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Theorem cdlemk12 31112
Description: Part of proof of Lemma K of [Crawley] p. 118. Eq. 4, line 10, p. 119. (Contributed by NM, 30-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 ) ) ) ) ) )
Assertion
Ref Expression
cdlemk12  |-  ( ( ( ( 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 `  X
) ) )  -> 
( ( S `  G ) `  P
)  =  ( ( P  .\/  ( G `
 P ) ) 
./\  ( ( ( S `  X ) `
 P )  .\/  ( R `  ( X  o.  `' G ) ) ) ) )
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)

Proof of Theorem cdlemk12
StepHypRef Expression
1 simp11l 1066 . 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
) )  /\  ( R `  G )  =/=  ( R `  X
) ) )  ->  K  e.  HL )
2 simp22l 1074 . 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
) )  /\  ( R `  G )  =/=  ( R `  X
) ) )  ->  P  e.  A )
3 simp11 985 . . 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 `  X
) ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
4 simp13 987 . . 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 `  X
) ) )  ->  G  e.  T )
5 cdlemk.l . . . 4  |-  .<_  =  ( le `  K )
6 cdlemk.a . . . 4  |-  A  =  ( Atoms `  K )
7 cdlemk.h . . . 4  |-  H  =  ( LHyp `  K
)
8 cdlemk.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
95, 6, 7, 8ltrnat 30402 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T  /\  P  e.  A
)  ->  ( G `  P )  e.  A
)
103, 4, 2, 9syl3anc 1182 . 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
) )  /\  ( R `  G )  =/=  ( R `  X
) ) )  -> 
( G `  P
)  e.  A )
11 simp12 986 . . . 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 `  X
) ) )  ->  F  e.  T )
12 simp21r 1073 . . . 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 `  X
) ) )  ->  X  e.  T )
133, 11, 123jca 1132 . . 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 `  X
) ) )  -> 
( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  X  e.  T ) )
14 simp21l 1072 . . . 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 `  X
) ) )  ->  N  e.  T )
15 simp22 989 . . . 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 `  X
) ) )  -> 
( P  e.  A  /\  -.  P  .<_  W ) )
16 simp23 990 . . . 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 `  X
) ) )  -> 
( R `  F
)  =  ( R `
 N ) )
1714, 15, 163jca 1132 . . 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 `  X
) ) )  -> 
( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) ) )
18 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
) )  /\  ( R `  G )  =/=  ( R `  X
) ) )  ->  F  =/=  (  _I  |`  B ) )
19 simp313 1104 . . 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 `  X
) ) )  ->  X  =/=  (  _I  |`  B ) )
20 simp32r 1081 . . 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 `  X
) ) )  -> 
( R `  X
)  =/=  ( R `
 F ) )
21 cdlemk.b . . . 4  |-  B  =  ( Base `  K
)
22 cdlemk.j . . . 4  |-  .\/  =  ( join `  K )
23 cdlemk.r . . . 4  |-  R  =  ( ( trL `  K
) `  W )
24 cdlemk.m . . . 4  |-  ./\  =  ( meet `  K )
25 cdlemk.s . . . 4  |-  S  =  ( f  e.  T  |->  ( iota_ i  e.  T
( i `  P
)  =  ( ( P  .\/  ( R `
 f ) ) 
./\  ( ( N `
 P )  .\/  ( R `  ( f  o.  `' F ) ) ) ) ) )
2621, 5, 22, 6, 7, 8, 23, 24, 25cdlemksat 31108 . . 3  |-  ( ( ( ( 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
)
2713, 17, 18, 19, 20, 26syl113anc 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
) )  /\  ( R `  G )  =/=  ( R `  X
) ) )  -> 
( ( S `  X ) `  P
)  e.  A )
28 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 `  G )  =/=  ( R `  X
) ) )  -> 
( R `  G
)  =/=  ( R `
 X ) )
2928necomd 2531 . . 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 `  X
) ) )  -> 
( R `  X
)  =/=  ( R `
 G ) )
306, 7, 8, 23trlcocnvat 30986 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  T  /\  G  e.  T )  /\  ( R `  X )  =/=  ( R `  G
) )  ->  ( R `  ( X  o.  `' G ) )  e.  A )
313, 12, 4, 29, 30syl121anc 1187 . 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
) )  /\  ( R `  G )  =/=  ( R `  X
) ) )  -> 
( R `  ( X  o.  `' G
) )  e.  A
)
32 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
) )  /\  ( R `  G )  =/=  ( R `  X
) ) )  -> 
( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T ) )
33 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
) )  /\  ( R `  G )  =/=  ( R `  X
) ) )  ->  G  =/=  (  _I  |`  B ) )
34 simp32l 1080 . . 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 `  X
) ) )  -> 
( R `  G
)  =/=  ( R `
 F ) )
3521, 5, 22, 6, 7, 8, 23, 24, 25cdlemksat 31108 . . 3  |-  ( ( ( ( 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 )  e.  A
)
3632, 17, 18, 33, 34, 35syl113anc 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
) )  /\  ( R `  G )  =/=  ( R `  X
) ) )  -> 
( ( S `  G ) `  P
)  e.  A )
3721, 5, 22, 6, 7, 8, 23, 24, 25cdlemksv2 31109 . . . . 5  |-  ( ( ( ( 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
) ) ) ) )
3832, 17, 18, 33, 34, 37syl113anc 1194 . . . 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 `  X
) ) )  -> 
( ( S `  G ) `  P
)  =  ( ( P  .\/  ( R `
 G ) ) 
./\  ( ( N `
 P )  .\/  ( R `  ( G  o.  `' F ) ) ) ) )
39 hllat 29626 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
401, 39syl 15 . . . . 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 `  G )  =/=  ( R `  X
) ) )  ->  K  e.  Lat )
4121, 6, 7, 8, 23trlnidat 30435 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T  /\  G  =/=  (  _I  |`  B ) )  ->  ( R `  G )  e.  A
)
423, 4, 33, 41syl3anc 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
) )  /\  ( R `  G )  =/=  ( R `  X
) ) )  -> 
( R `  G
)  e.  A )
4321, 22, 6hlatjcl 29629 . . . . . 6  |-  ( ( K  e.  HL  /\  P  e.  A  /\  ( R `  G )  e.  A )  -> 
( P  .\/  ( R `  G )
)  e.  B )
441, 2, 42, 43syl3anc 1182 . . . . 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 `  G )  =/=  ( R `  X
) ) )  -> 
( P  .\/  ( R `  G )
)  e.  B )
455, 6, 7, 8ltrnat 30402 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  N  e.  T  /\  P  e.  A
)  ->  ( N `  P )  e.  A
)
463, 14, 2, 45syl3anc 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
) )  /\  ( R `  G )  =/=  ( R `  X
) ) )  -> 
( N `  P
)  e.  A )
476, 7, 8, 23trlcocnvat 30986 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( G  e.  T  /\  F  e.  T )  /\  ( R `  G )  =/=  ( R `  F
) )  ->  ( R `  ( G  o.  `' F ) )  e.  A )
483, 4, 11, 34, 47syl121anc 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
) )  /\  ( R `  G )  =/=  ( R `  X
) ) )  -> 
( R `  ( G  o.  `' F
) )  e.  A
)
4921, 22, 6hlatjcl 29629 . . . . . 6  |-  ( ( K  e.  HL  /\  ( N `  P )  e.  A  /\  ( R `  ( G  o.  `' F ) )  e.  A )  ->  (
( N `  P
)  .\/  ( R `  ( G  o.  `' F ) ) )  e.  B )
501, 46, 48, 49syl3anc 1182 . . . . 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 `  G )  =/=  ( R `  X
) ) )  -> 
( ( N `  P )  .\/  ( R `  ( G  o.  `' F ) ) )  e.  B )
5121, 5, 24latmle1 14184 . . . . 5  |-  ( ( K  e.  Lat  /\  ( P  .\/  ( R `
 G ) )  e.  B  /\  (
( N `  P
)  .\/  ( R `  ( G  o.  `' F ) ) )  e.  B )  -> 
( ( P  .\/  ( R `  G ) )  ./\  ( ( N `  P )  .\/  ( R `  ( G  o.  `' F
) ) ) ) 
.<_  ( P  .\/  ( R `  G )
) )
5240, 44, 50, 51syl3anc 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
) )  /\  ( R `  G )  =/=  ( R `  X
) ) )  -> 
( ( P  .\/  ( R `  G ) )  ./\  ( ( N `  P )  .\/  ( R `  ( G  o.  `' F
) ) ) ) 
.<_  ( P  .\/  ( R `  G )
) )
5338, 52eqbrtrd 4045 . . 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 `  X
) ) )  -> 
( ( S `  G ) `  P
)  .<_  ( P  .\/  ( R `  G ) ) )
545, 22, 6, 7, 8, 23trljat1 30428 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( P  .\/  ( R `  G
) )  =  ( P  .\/  ( G `
 P ) ) )
553, 4, 15, 54syl3anc 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
) )  /\  ( R `  G )  =/=  ( R `  X
) ) )  -> 
( P  .\/  ( R `  G )
)  =  ( P 
.\/  ( G `  P ) ) )
5653, 55breqtrd 4049 . 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
) )  /\  ( R `  G )  =/=  ( R `  X
) ) )  -> 
( ( S `  G ) `  P
)  .<_  ( P  .\/  ( G `  P ) ) )
57 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
) )  /\  ( R `  G )  =/=  ( R `  X
) ) )  -> 
( ( N  e.  T  /\  X  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) ) )
58 simp31 991 . . 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 `  X
) ) )  -> 
( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  X  =/=  (  _I  |`  B ) ) )
59 eqid 2285 . . . 4  |-  ( ( ( G `  P
)  .\/  ( X `  P ) )  ./\  ( ( R `  ( G  o.  `' F ) )  .\/  ( R `  ( X  o.  `' F ) ) ) )  =  ( ( ( G `
 P )  .\/  ( X `  P ) )  ./\  ( ( R `  ( G  o.  `' F ) )  .\/  ( R `  ( X  o.  `' F ) ) ) )
6021, 5, 22, 6, 7, 8, 23, 24, 25, 59cdlemk11 31111 . . 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 )  .<_  ( ( ( S `
 X ) `  P )  .\/  ( R `  ( X  o.  `' G ) ) ) )
6132, 57, 58, 34, 20, 60syl113anc 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
) )  /\  ( R `  G )  =/=  ( R `  X
) ) )  -> 
( ( S `  G ) `  P
)  .<_  ( ( ( S `  X ) `
 P )  .\/  ( R `  ( X  o.  `' G ) ) ) )
625, 22, 6hlatlej2 29638 . . . . 5  |-  ( ( K  e.  HL  /\  P  e.  A  /\  ( R `  G )  e.  A )  -> 
( R `  G
)  .<_  ( P  .\/  ( R `  G ) ) )
631, 2, 42, 62syl3anc 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
) )  /\  ( R `  G )  =/=  ( R `  X
) ) )  -> 
( R `  G
)  .<_  ( P  .\/  ( R `  G ) ) )
6463, 55breqtrd 4049 . . 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 `  X
) ) )  -> 
( R `  G
)  .<_  ( P  .\/  ( G `  P ) ) )
6521, 5, 22, 6, 7, 8, 23, 24, 25cdlemksel 31107 . . . . . 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 )  e.  T
)
6613, 17, 18, 19, 20, 65syl113anc 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
) )  /\  ( R `  G )  =/=  ( R `  X
) ) )  -> 
( S `  X
)  e.  T )
675, 6, 7, 8ltrnel 30401 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S `  X )  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( (
( S `  X
) `  P )  e.  A  /\  -.  (
( S `  X
) `  P )  .<_  W ) )
683, 66, 15, 67syl3anc 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
) )  /\  ( R `  G )  =/=  ( R `  X
) ) )  -> 
( ( ( S `
 X ) `  P )  e.  A  /\  -.  ( ( S `
 X ) `  P )  .<_  W ) )
697, 8ltrncnv 30408 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T
)  ->  `' G  e.  T )
703, 4, 69syl2anc 642 . . . . . . 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 `  G )  =/=  ( R `  X
) ) )  ->  `' G  e.  T
)
717, 8, 23trlcnv 30427 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T
)  ->  ( R `  `' G )  =  ( R `  G ) )
723, 4, 71syl2anc 642 . . . . . . . 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
) )  /\  ( R `  G )  =/=  ( R `  X
) ) )  -> 
( R `  `' G )  =  ( R `  G ) )
7372, 28eqnetrd 2466 . . . . . . 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 `  G )  =/=  ( R `  X
) ) )  -> 
( R `  `' G )  =/=  ( R `  X )
)
7421, 7, 8, 23trlcone 30990 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( `' G  e.  T  /\  X  e.  T )  /\  (
( R `  `' G )  =/=  ( R `  X )  /\  X  =/=  (  _I  |`  B ) ) )  ->  ( R `  `' G )  =/=  ( R `  ( `' G  o.  X )
) )
753, 70, 12, 73, 19, 74syl122anc 1191 . . . . . 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
) )  /\  ( R `  G )  =/=  ( R `  X
) ) )  -> 
( R `  `' G )  =/=  ( R `  ( `' G  o.  X )
) )
7675necomd 2531 . . . . 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 `  G )  =/=  ( R `  X
) ) )  -> 
( R `  ( `' G  o.  X
) )  =/=  ( R `  `' G
) )
777, 8ltrncom 31000 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  `' G  e.  T  /\  X  e.  T )  ->  ( `' G  o.  X
)  =  ( X  o.  `' G ) )
783, 70, 12, 77syl3anc 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
) )  /\  ( R `  G )  =/=  ( R `  X
) ) )  -> 
( `' G  o.  X )  =  ( X  o.  `' G
) )
7978fveq2d 5531 . . . . 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 `  G )  =/=  ( R `  X
) ) )  -> 
( R `  ( `' G  o.  X
) )  =  ( R `  ( X  o.  `' G ) ) )
8076, 79, 723netr3d 2474 . . . 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 `  X
) ) )  -> 
( R `  ( X  o.  `' G
) )  =/=  ( R `  G )
)
817, 8ltrnco 30981 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  T  /\  `' G  e.  T
)  ->  ( X  o.  `' G )  e.  T
)
823, 12, 70, 81syl3anc 1182 . . . . 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 `  G )  =/=  ( R `  X
) ) )  -> 
( X  o.  `' G )  e.  T
)
835, 7, 8, 23trlle 30446 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  o.  `' G )  e.  T
)  ->  ( R `  ( X  o.  `' G ) )  .<_  W )
843, 82, 83syl2anc 642 . . . 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 `  X
) ) )  -> 
( R `  ( X  o.  `' G
) )  .<_  W )
855, 7, 8, 23trlle 30446 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T
)  ->  ( R `  G )  .<_  W )
863, 4, 85syl2anc 642 . . . 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 `  X
) ) )  -> 
( R `  G
)  .<_  W )
875, 22, 6, 7lhp2atnle 30295 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( ( S `  X ) `  P
)  e.  A  /\  -.  ( ( S `  X ) `  P
)  .<_  W )  /\  ( R `  ( X  o.  `' G ) )  =/=  ( R `
 G ) )  /\  ( ( R `
 ( X  o.  `' G ) )  e.  A  /\  ( R `
 ( X  o.  `' G ) )  .<_  W )  /\  (
( R `  G
)  e.  A  /\  ( R `  G ) 
.<_  W ) )  ->  -.  ( R `  G
)  .<_  ( ( ( S `  X ) `
 P )  .\/  ( R `  ( X  o.  `' G ) ) ) )
883, 68, 80, 31, 84, 42, 86, 87syl322anc 1210 . . 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 `  X
) ) )  ->  -.  ( R `  G
)  .<_  ( ( ( S `  X ) `
 P )  .\/  ( R `  ( X  o.  `' G ) ) ) )
89 nbrne1 4042 . . 3  |-  ( ( ( R `  G
)  .<_  ( P  .\/  ( G `  P ) )  /\  -.  ( R `  G )  .<_  ( ( ( S `
 X ) `  P )  .\/  ( R `  ( X  o.  `' G ) ) ) )  ->  ( P  .\/  ( G `  P
) )  =/=  (
( ( S `  X ) `  P
)  .\/  ( R `  ( X  o.  `' G ) ) ) )
9064, 88, 89syl2anc 642 . 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
) )  /\  ( R `  G )  =/=  ( R `  X
) ) )  -> 
( P  .\/  ( G `  P )
)  =/=  ( ( ( S `  X
) `  P )  .\/  ( R `  ( X  o.  `' G
) ) ) )
915, 22, 24, 62atm 29789 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  ( G `  P )  e.  A )  /\  ( ( ( S `
 X ) `  P )  e.  A  /\  ( R `  ( X  o.  `' G
) )  e.  A  /\  ( ( S `  G ) `  P
)  e.  A )  /\  ( ( ( S `  G ) `
 P )  .<_  ( P  .\/  ( G `
 P ) )  /\  ( ( S `
 G ) `  P )  .<_  ( ( ( S `  X
) `  P )  .\/  ( R `  ( X  o.  `' G
) ) )  /\  ( P  .\/  ( G `
 P ) )  =/=  ( ( ( S `  X ) `
 P )  .\/  ( R `  ( X  o.  `' G ) ) ) ) )  ->  ( ( S `
 G ) `  P )  =  ( ( P  .\/  ( G `  P )
)  ./\  ( (
( S `  X
) `  P )  .\/  ( R `  ( X  o.  `' G
) ) ) ) )
921, 2, 10, 27, 31, 36, 56, 61, 90, 91syl333anc 1214 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
) )  /\  ( R `  G )  =/=  ( R `  X
) ) )  -> 
( ( S `  G ) `  P
)  =  ( ( P  .\/  ( G `
 P ) ) 
./\  ( ( ( S `  X ) `
 P )  .\/  ( R `  ( X  o.  `' G ) ) ) ) )
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:  cdlemk21N  31135  cdlemk20  31136
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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