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Theorem cdlemk20 36340
Description: Part of proof of Lemma K of [Crawley] p. 118. Line 22, p. 119 for the i=2, j=1 case. Note typo on line 22: f should be fi. Our  D,  C,  O,  Q,  U,  V represent their f1, f2, k1, k2, sigma1, sigma2. (Contributed by NM, 5-Jul-2013.)
Hypotheses
Ref Expression
cdlemk1.b  |-  B  =  ( Base `  K
)
cdlemk1.l  |-  .<_  =  ( le `  K )
cdlemk1.j  |-  .\/  =  ( join `  K )
cdlemk1.m  |-  ./\  =  ( meet `  K )
cdlemk1.a  |-  A  =  ( Atoms `  K )
cdlemk1.h  |-  H  =  ( LHyp `  K
)
cdlemk1.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemk1.r  |-  R  =  ( ( trL `  K
) `  W )
cdlemk1.s  |-  S  =  ( f  e.  T  |->  ( iota_ i  e.  T  ( i `  P
)  =  ( ( P  .\/  ( R `
 f ) ) 
./\  ( ( N `
 P )  .\/  ( R `  ( f  o.  `' F ) ) ) ) ) )
cdlemk1.o  |-  O  =  ( S `  D
)
cdlemk1.u  |-  U  =  ( e  e.  T  |->  ( iota_ j  e.  T  ( j `  P
)  =  ( ( P  .\/  ( R `
 e ) ) 
./\  ( ( O `
 P )  .\/  ( R `  ( e  o.  `' D ) ) ) ) ) )
cdlemk2a.q  |-  Q  =  ( S `  C
)
Assertion
Ref Expression
cdlemk20  |-  ( ( ( ( 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 ) )
Distinct variable groups:    f, i,  ./\    .<_ , i    .\/ , f, i    A, i    D, f, i    f, F, i    i, H    i, K    f, N, i    P, f, i    R, f, i    T, f, i    f, W, i    ./\ , e    .\/ , e    D, e, j    e, O    P, e    R, e    T, e   
e, W    ./\ , j    .<_ , j    .\/ , j    A, j    D, j   
j, F    j, H    j, K    j, N    j, O    P, j    R, j    T, j    j, W    e, F, f, i    C, e   
f, j, C, i
Allowed substitution hints:    A( e, f)    B( e, f, i, j)    Q( e, f, i, j)    S( e, f, i, j)    U( e, f, i, j)    H( e, f)    K( e, f)    .<_ ( e, f)    N( e)    O( f, i)

Proof of Theorem cdlemk20
StepHypRef Expression
1 simp11 1027 . . 3  |-  ( ( ( ( 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 )
) ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
2 simp23 1032 . . 3  |-  ( ( ( ( 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 )
) ) )  -> 
( R `  F
)  =  ( R `
 N ) )
3 simp21r 1115 . . 3  |-  ( ( ( ( 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 )
) ) )  ->  C  e.  T )
4 simp12 1028 . . 3  |-  ( ( ( ( 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 )
) ) )  ->  F  e.  T )
5 simp13 1029 . . 3  |-  ( ( ( ( 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 )
) ) )  ->  D  e.  T )
6 simp21l 1114 . . 3  |-  ( ( ( ( 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 )
) ) )  ->  N  e.  T )
7 simp3r1 1105 . . . 4  |-  ( ( ( ( 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 )
) ) )  -> 
( R `  D
)  =/=  ( R `
 F ) )
8 simp3r3 1107 . . . . 5  |-  ( ( ( ( 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 )
) ) )  -> 
( R `  C
)  =/=  ( R `
 D ) )
98necomd 2714 . . . 4  |-  ( ( ( ( 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 )
) ) )  -> 
( R `  D
)  =/=  ( R `
 C ) )
107, 9jca 532 . . 3  |-  ( ( ( ( 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 )
) ) )  -> 
( ( R `  D )  =/=  ( R `  F )  /\  ( R `  D
)  =/=  ( R `
 C ) ) )
11 simp3l1 1102 . . . 4  |-  ( ( ( ( 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 )
) ) )  ->  F  =/=  (  _I  |`  B ) )
12 simp3l3 1104 . . . 4  |-  ( ( ( ( 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 )
) ) )  ->  C  =/=  (  _I  |`  B ) )
13 simp3l2 1103 . . . 4  |-  ( ( ( ( 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 )
) ) )  ->  D  =/=  (  _I  |`  B ) )
1411, 12, 133jca 1177 . . 3  |-  ( ( ( ( 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 )
) ) )  -> 
( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) ) )
15 simp22 1031 . . 3  |-  ( ( ( ( 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 )
) ) )  -> 
( P  e.  A  /\  -.  P  .<_  W ) )
16 cdlemk1.b . . . 4  |-  B  =  ( Base `  K
)
17 cdlemk1.l . . . 4  |-  .<_  =  ( le `  K )
18 cdlemk1.j . . . 4  |-  .\/  =  ( join `  K )
19 cdlemk1.m . . . 4  |-  ./\  =  ( meet `  K )
20 cdlemk1.a . . . 4  |-  A  =  ( Atoms `  K )
21 cdlemk1.h . . . 4  |-  H  =  ( LHyp `  K
)
22 cdlemk1.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
23 cdlemk1.r . . . 4  |-  R  =  ( ( trL `  K
) `  W )
24 cdlemk1.s . . . 4  |-  S  =  ( f  e.  T  |->  ( iota_ i  e.  T  ( i `  P
)  =  ( ( P  .\/  ( R `
 f ) ) 
./\  ( ( N `
 P )  .\/  ( R `  ( f  o.  `' F ) ) ) ) ) )
25 cdlemk1.o . . . 4  |-  O  =  ( S `  D
)
26 cdlemk1.u . . . 4  |-  U  =  ( e  e.  T  |->  ( iota_ j  e.  T  ( j `  P
)  =  ( ( P  .\/  ( R `
 e ) ) 
./\  ( ( O `
 P )  .\/  ( R `  ( e  o.  `' D ) ) ) ) ) )
2716, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26cdlemkuv2 36333 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N )  /\  C  e.  T )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T )  /\  ( ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  D )  =/=  ( R `  C )
)  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( ( U `  C ) `  P )  =  ( ( P  .\/  ( R `  C )
)  ./\  ( ( O `  P )  .\/  ( R `  ( C  o.  `' D
) ) ) ) )
281, 2, 3, 4, 5, 6, 10, 14, 15, 27syl333anc 1261 . 2  |-  ( ( ( ( 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
)  =  ( ( P  .\/  ( R `
 C ) ) 
./\  ( ( O `
 P )  .\/  ( R `  ( C  o.  `' D ) ) ) ) )
2917, 18, 20, 21, 22, 23trljat1 35631 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  C  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( P  .\/  ( R `  C
) )  =  ( P  .\/  ( C `
 P ) ) )
301, 3, 15, 29syl3anc 1229 . . 3  |-  ( ( ( ( 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 )
) ) )  -> 
( P  .\/  ( R `  C )
)  =  ( P 
.\/  ( C `  P ) ) )
3125fveq1i 5857 . . . . 5  |-  ( O `
 P )  =  ( ( S `  D ) `  P
)
3231a1i 11 . . . 4  |-  ( ( ( ( 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 )
) ) )  -> 
( O `  P
)  =  ( ( S `  D ) `
 P ) )
3321, 22, 23trlcocnv 36186 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  C  e.  T  /\  D  e.  T
)  ->  ( R `  ( C  o.  `' D ) )  =  ( R `  ( D  o.  `' C
) ) )
341, 3, 5, 33syl3anc 1229 . . . 4  |-  ( ( ( ( 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 )
) ) )  -> 
( R `  ( C  o.  `' D
) )  =  ( R `  ( D  o.  `' C ) ) )
3532, 34oveq12d 6299 . . 3  |-  ( ( ( ( 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 )
) ) )  -> 
( ( O `  P )  .\/  ( R `  ( C  o.  `' D ) ) )  =  ( ( ( S `  D ) `
 P )  .\/  ( R `  ( D  o.  `' C ) ) ) )
3630, 35oveq12d 6299 . 2  |-  ( ( ( ( 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 )
) ) )  -> 
( ( P  .\/  ( R `  C ) )  ./\  ( ( O `  P )  .\/  ( R `  ( C  o.  `' D
) ) ) )  =  ( ( P 
.\/  ( C `  P ) )  ./\  ( ( ( S `
 D ) `  P )  .\/  ( R `  ( D  o.  `' C ) ) ) ) )
37 cdlemk2a.q . . . 4  |-  Q  =  ( S `  C
)
3837fveq1i 5857 . . 3  |-  ( Q `
 P )  =  ( ( S `  C ) `  P
)
396, 5jca 532 . . . 4  |-  ( ( ( ( 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 )
) ) )  -> 
( N  e.  T  /\  D  e.  T
) )
40 simp3r2 1106 . . . . 5  |-  ( ( ( ( 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 )
) ) )  -> 
( R `  C
)  =/=  ( R `
 F ) )
4140, 7jca 532 . . . 4  |-  ( ( ( ( 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 )
) ) )  -> 
( ( R `  C )  =/=  ( R `  F )  /\  ( R `  D
)  =/=  ( R `
 F ) ) )
4216, 17, 18, 20, 21, 22, 23, 19, 24cdlemk12 36316 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  C  e.  T )  /\  (
( N  e.  T  /\  D  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( ( R `  C )  =/=  ( R `  F )  /\  ( R `  D )  =/=  ( R `  F
) )  /\  ( R `  C )  =/=  ( R `  D
) ) )  -> 
( ( S `  C ) `  P
)  =  ( ( P  .\/  ( C `
 P ) ) 
./\  ( ( ( S `  D ) `
 P )  .\/  ( R `  ( D  o.  `' C ) ) ) ) )
431, 4, 3, 39, 15, 2, 14, 41, 8, 42syl333anc 1261 . . 3  |-  ( ( ( ( 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 )
) ) )  -> 
( ( S `  C ) `  P
)  =  ( ( P  .\/  ( C `
 P ) ) 
./\  ( ( ( S `  D ) `
 P )  .\/  ( R `  ( D  o.  `' C ) ) ) ) )
4438, 43syl5req 2497 . 2  |-  ( ( ( ( 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 )
) ) )  -> 
( ( P  .\/  ( C `  P ) )  ./\  ( (
( S `  D
) `  P )  .\/  ( R `  ( D  o.  `' C
) ) ) )  =  ( Q `  P ) )
4528, 36, 443eqtrd 2488 1  |-  ( ( ( ( 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 ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804    =/= wne 2638   class class class wbr 4437    |-> cmpt 4495    _I cid 4780   `'ccnv 4988    |` cres 4991    o. ccom 4993   ` cfv 5578   iota_crio 6241  (class class class)co 6281   Basecbs 14613   lecple 14685   joincjn 15551   meetcmee 15552   Atomscatm 34728   HLchlt 34815   LHypclh 35448   LTrncltrn 35565   trLctrl 35623
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-riotaBAD 34424
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-iin 4318  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-1st 6785  df-2nd 6786  df-undef 7004  df-map 7424  df-preset 15535  df-poset 15553  df-plt 15566  df-lub 15582  df-glb 15583  df-join 15584  df-meet 15585  df-p0 15647  df-p1 15648  df-lat 15654  df-clat 15716  df-oposet 34641  df-ol 34643  df-oml 34644  df-covers 34731  df-ats 34732  df-atl 34763  df-cvlat 34787  df-hlat 34816  df-llines 34962  df-lplanes 34963  df-lvols 34964  df-lines 34965  df-psubsp 34967  df-pmap 34968  df-padd 35260  df-lhyp 35452  df-laut 35453  df-ldil 35568  df-ltrn 35569  df-trl 35624
This theorem is referenced by:  cdlemk20-2N  36358  cdlemk22  36359
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