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Theorem cdlemk7 31110
Description: Part of proof of Lemma K of [Crawley] p. 118. Line 5, p. 119. (Contributed by NM, 27-Jun-2013.)
Hypotheses
Ref Expression
cdlemk.b  |-  B  =  ( Base `  K
)
cdlemk.l  |-  .<_  =  ( le `  K )
cdlemk.j  |-  .\/  =  ( join `  K )
cdlemk.a  |-  A  =  ( Atoms `  K )
cdlemk.h  |-  H  =  ( LHyp `  K
)
cdlemk.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemk.r  |-  R  =  ( ( trL `  K
) `  W )
cdlemk.m  |-  ./\  =  ( meet `  K )
cdlemk.s  |-  S  =  ( f  e.  T  |->  ( iota_ i  e.  T
( i `  P
)  =  ( ( P  .\/  ( R `
 f ) ) 
./\  ( ( N `
 P )  .\/  ( R `  ( f  o.  `' F ) ) ) ) ) )
cdlemk.v  |-  V  =  ( ( ( G `
 P )  .\/  ( X `  P ) )  ./\  ( ( R `  ( G  o.  `' F ) )  .\/  ( R `  ( X  o.  `' F ) ) ) )
Assertion
Ref Expression
cdlemk7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  X  =/=  (  _I  |`  B ) )  /\  ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) )  ->  (
( S `  G
) `  P )  .<_  ( ( ( S `
 X ) `  P )  .\/  V
) )
Distinct variable groups:    ./\ , f    .\/ , f    f, F, i    f, G, i    f, N    P, f    R, f    T, f   
f, W    ./\ , i    .<_ , i    .\/ , i    A, i    i, F   
i, H    i, K    i, N    P, i    R, i    T, i    i, W    f, X, i
Allowed substitution hints:    A( f)    B( f, i)    S( f, i)    H( f)    K( f)    .<_ ( f)    V( f, i)

Proof of Theorem cdlemk7
StepHypRef Expression
1 simp1 955 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  X  =/=  (  _I  |`  B ) )  /\  ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) )  ->  (
( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T
) )
2 simp2 956 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  X  =/=  (  _I  |`  B ) )  /\  ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) )  ->  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) ) )
3 simp311 1102 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  X  =/=  (  _I  |`  B ) )  /\  ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) )  ->  F  =/=  (  _I  |`  B ) )
4 simp312 1103 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  X  =/=  (  _I  |`  B ) )  /\  ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) )  ->  G  =/=  (  _I  |`  B ) )
5 simp32 992 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  X  =/=  (  _I  |`  B ) )  /\  ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) )  ->  ( R `  G )  =/=  ( R `  F
) )
6 simp33 993 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  X  =/=  (  _I  |`  B ) )  /\  ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) )  ->  ( R `  X )  =/=  ( R `  F
) )
75, 6jca 518 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  X  =/=  (  _I  |`  B ) )  /\  ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) )  ->  (
( R `  G
)  =/=  ( R `
 F )  /\  ( R `  X )  =/=  ( R `  F ) ) )
8 cdlemk.b . . . 4  |-  B  =  ( Base `  K
)
9 cdlemk.l . . . 4  |-  .<_  =  ( le `  K )
10 cdlemk.j . . . 4  |-  .\/  =  ( join `  K )
11 cdlemk.a . . . 4  |-  A  =  ( Atoms `  K )
12 cdlemk.h . . . 4  |-  H  =  ( LHyp `  K
)
13 cdlemk.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
14 cdlemk.r . . . 4  |-  R  =  ( ( trL `  K
) `  W )
15 cdlemk.m . . . 4  |-  ./\  =  ( meet `  K )
168, 9, 10, 11, 12, 13, 14, 15cdlemk6 31099 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  ( ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) ) )  -> 
( ( P  .\/  ( G `  P ) )  ./\  ( ( N `  P )  .\/  ( R `  ( G  o.  `' F
) ) ) ) 
.<_  ( ( ( ( G `  P ) 
.\/  ( X `  P ) )  ./\  ( ( R `  ( G  o.  `' F ) )  .\/  ( R `  ( X  o.  `' F ) ) ) )  .\/  ( ( ( X `
 P )  .\/  P )  ./\  ( ( R `  ( X  o.  `' F ) )  .\/  ( N `  P ) ) ) ) )
171, 2, 3, 4, 7, 16syl113anc 1194 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  X  =/=  (  _I  |`  B ) )  /\  ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) )  ->  (
( P  .\/  ( G `  P )
)  ./\  ( ( N `  P )  .\/  ( R `  ( G  o.  `' F
) ) ) ) 
.<_  ( ( ( ( G `  P ) 
.\/  ( X `  P ) )  ./\  ( ( R `  ( G  o.  `' F ) )  .\/  ( R `  ( X  o.  `' F ) ) ) )  .\/  ( ( ( X `
 P )  .\/  P )  ./\  ( ( R `  ( X  o.  `' F ) )  .\/  ( N `  P ) ) ) ) )
18 simp21l 1072 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  X  =/=  (  _I  |`  B ) )  /\  ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) )  ->  N  e.  T )
19 simp22 989 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  X  =/=  (  _I  |`  B ) )  /\  ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
20 simp23 990 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  X  =/=  (  _I  |`  B ) )  /\  ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) )  ->  ( R `  F )  =  ( R `  N ) )
2118, 19, 203jca 1132 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  X  =/=  (  _I  |`  B ) )  /\  ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) )  ->  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) ) )
22 cdlemk.s . . . . 5  |-  S  =  ( f  e.  T  |->  ( iota_ i  e.  T
( i `  P
)  =  ( ( P  .\/  ( R `
 f ) ) 
./\  ( ( N `
 P )  .\/  ( R `  ( f  o.  `' F ) ) ) ) ) )
238, 9, 10, 11, 12, 13, 14, 15, 22cdlemksv2 31109 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  F ) ) )  ->  ( ( S `
 G ) `  P )  =  ( ( P  .\/  ( R `  G )
)  ./\  ( ( N `  P )  .\/  ( R `  ( G  o.  `' F
) ) ) ) )
241, 21, 3, 4, 5, 23syl113anc 1194 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  X  =/=  (  _I  |`  B ) )  /\  ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) )  ->  (
( S `  G
) `  P )  =  ( ( P 
.\/  ( R `  G ) )  ./\  ( ( N `  P )  .\/  ( R `  ( G  o.  `' F ) ) ) ) )
25 simp11 985 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  X  =/=  (  _I  |`  B ) )  /\  ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) )  ->  ( K  e.  HL  /\  W  e.  H ) )
26 simp13 987 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  X  =/=  (  _I  |`  B ) )  /\  ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) )  ->  G  e.  T )
279, 10, 11, 12, 13, 14trljat1 30428 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( P  .\/  ( R `  G
) )  =  ( P  .\/  ( G `
 P ) ) )
2825, 26, 19, 27syl3anc 1182 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  X  =/=  (  _I  |`  B ) )  /\  ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) )  ->  ( P  .\/  ( R `  G ) )  =  ( P  .\/  ( G `  P )
) )
2928oveq1d 5875 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  X  =/=  (  _I  |`  B ) )  /\  ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) )  ->  (
( P  .\/  ( R `  G )
)  ./\  ( ( N `  P )  .\/  ( R `  ( G  o.  `' F
) ) ) )  =  ( ( P 
.\/  ( G `  P ) )  ./\  ( ( N `  P )  .\/  ( R `  ( G  o.  `' F ) ) ) ) )
3024, 29eqtrd 2317 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  X  =/=  (  _I  |`  B ) )  /\  ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) )  ->  (
( S `  G
) `  P )  =  ( ( P 
.\/  ( G `  P ) )  ./\  ( ( N `  P )  .\/  ( R `  ( G  o.  `' F ) ) ) ) )
31 simp11l 1066 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  X  =/=  (  _I  |`  B ) )  /\  ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) )  ->  K  e.  HL )
32 hllat 29626 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
3331, 32syl 15 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  X  =/=  (  _I  |`  B ) )  /\  ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) )  ->  K  e.  Lat )
34 simp12 986 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  X  =/=  (  _I  |`  B ) )  /\  ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) )  ->  F  e.  T )
35 simp21r 1073 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  X  =/=  (  _I  |`  B ) )  /\  ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) )  ->  X  e.  T )
3625, 34, 353jca 1132 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  X  =/=  (  _I  |`  B ) )  /\  ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) )  ->  (
( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  X  e.  T
) )
37 simp313 1104 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  X  =/=  (  _I  |`  B ) )  /\  ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) )  ->  X  =/=  (  _I  |`  B ) )
388, 9, 10, 11, 12, 13, 14, 15, 22cdlemksat 31108 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  X  e.  T )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  =/=  (  _I  |`  B )  /\  X  =/=  (  _I  |`  B )  /\  ( R `  X )  =/=  ( R `  F ) ) )  ->  ( ( S `
 X ) `  P )  e.  A
)
3936, 21, 3, 37, 6, 38syl113anc 1194 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  X  =/=  (  _I  |`  B ) )  /\  ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) )  ->  (
( S `  X
) `  P )  e.  A )
408, 11atbase 29552 . . . . 5  |-  ( ( ( S `  X
) `  P )  e.  A  ->  ( ( S `  X ) `
 P )  e.  B )
4139, 40syl 15 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  X  =/=  (  _I  |`  B ) )  /\  ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) )  ->  (
( S `  X
) `  P )  e.  B )
42 simp11r 1067 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  X  =/=  (  _I  |`  B ) )  /\  ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) )  ->  W  e.  H )
43 simp22l 1074 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  X  =/=  (  _I  |`  B ) )  /\  ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) )  ->  P  e.  A )
44 cdlemk.v . . . . . 6  |-  V  =  ( ( ( G `
 P )  .\/  ( X `  P ) )  ./\  ( ( R `  ( G  o.  `' F ) )  .\/  ( R `  ( X  o.  `' F ) ) ) )
458, 9, 10, 11, 12, 13, 14, 15, 44cdlemkvcl 31104 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T  /\  X  e.  T )  /\  P  e.  A )  ->  V  e.  B )
4631, 42, 34, 26, 35, 43, 45syl231anc 1202 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  X  =/=  (  _I  |`  B ) )  /\  ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) )  ->  V  e.  B )
478, 10latjcom 14167 . . . 4  |-  ( ( K  e.  Lat  /\  ( ( S `  X ) `  P
)  e.  B  /\  V  e.  B )  ->  ( ( ( S `
 X ) `  P )  .\/  V
)  =  ( V 
.\/  ( ( S `
 X ) `  P ) ) )
4833, 41, 46, 47syl3anc 1182 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  X  =/=  (  _I  |`  B ) )  /\  ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) )  ->  (
( ( S `  X ) `  P
)  .\/  V )  =  ( V  .\/  ( ( S `  X ) `  P
) ) )
4944a1i 10 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  X  =/=  (  _I  |`  B ) )  /\  ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) )  ->  V  =  ( ( ( G `  P ) 
.\/  ( X `  P ) )  ./\  ( ( R `  ( G  o.  `' F ) )  .\/  ( R `  ( X  o.  `' F ) ) ) ) )
508, 9, 10, 11, 12, 13, 14, 15, 22cdlemksv2 31109 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  X  e.  T )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  =/=  (  _I  |`  B )  /\  X  =/=  (  _I  |`  B )  /\  ( R `  X )  =/=  ( R `  F ) ) )  ->  ( ( S `
 X ) `  P )  =  ( ( P  .\/  ( R `  X )
)  ./\  ( ( N `  P )  .\/  ( R `  ( X  o.  `' F
) ) ) ) )
5136, 21, 3, 37, 6, 50syl113anc 1194 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  X  =/=  (  _I  |`  B ) )  /\  ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) )  ->  (
( S `  X
) `  P )  =  ( ( P 
.\/  ( R `  X ) )  ./\  ( ( N `  P )  .\/  ( R `  ( X  o.  `' F ) ) ) ) )
529, 10, 11, 12, 13, 14trljat1 30428 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( P  .\/  ( R `  X
) )  =  ( P  .\/  ( X `
 P ) ) )
5325, 35, 19, 52syl3anc 1182 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  X  =/=  (  _I  |`  B ) )  /\  ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) )  ->  ( P  .\/  ( R `  X ) )  =  ( P  .\/  ( X `  P )
) )
549, 11, 12, 13ltrnat 30402 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  T  /\  P  e.  A
)  ->  ( X `  P )  e.  A
)
5525, 35, 43, 54syl3anc 1182 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  X  =/=  (  _I  |`  B ) )  /\  ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) )  ->  ( X `  P )  e.  A )
5610, 11hlatjcom 29630 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( X `  P )  e.  A  /\  P  e.  A )  ->  (
( X `  P
)  .\/  P )  =  ( P  .\/  ( X `  P ) ) )
5731, 55, 43, 56syl3anc 1182 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  X  =/=  (  _I  |`  B ) )  /\  ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) )  ->  (
( X `  P
)  .\/  P )  =  ( P  .\/  ( X `  P ) ) )
5853, 57eqtr4d 2320 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  X  =/=  (  _I  |`  B ) )  /\  ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) )  ->  ( P  .\/  ( R `  X ) )  =  ( ( X `  P )  .\/  P
) )
599, 11, 12, 13ltrnat 30402 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  N  e.  T  /\  P  e.  A
)  ->  ( N `  P )  e.  A
)
6025, 18, 43, 59syl3anc 1182 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  X  =/=  (  _I  |`  B ) )  /\  ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) )  ->  ( N `  P )  e.  A )
6135, 34jca 518 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  X  =/=  (  _I  |`  B ) )  /\  ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) )  ->  ( X  e.  T  /\  F  e.  T )
)
6211, 12, 13, 14trlcocnvat 30986 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  T  /\  F  e.  T )  /\  ( R `  X )  =/=  ( R `  F
) )  ->  ( R `  ( X  o.  `' F ) )  e.  A )
6325, 61, 6, 62syl3anc 1182 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  X  =/=  (  _I  |`  B ) )  /\  ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) )  ->  ( R `  ( X  o.  `' F ) )  e.  A )
6410, 11hlatjcom 29630 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( N `  P )  e.  A  /\  ( R `  ( X  o.  `' F ) )  e.  A )  ->  (
( N `  P
)  .\/  ( R `  ( X  o.  `' F ) ) )  =  ( ( R `
 ( X  o.  `' F ) )  .\/  ( N `  P ) ) )
6531, 60, 63, 64syl3anc 1182 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  X  =/=  (  _I  |`  B ) )  /\  ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) )  ->  (
( N `  P
)  .\/  ( R `  ( X  o.  `' F ) ) )  =  ( ( R `
 ( X  o.  `' F ) )  .\/  ( N `  P ) ) )
6658, 65oveq12d 5878 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  X  =/=  (  _I  |`  B ) )  /\  ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) )  ->  (
( P  .\/  ( R `  X )
)  ./\  ( ( N `  P )  .\/  ( R `  ( X  o.  `' F
) ) ) )  =  ( ( ( X `  P ) 
.\/  P )  ./\  ( ( R `  ( X  o.  `' F ) )  .\/  ( N `  P ) ) ) )
6751, 66eqtrd 2317 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  X  =/=  (  _I  |`  B ) )  /\  ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) )  ->  (
( S `  X
) `  P )  =  ( ( ( X `  P ) 
.\/  P )  ./\  ( ( R `  ( X  o.  `' F ) )  .\/  ( N `  P ) ) ) )
6849, 67oveq12d 5878 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  X  =/=  (  _I  |`  B ) )  /\  ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) )  ->  ( V  .\/  ( ( S `
 X ) `  P ) )  =  ( ( ( ( G `  P ) 
.\/  ( X `  P ) )  ./\  ( ( R `  ( G  o.  `' F ) )  .\/  ( R `  ( X  o.  `' F ) ) ) )  .\/  ( ( ( X `
 P )  .\/  P )  ./\  ( ( R `  ( X  o.  `' F ) )  .\/  ( N `  P ) ) ) ) )
6948, 68eqtrd 2317 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  X  =/=  (  _I  |`  B ) )  /\  ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) )  ->  (
( ( S `  X ) `  P
)  .\/  V )  =  ( ( ( ( G `  P
)  .\/  ( X `  P ) )  ./\  ( ( R `  ( G  o.  `' F ) )  .\/  ( R `  ( X  o.  `' F ) ) ) )  .\/  ( ( ( X `
 P )  .\/  P )  ./\  ( ( R `  ( X  o.  `' F ) )  .\/  ( N `  P ) ) ) ) )
7017, 30, 693brtr4d 4055 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  X  =/=  (  _I  |`  B ) )  /\  ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) )  ->  (
( S `  G
) `  P )  .<_  ( ( ( S `
 X ) `  P )  .\/  V
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1625    e. wcel 1686    =/= wne 2448   class class class wbr 4025    e. cmpt 4079    _I cid 4306   `'ccnv 4690    |` cres 4693    o. ccom 4695   ` cfv 5257  (class class class)co 5860   iota_crio 6299   Basecbs 13150   lecple 13217   joincjn 14080   meetcmee 14081   Latclat 14153   Atomscatm 29526   HLchlt 29613   LHypclh 30246   LTrncltrn 30363   trLctrl 30420
This theorem is referenced by:  cdlemk11  31111
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514
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 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-iun 3909  df-iin 3910  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-1st 6124  df-2nd 6125  df-undef 6300  df-riota 6306  df-map 6776  df-poset 14082  df-plt 14094  df-lub 14110  df-glb 14111  df-join 14112  df-meet 14113  df-p0 14147  df-p1 14148  df-lat 14154  df-clat 14216  df-oposet 29439  df-ol 29441  df-oml 29442  df-covers 29529  df-ats 29530  df-atl 29561  df-cvlat 29585  df-hlat 29614  df-llines 29760  df-lplanes 29761  df-lvols 29762  df-lines 29763  df-psubsp 29765  df-pmap 29766  df-padd 30058  df-lhyp 30250  df-laut 30251  df-ldil 30366  df-ltrn 30367  df-trl 30421
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