16188   meetcmee 16189   Atomscatm 32798   HLchlt 32885   LHypclh 33518   LTrncltrn 33635   trLctrl 33693
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 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597  ax-riotaBAD 32494
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 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rmo 2779  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-iun 4301  df-iin 4302  df-br 4424  df-opab 4483  df-mpt 4484  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-1st 6807  df-2nd 6808  df-undef 7031  df-map 7485  df-preset 16172  df-poset 16190  df-plt 16203  df-lub 16219  df-glb 16220  df-join 16221  df-meet 16222  df-p0 16284  df-p1 16285  df-lat 16291  df-clat 16353  df-oposet 32711  df-ol 32713  df-oml 32714  df-covers 32801  df-ats 32802  df-atl 32833  df-cvlat 32857  df-hlat 32886  df-llines 33032  df-lplanes 33033  df-lvols 33034  df-lines 33035  df-psubsp 33037  df-pmap 33038  df-padd 33330  df-lhyp 33522  df-laut 33523  df-ldil 33638  df-ltrn 33639  df-trl 33694
This theorem is referenced by:  cdlemk22-3  34437
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