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Theorem cdlemk12u 34414
Description: Part of proof of Lemma K of [Crawley] p. 118. Line 18, p. 119, showing Eq. 4 (line 10, p. 119) for the sigma1 ( U) case. (Contributed by NM, 4-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 ) ) ) ) ) )
Assertion
Ref Expression
cdlemk12u  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( X  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  X )
)  /\  ( ( R `  D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  (
( U `  G
) `  P )  =  ( ( P 
.\/  ( G `  P ) )  ./\  ( ( ( U `
 X ) `  P )  .\/  ( R `  ( X  o.  `' G ) ) ) ) )
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, G, 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    e, X, j
Allowed substitution hints:    A( e, f)    B( e, f, i, j)    S( e, f, i, j)    U( e, f, i, j)    G( f, i)    H( e, f)    K( e, f)    .<_ ( e, f)    N( e)    O( f, i)    X( f, i)

Proof of Theorem cdlemk12u
StepHypRef Expression
1 simp11l 1117 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( X  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  X )
)  /\  ( ( R `  D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  K  e.  HL )
2 simp22l 1125 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( X  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  X )
)  /\  ( ( R `  D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  P  e.  A )
3 simp11 1036 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( X  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  X )
)  /\  ( ( R `  D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
4 simp212 1145 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( X  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  X )
)  /\  ( ( R `  D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  G  e.  T )
5 cdlemk1.l . . . 4  |-  .<_  =  ( le `  K )
6 cdlemk1.a . . . 4  |-  A  =  ( Atoms `  K )
7 cdlemk1.h . . . 4  |-  H  =  ( LHyp `  K
)
8 cdlemk1.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
95, 6, 7, 8ltrnat 33680 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T  /\  P  e.  A
)  ->  ( G `  P )  e.  A
)
103, 4, 2, 9syl3anc 1265 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( X  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  X )
)  /\  ( ( R `  D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  ( G `  P )  e.  A )
11 simp23 1041 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( X  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  X )
)  /\  ( ( R `  D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  ( R `  F )  =  ( R `  N ) )
12 simp213 1146 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( X  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  X )
)  /\  ( ( R `  D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  X  e.  T )
13 simp12 1037 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( X  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  X )
)  /\  ( ( R `  D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  F  e.  T )
14 simp13 1038 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( X  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  X )
)  /\  ( ( R `  D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  D  e.  T )
15 simp211 1144 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( X  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  X )
)  /\  ( ( R `  D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  N  e.  T )
16 simp331 1159 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( X  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  X )
)  /\  ( ( R `  D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  ( R `  D )  =/=  ( R `  F
) )
17 simp333 1161 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( X  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  X )
)  /\  ( ( R `  D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  ( R `  X )  =/=  ( R `  D
) )
1817necomd 2696 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( X  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  X )
)  /\  ( ( R `  D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  ( R `  D )  =/=  ( R `  X
) )
1916, 18jca 535 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( X  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  X )
)  /\  ( ( R `  D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  (
( R `  D
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  X ) ) )
20 simp311 1153 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( X  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  X )
)  /\  ( ( R `  D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  F  =/=  (  _I  |`  B ) )
21 simp32l 1131 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( X  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  X )
)  /\  ( ( R `  D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  X  =/=  (  _I  |`  B ) )
22 simp312 1154 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( X  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  X )
)  /\  ( ( R `  D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  D  =/=  (  _I  |`  B ) )
2320, 21, 223jca 1186 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( X  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  X )
)  /\  ( ( R `  D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  ( F  =/=  (  _I  |`  B )  /\  X  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) ) )
24 simp22 1040 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( X  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  X )
)  /\  ( ( R `  D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
25 cdlemk1.b . . . 4  |-  B  =  ( Base `  K
)
26 cdlemk1.j . . . 4  |-  .\/  =  ( join `  K )
27 cdlemk1.m . . . 4  |-  ./\  =  ( meet `  K )
28 cdlemk1.r . . . 4  |-  R  =  ( ( trL `  K
) `  W )
29 cdlemk1.s . . . 4  |-  S  =  ( f  e.  T  |->  ( iota_ i  e.  T  ( i `  P
)  =  ( ( P  .\/  ( R `
 f ) ) 
./\  ( ( N `
 P )  .\/  ( R `  ( f  o.  `' F ) ) ) ) ) )
30 cdlemk1.o . . . 4  |-  O  =  ( S `  D
)
31 cdlemk1.u . . . 4  |-  U  =  ( e  e.  T  |->  ( iota_ j  e.  T  ( j `  P
)  =  ( ( P  .\/  ( R `
 e ) ) 
./\  ( ( O `
 P )  .\/  ( R `  ( e  o.  `' D ) ) ) ) ) )
3225, 5, 26, 27, 6, 7, 8, 28, 29, 30, 31cdlemkuat 34408 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N )  /\  X  e.  T )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T )  /\  ( ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  D )  =/=  ( R `  X )
)  /\  ( F  =/=  (  _I  |`  B )  /\  X  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( ( U `  X ) `  P )  e.  A
)
333, 11, 12, 13, 14, 15, 19, 23, 24, 32syl333anc 1297 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( X  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  X )
)  /\  ( ( R `  D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  (
( U `  X
) `  P )  e.  A )
34 simp32r 1132 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( X  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  X )
)  /\  ( ( R `  D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  ( R `  G )  =/=  ( R `  X
) )
3534necomd 2696 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( X  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  X )
)  /\  ( ( R `  D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  ( R `  X )  =/=  ( R `  G
) )
366, 7, 8, 28trlcocnvat 34266 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  T  /\  G  e.  T )  /\  ( R `  X )  =/=  ( R `  G
) )  ->  ( R `  ( X  o.  `' G ) )  e.  A )
373, 12, 4, 35, 36syl121anc 1270 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( X  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  X )
)  /\  ( ( R `  D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  ( R `  ( X  o.  `' G ) )  e.  A )
38 simp332 1160 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( X  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  X )
)  /\  ( ( R `  D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  ( R `  G )  =/=  ( R `  D
) )
3938necomd 2696 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( X  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  X )
)  /\  ( ( R `  D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  ( R `  D )  =/=  ( R `  G
) )
4016, 39jca 535 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( X  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  X )
)  /\  ( ( R `  D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  (
( R `  D
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  G ) ) )
41 simp313 1155 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( X  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  X )
)  /\  ( ( R `  D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  G  =/=  (  _I  |`  B ) )
4220, 41, 223jca 1186 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( X  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  X )
)  /\  ( ( R `  D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) ) )
4325, 5, 26, 27, 6, 7, 8, 28, 29, 30, 31cdlemkuat 34408 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N )  /\  G  e.  T )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T )  /\  ( ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  D )  =/=  ( R `  G )
)  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( ( U `  G ) `  P )  e.  A
)
443, 11, 4, 13, 14, 15, 40, 42, 24, 43syl333anc 1297 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( X  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  X )
)  /\  ( ( R `  D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  (
( U `  G
) `  P )  e.  A )
4525, 5, 26, 27, 6, 7, 8, 28, 29, 30, 31cdlemkuv2 34409 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N )  /\  G  e.  T )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T )  /\  ( ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  D )  =/=  ( R `  G )
)  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( ( U `  G ) `  P )  =  ( ( P  .\/  ( R `  G )
)  ./\  ( ( O `  P )  .\/  ( R `  ( G  o.  `' D
) ) ) ) )
463, 11, 4, 13, 14, 15, 40, 42, 24, 45syl333anc 1297 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( X  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  X )
)  /\  ( ( R `  D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  (
( U `  G
) `  P )  =  ( ( P 
.\/  ( R `  G ) )  ./\  ( ( O `  P )  .\/  ( R `  ( G  o.  `' D ) ) ) ) )
47 hllat 32904 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
481, 47syl 17 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( X  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  X )
)  /\  ( ( R `  D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  K  e.  Lat )
4925, 6, 7, 8, 28trlnidat 33714 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T  /\  G  =/=  (  _I  |`  B ) )  ->  ( R `  G )  e.  A
)
503, 4, 41, 49syl3anc 1265 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( X  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  X )
)  /\  ( ( R `  D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  ( R `  G )  e.  A )
5125, 26, 6hlatjcl 32907 . . . . . 6  |-  ( ( K  e.  HL  /\  P  e.  A  /\  ( R `  G )  e.  A )  -> 
( P  .\/  ( R `  G )
)  e.  B )
521, 2, 50, 51syl3anc 1265 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( X  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  X )
)  /\  ( ( R `  D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  ( P  .\/  ( R `  G ) )  e.  B )
53 simp1 1006 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( X  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  X )
)  /\  ( ( R `  D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  (
( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T
) )
5415, 24, 113jca 1186 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( X  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  X )
)  /\  ( ( R `  D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) ) )
5525, 5, 26, 27, 6, 7, 8, 28, 29, 30cdlemkoatnle 34393 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  ( R `  D )  =/=  ( R `  F ) ) )  ->  ( ( O `
 P )  e.  A  /\  -.  ( O `  P )  .<_  W ) )
5655simpld 461 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  ( R `  D )  =/=  ( R `  F ) ) )  ->  ( O `  P )  e.  A
)
5753, 54, 20, 22, 16, 56syl113anc 1277 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( X  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  X )
)  /\  ( ( R `  D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  ( O `  P )  e.  A )
586, 7, 8, 28trlcocnvat 34266 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( G  e.  T  /\  D  e.  T )  /\  ( R `  G )  =/=  ( R `  D
) )  ->  ( R `  ( G  o.  `' D ) )  e.  A )
593, 4, 14, 38, 58syl121anc 1270 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( X  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  X )
)  /\  ( ( R `  D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  ( R `  ( G  o.  `' D ) )  e.  A )
6025, 26, 6hlatjcl 32907 . . . . . 6  |-  ( ( K  e.  HL  /\  ( O `  P )  e.  A  /\  ( R `  ( G  o.  `' D ) )  e.  A )  ->  (
( O `  P
)  .\/  ( R `  ( G  o.  `' D ) ) )  e.  B )
611, 57, 59, 60syl3anc 1265 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( X  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  X )
)  /\  ( ( R `  D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  (
( O `  P
)  .\/  ( R `  ( G  o.  `' D ) ) )  e.  B )
6225, 5, 27latmle1 16327 . . . . 5  |-  ( ( K  e.  Lat  /\  ( P  .\/  ( R `
 G ) )  e.  B  /\  (
( O `  P
)  .\/  ( R `  ( G  o.  `' D ) ) )  e.  B )  -> 
( ( P  .\/  ( R `  G ) )  ./\  ( ( O `  P )  .\/  ( R `  ( G  o.  `' D
) ) ) ) 
.<_  ( P  .\/  ( R `  G )
) )
6348, 52, 61, 62syl3anc 1265 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( X  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  X )
)  /\  ( ( R `  D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  (
( P  .\/  ( R `  G )
)  ./\  ( ( O `  P )  .\/  ( R `  ( G  o.  `' D
) ) ) ) 
.<_  ( P  .\/  ( R `  G )
) )
6446, 63eqbrtrd 4450 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( X  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  X )
)  /\  ( ( R `  D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  (
( U `  G
) `  P )  .<_  ( P  .\/  ( R `  G )
) )
655, 26, 6, 7, 8, 28trljat1 33707 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( P  .\/  ( R `  G
) )  =  ( P  .\/  ( G `
 P ) ) )
663, 4, 24, 65syl3anc 1265 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( X  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  X )
)  /\  ( ( R `  D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  ( P  .\/  ( R `  G ) )  =  ( P  .\/  ( G `  P )
) )
6764, 66breqtrd 4454 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( X  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  X )
)  /\  ( ( R `  D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  (
( U `  G
) `  P )  .<_  ( P  .\/  ( G `  P )
) )
68 simp2 1007 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( X  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  X )
)  /\  ( ( R `  D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) ) )
69 simp31 1042 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( X  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  X )
)  /\  ( ( R `  D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) ) )
70 simp33 1044 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( X  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  X )
)  /\  ( ( R `  D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  (
( R `  D
)  =/=  ( R `
 F )  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X )  =/=  ( R `  D
) ) )
71 eqid 2423 . . . 4  |-  ( ( ( G `  P
)  .\/  ( X `  P ) )  ./\  ( ( R `  ( G  o.  `' D ) )  .\/  ( R `  ( X  o.  `' D ) ) ) )  =  ( ( ( G `
 P )  .\/  ( X `  P ) )  ./\  ( ( R `  ( G  o.  `' D ) )  .\/  ( R `  ( X  o.  `' D ) ) ) )
7225, 5, 26, 27, 6, 7, 8, 28, 29, 30, 31, 71cdlemk11u 34413 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  X  =/=  (  _I  |`  B )  /\  ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  (
( U `  G
) `  P )  .<_  ( ( ( U `
 X ) `  P )  .\/  ( R `  ( X  o.  `' G ) ) ) )
7353, 68, 69, 21, 70, 72syl113anc 1277 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( X  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  X )
)  /\  ( ( R `  D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  (
( U `  G
) `  P )  .<_  ( ( ( U `
 X ) `  P )  .\/  ( R `  ( X  o.  `' G ) ) ) )
745, 26, 6hlatlej2 32916 . . . . 5  |-  ( ( K  e.  HL  /\  P  e.  A  /\  ( R `  G )  e.  A )  -> 
( R `  G
)  .<_  ( P  .\/  ( R `  G ) ) )
751, 2, 50, 74syl3anc 1265 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( X  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  X )
)  /\  ( ( R `  D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  ( R `  G )  .<_  ( P  .\/  ( R `  G )
) )
7675, 66breqtrd 4454 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( X  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  X )
)  /\  ( ( R `  D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  ( R `  G )  .<_  ( P  .\/  ( G `  P )
) )
7725, 5, 26, 27, 6, 7, 8, 28, 29, 30, 31cdlemkuel 34407 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N )  /\  X  e.  T )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T )  /\  ( ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  D )  =/=  ( R `  X )
)  /\  ( F  =/=  (  _I  |`  B )  /\  X  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( U `  X )  e.  T
)
783, 11, 12, 13, 14, 15, 19, 23, 24, 77syl333anc 1297 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( X  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  X )
)  /\  ( ( R `  D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  ( U `  X )  e.  T )
795, 6, 7, 8ltrnel 33679 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U `  X )  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( (
( U `  X
) `  P )  e.  A  /\  -.  (
( U `  X
) `  P )  .<_  W ) )
803, 78, 24, 79syl3anc 1265 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( X  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  X )
)  /\  ( ( R `  D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  (
( ( U `  X ) `  P
)  e.  A  /\  -.  ( ( U `  X ) `  P
)  .<_  W ) )
817, 8ltrncnv 33686 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T
)  ->  `' G  e.  T )
823, 4, 81syl2anc 666 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( X  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  X )
)  /\  ( ( R `  D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  `' G  e.  T )
837, 8, 28trlcnv 33706 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T
)  ->  ( R `  `' G )  =  ( R `  G ) )
843, 4, 83syl2anc 666 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( X  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  X )
)  /\  ( ( R `  D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  ( R `  `' G
)  =  ( R `
 G ) )
8584, 34eqnetrd 2718 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( X  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  X )
)  /\  ( ( R `  D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  ( R `  `' G
)  =/=  ( R `
 X ) )
8625, 7, 8, 28trlcone 34270 . . . . . . 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 )
) )
873, 82, 12, 85, 21, 86syl122anc 1274 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( X  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  X )
)  /\  ( ( R `  D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  ( R `  `' G
)  =/=  ( R `
 ( `' G  o.  X ) ) )
8887necomd 2696 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( X  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  X )
)  /\  ( ( R `  D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  ( R `  ( `' G  o.  X )
)  =/=  ( R `
 `' G ) )
897, 8ltrncom 34280 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  `' G  e.  T  /\  X  e.  T )  ->  ( `' G  o.  X
)  =  ( X  o.  `' G ) )
903, 82, 12, 89syl3anc 1265 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( X  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  X )
)  /\  ( ( R `  D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  ( `' G  o.  X
)  =  ( X  o.  `' G ) )
9190fveq2d 5891 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( X  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  X )
)  /\  ( ( R `  D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  ( R `  ( `' G  o.  X )
)  =  ( R `
 ( X  o.  `' G ) ) )
9288, 91, 843netr3d 2728 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( X  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  X )
)  /\  ( ( R `  D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  ( R `  ( X  o.  `' G ) )  =/=  ( R `  G
) )
937, 8ltrnco 34261 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  T  /\  `' G  e.  T
)  ->  ( X  o.  `' G )  e.  T
)
943, 12, 82, 93syl3anc 1265 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( X  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  X )
)  /\  ( ( R `  D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  ( X  o.  `' G
)  e.  T )
955, 7, 8, 28trlle 33725 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  o.  `' G )  e.  T
)  ->  ( R `  ( X  o.  `' G ) )  .<_  W )
963, 94, 95syl2anc 666 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( X  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  X )
)  /\  ( ( R `  D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  ( R `  ( X  o.  `' G ) )  .<_  W )
975, 7, 8, 28trlle 33725 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T
)  ->  ( R `  G )  .<_  W )
983, 4, 97syl2anc 666 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( X  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  X )
)  /\  ( ( R `  D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  ( R `  G )  .<_  W )
995, 26, 6, 7lhp2atnle 33573 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( ( U `  X ) `  P
)  e.  A  /\  -.  ( ( U `  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
)  .<_  ( ( ( U `  X ) `
 P )  .\/  ( R `  ( X  o.  `' G ) ) ) )
1003, 80, 92, 37, 96, 50, 98, 99syl322anc 1293 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( X  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  X )
)  /\  ( ( R `  D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  -.  ( R `  G ) 
.<_  ( ( ( U `
 X ) `  P )  .\/  ( R `  ( X  o.  `' G ) ) ) )
101 nbrne1 4447 . . 3  |-  ( ( ( R `  G
)  .<_  ( P  .\/  ( G `  P ) )  /\  -.  ( R `  G )  .<_  ( ( ( U `
 X ) `  P )  .\/  ( R `  ( X  o.  `' G ) ) ) )  ->  ( P  .\/  ( G `  P
) )  =/=  (
( ( U `  X ) `  P
)  .\/  ( R `  ( X  o.  `' G ) ) ) )
10276, 100, 101syl2anc 666 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( X  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  X )
)  /\  ( ( R `  D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  ( P  .\/  ( G `  P ) )  =/=  ( ( ( U `
 X ) `  P )  .\/  ( R `  ( X  o.  `' G ) ) ) )
1035, 26, 27, 62atm 33067 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  ( G `  P )  e.  A )  /\  ( ( ( U `
 X ) `  P )  e.  A  /\  ( R `  ( X  o.  `' G
) )  e.  A  /\  ( ( U `  G ) `  P
)  e.  A )  /\  ( ( ( U `  G ) `
 P )  .<_  ( P  .\/  ( G `
 P ) )  /\  ( ( U `
 G ) `  P )  .<_  ( ( ( U `  X
) `  P )  .\/  ( R `  ( X  o.  `' G
) ) )  /\  ( P  .\/  ( G `
 P ) )  =/=  ( ( ( U `  X ) `
 P )  .\/  ( R `  ( X  o.  `' G ) ) ) ) )  ->  ( ( U `
 G ) `  P )  =  ( ( P  .\/  ( G `  P )
)  ./\  ( (
( U `  X
) `  P )  .\/  ( R `  ( X  o.  `' G
) ) ) ) )
1041, 2, 10, 33, 37, 44, 67, 73, 102, 103syl333anc 1297 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( X  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  X )
)  /\  ( ( R `  D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  (
( U `  G
) `  P )  =  ( ( P 
.\/  ( G `  P ) )  ./\  ( ( ( U `
 X ) `  P )  .\/  ( R `  ( X  o.  `' G ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 371    /\ w3a 983    = wceq 1438    e. wcel 1873    =/= wne 2619   class class class wbr 4429    |-> cmpt 4488    _I cid 4769   `'ccnv 4858    |` cres 4861    o. ccom 4863   ` cfv 5607   iota_crio 6272  (class class class)co 6311   Basecbs 15126   lecple 15202   joincjn 16194   meetcmee 16195   Latclat 16296   Atomscatm 32804   HLchlt 32891   LHypclh 33524   LTrncltrn 33641   trLctrl 33699
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1664  ax-4 1677  ax-5 1753  ax-6 1799  ax-7 1844  ax-8 1875  ax-9 1877  ax-10 1892  ax-11 1897  ax-12 1910  ax-13 2058  ax-ext 2402  ax-rep 4542  ax-sep 4552  ax-nul 4561  ax-pow 4608  ax-pr 4666  ax-un 6603  ax-riotaBAD 32500
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-ex 1659  df-nf 1663  df-sb 1792  df-eu 2274  df-mo 2275  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-nel 2622  df-ral 2781  df-rex 2782  df-reu 2783  df-rmo 2784  df-rab 2785  df-v 3087  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3918  df-pw 3989  df-sn 4005  df-pr 4007  df-op 4011  df-uni 4226  df-iun 4307  df-iin 4308  df-br 4430  df-opab 4489  df-mpt 4490  df-id 4774  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-rn 4870  df-res 4871  df-ima 4872  df-iota 5571  df-fun 5609  df-fn 5610  df-f 5611  df-f1 5612  df-fo 5613  df-f1o 5614  df-fv 5615  df-riota 6273  df-ov 6314  df-oprab 6315  df-mpt2 6316  df-1st 6813  df-2nd 6814  df-undef 7037  df-map 7491  df-preset 16178  df-poset 16196  df-plt 16209  df-lub 16225  df-glb 16226  df-join 16227  df-meet 16228  df-p0 16290  df-p1 16291  df-lat 16297  df-clat 16359  df-oposet 32717  df-ol 32719  df-oml 32720  df-covers 32807  df-ats 32808  df-atl 32839  df-cvlat 32863  df-hlat 32892  df-llines 33038  df-lplanes 33039  df-lvols 33040  df-lines 33041  df-psubsp 33043  df-pmap 33044  df-padd 33336  df-lhyp 33528  df-laut 33529  df-ldil 33644  df-ltrn 33645  df-trl 33700
This theorem is referenced by:  cdlemk12u-2N  34432  cdlemk22  34435
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