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Theorem cdlemk7 30167
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 960 . . 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 961 . . 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 1107 . . 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 1108 . . 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 997 . . . 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 998 . . . 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 520 . . 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 30156 . . 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 1199 . 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 1077 . . . . 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 994 . . . . 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 995 . . . . 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 1137 . . . 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 30166 . . . 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 1199 . . 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 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 )
) )  ->  ( K  e.  HL  /\  W  e.  H ) )
26 simp13 992 . . . . 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 29485 . . . . 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 1187 . . . 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 5772 . . 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 2288 . 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 1071 . . . . 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 28683 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
3331, 32syl 17 . . . 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 991 . . . . . . 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 1078 . . . . . . 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 1137 . . . . . 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 1109 . . . . . 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 30165 . . . . . 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 1199 . . . . 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 28609 . . . . 5  |-  ( ( ( S `  X
) `  P )  e.  A  ->  ( ( S `  X ) `
 P )  e.  B )
4139, 40syl 17 . . . 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 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 )
) )  ->  W  e.  H )
43 simp22l 1079 . . . . 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 30161 . . . . 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 1207 . . . 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 14092 . . . 4  |-  ( ( K  e.  Lat  /\  ( ( S `  X ) `  P
)  e.  B  /\  V  e.  B )  ->  ( ( ( S `
 X ) `  P )  .\/  V
)  =  ( V 
.\/  ( ( S `
 X ) `  P ) ) )
4833, 41, 46, 47syl3anc 1187 . . 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 12 . . . 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 30166 . . . . . 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 1199 . . . . 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 29485 . . . . . . . 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 1187 . . . . . . 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 29459 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  T  /\  P  e.  A
)  ->  ( X `  P )  e.  A
)
5525, 35, 43, 54syl3anc 1187 . . . . . . . 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 28687 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( X `  P )  e.  A  /\  P  e.  A )  ->  (
( X `  P
)  .\/  P )  =  ( P  .\/  ( X `  P ) ) )
5731, 55, 43, 56syl3anc 1187 . . . . . . 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 2291 . . . . . 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 29459 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  N  e.  T  /\  P  e.  A
)  ->  ( N `  P )  e.  A
)
6025, 18, 43, 59syl3anc 1187 . . . . . . 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 520 . . . . . . . 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 30043 . . . . . . . 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 1187 . . . . . . 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 28687 . . . . . . 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 1187 . . . . . 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 5775 . . . . 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 2288 . . . 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 5775 . . 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 2288 . 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 3993 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 5    -> wi 6    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2419   class class class wbr 3963    e. cmpt 4017    _I cid 4241   `'ccnv 4625    |` cres 4628    o. ccom 4630   ` cfv 4638  (class class class)co 5757   iota_crio 6228   Basecbs 13075   lecple 13142   joincjn 14005   meetcmee 14006   Latclat 14078   Atomscatm 28583   HLchlt 28670   LHypclh 29303   LTrncltrn 29420   trLctrl 29477
This theorem is referenced by:  cdlemk11  30168
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4071  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3769  df-iun 3848  df-iin 3849  df-br 3964  df-opab 4018  df-mpt 4019  df-id 4246  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-1st 6021  df-2nd 6022  df-iota 6190  df-undef 6229  df-riota 6237  df-map 6707  df-poset 14007  df-plt 14019  df-lub 14035  df-glb 14036  df-join 14037  df-meet 14038  df-p0 14072  df-p1 14073  df-lat 14079  df-clat 14141  df-oposet 28496  df-ol 28498  df-oml 28499  df-covers 28586  df-ats 28587  df-atl 28618  df-cvlat 28642  df-hlat 28671  df-llines 28817  df-lplanes 28818  df-lvols 28819  df-lines 28820  df-psubsp 28822  df-pmap 28823  df-padd 29115  df-lhyp 29307  df-laut 29308  df-ldil 29423  df-ltrn 29424  df-trl 29478
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