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Theorem cdlemk20 31063
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 985 . . 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 990 . . 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 1073 . . 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 986 . . 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 987 . . 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 1072 . . 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 1063 . . . 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 1065 . . . . 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 2529 . . . 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 518 . . 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 1060 . . . 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 1062 . . . 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 1061 . . . 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 1132 . . 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 989 . . 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 31056 . . 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 1214 . 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 30355 . . . 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 1182 . . 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 5526 . . . . 5  |-  ( O `
 P )  =  ( ( S `  D ) `  P
)
3231a1i 10 . . . 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 30909 . . . . 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 1182 . . . 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 5876 . . 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 5876 . 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 5526 . . 3  |-  ( Q `
 P )  =  ( ( S `  C ) `  P
)
396, 5jca 518 . . . 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 1064 . . . . 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 518 . . . 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 31039 . . . 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 1214 . . 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 2328 . 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 2319 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 set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   class class class wbr 4023    e. cmpt 4077    _I cid 4304   `'ccnv 4688    |` cres 4691    o. ccom 4693   ` cfv 5255  (class class class)co 5858   iota_crio 6297   Basecbs 13148   lecple 13215   joincjn 14078   meetcmee 14079   Atomscatm 29453   HLchlt 29540   LHypclh 30173   LTrncltrn 30290   trLctrl 30347
This theorem is referenced by:  cdlemk20-2N  31081  cdlemk22  31082
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-map 6774  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-p1 14146  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-llines 29687  df-lplanes 29688  df-lvols 29689  df-lines 29690  df-psubsp 29692  df-pmap 29693  df-padd 29985  df-lhyp 30177  df-laut 30178  df-ldil 30293  df-ltrn 30294  df-trl 30348
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