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Theorem cdlemg17h 30857
Description: TODO: fix comment. (Contributed by NM, 10-May-2013.)
Hypotheses
Ref Expression
cdlemg12.l  |-  .<_  =  ( le `  K )
cdlemg12.j  |-  .\/  =  ( join `  K )
cdlemg12.m  |-  ./\  =  ( meet `  K )
cdlemg12.a  |-  A  =  ( Atoms `  K )
cdlemg12.h  |-  H  =  ( LHyp `  K
)
cdlemg12.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemg12b.r  |-  R  =  ( ( trL `  K
) `  W )
Assertion
Ref Expression
cdlemg17h  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( P  =/=  Q  /\  S  .<_  ( ( F `  P ) 
.\/  ( F `  Q ) ) ) )  /\  ( ( G `  P )  =/=  P  /\  ( R `  G )  .<_  ( P  .\/  Q
)  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  ( S  =  ( F `  P
)  \/  S  =  ( F `  Q
) ) )
Distinct variable groups:    A, r    G, r    .\/ , r    .<_ , r    P, r    Q, r    W, r    F, r    S, r
Allowed substitution hints:    R( r)    T( r)    H( r)    K( r)    ./\ ( r)

Proof of Theorem cdlemg17h
StepHypRef Expression
1 simp11l 1066 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( P  =/=  Q  /\  S  .<_  ( ( F `  P ) 
.\/  ( F `  Q ) ) ) )  /\  ( ( G `  P )  =/=  P  /\  ( R `  G )  .<_  ( P  .\/  Q
)  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  K  e.  HL )
2 simp23r 1077 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( P  =/=  Q  /\  S  .<_  ( ( F `  P ) 
.\/  ( F `  Q ) ) ) )  /\  ( ( G `  P )  =/=  P  /\  ( R `  G )  .<_  ( P  .\/  Q
)  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  S  .<_  ( ( F `  P ) 
.\/  ( F `  Q ) ) )
3 simp11 985 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( P  =/=  Q  /\  S  .<_  ( ( F `  P ) 
.\/  ( F `  Q ) ) ) )  /\  ( ( G `  P )  =/=  P  /\  ( R `  G )  .<_  ( P  .\/  Q
)  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
4 simp22l 1074 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( P  =/=  Q  /\  S  .<_  ( ( F `  P ) 
.\/  ( F `  Q ) ) ) )  /\  ( ( G `  P )  =/=  P  /\  ( R `  G )  .<_  ( P  .\/  Q
)  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  F  e.  T
)
5 simp21l 1072 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( P  =/=  Q  /\  S  .<_  ( ( F `  P ) 
.\/  ( F `  Q ) ) ) )  /\  ( ( G `  P )  =/=  P  /\  ( R `  G )  .<_  ( P  .\/  Q
)  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  S  e.  A
)
6 cdlemg12.l . . . . . . . . 9  |-  .<_  =  ( le `  K )
7 cdlemg12.a . . . . . . . . 9  |-  A  =  ( Atoms `  K )
8 cdlemg12.h . . . . . . . . 9  |-  H  =  ( LHyp `  K
)
9 cdlemg12.t . . . . . . . . 9  |-  T  =  ( ( LTrn `  K
) `  W )
106, 7, 8, 9ltrncnvat 30330 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  S  e.  A
)  ->  ( `' F `  S )  e.  A )
113, 4, 5, 10syl3anc 1182 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( P  =/=  Q  /\  S  .<_  ( ( F `  P ) 
.\/  ( F `  Q ) ) ) )  /\  ( ( G `  P )  =/=  P  /\  ( R `  G )  .<_  ( P  .\/  Q
)  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  ( `' F `  S )  e.  A
)
12 eqid 2283 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  K )
1312, 7atbase 29479 . . . . . . 7  |-  ( ( `' F `  S )  e.  A  ->  ( `' F `  S )  e.  ( Base `  K
) )
1411, 13syl 15 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( P  =/=  Q  /\  S  .<_  ( ( F `  P ) 
.\/  ( F `  Q ) ) ) )  /\  ( ( G `  P )  =/=  P  /\  ( R `  G )  .<_  ( P  .\/  Q
)  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  ( `' F `  S )  e.  (
Base `  K )
)
15 simp12l 1068 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( P  =/=  Q  /\  S  .<_  ( ( F `  P ) 
.\/  ( F `  Q ) ) ) )  /\  ( ( G `  P )  =/=  P  /\  ( R `  G )  .<_  ( P  .\/  Q
)  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  P  e.  A
)
16 simp13l 1070 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( P  =/=  Q  /\  S  .<_  ( ( F `  P ) 
.\/  ( F `  Q ) ) ) )  /\  ( ( G `  P )  =/=  P  /\  ( R `  G )  .<_  ( P  .\/  Q
)  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  Q  e.  A
)
17 cdlemg12.j . . . . . . . 8  |-  .\/  =  ( join `  K )
1812, 17, 7hlatjcl 29556 . . . . . . 7  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
191, 15, 16, 18syl3anc 1182 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( P  =/=  Q  /\  S  .<_  ( ( F `  P ) 
.\/  ( F `  Q ) ) ) )  /\  ( ( G `  P )  =/=  P  /\  ( R `  G )  .<_  ( P  .\/  Q
)  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  ( P  .\/  Q )  e.  ( Base `  K ) )
2012, 6, 8, 9ltrnle 30318 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( ( `' F `  S )  e.  (
Base `  K )  /\  ( P  .\/  Q
)  e.  ( Base `  K ) ) )  ->  ( ( `' F `  S ) 
.<_  ( P  .\/  Q
)  <->  ( F `  ( `' F `  S ) )  .<_  ( F `  ( P  .\/  Q
) ) ) )
213, 4, 14, 19, 20syl112anc 1186 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( P  =/=  Q  /\  S  .<_  ( ( F `  P ) 
.\/  ( F `  Q ) ) ) )  /\  ( ( G `  P )  =/=  P  /\  ( R `  G )  .<_  ( P  .\/  Q
)  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  ( ( `' F `  S ) 
.<_  ( P  .\/  Q
)  <->  ( F `  ( `' F `  S ) )  .<_  ( F `  ( P  .\/  Q
) ) ) )
2212, 8, 9ltrn1o 30313 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  F :
( Base `  K ) -1-1-onto-> ( Base `  K ) )
233, 4, 22syl2anc 642 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( P  =/=  Q  /\  S  .<_  ( ( F `  P ) 
.\/  ( F `  Q ) ) ) )  /\  ( ( G `  P )  =/=  P  /\  ( R `  G )  .<_  ( P  .\/  Q
)  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  F : (
Base `  K ) -1-1-onto-> ( Base `  K ) )
2412, 7atbase 29479 . . . . . . . 8  |-  ( S  e.  A  ->  S  e.  ( Base `  K
) )
255, 24syl 15 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( P  =/=  Q  /\  S  .<_  ( ( F `  P ) 
.\/  ( F `  Q ) ) ) )  /\  ( ( G `  P )  =/=  P  /\  ( R `  G )  .<_  ( P  .\/  Q
)  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  S  e.  (
Base `  K )
)
26 f1ocnvfv2 5793 . . . . . . 7  |-  ( ( F : ( Base `  K ) -1-1-onto-> ( Base `  K
)  /\  S  e.  ( Base `  K )
)  ->  ( F `  ( `' F `  S ) )  =  S )
2723, 25, 26syl2anc 642 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( P  =/=  Q  /\  S  .<_  ( ( F `  P ) 
.\/  ( F `  Q ) ) ) )  /\  ( ( G `  P )  =/=  P  /\  ( R `  G )  .<_  ( P  .\/  Q
)  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  ( F `  ( `' F `  S ) )  =  S )
2812, 7atbase 29479 . . . . . . . 8  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
2915, 28syl 15 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( P  =/=  Q  /\  S  .<_  ( ( F `  P ) 
.\/  ( F `  Q ) ) ) )  /\  ( ( G `  P )  =/=  P  /\  ( R `  G )  .<_  ( P  .\/  Q
)  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  P  e.  (
Base `  K )
)
3012, 7atbase 29479 . . . . . . . 8  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
3116, 30syl 15 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( P  =/=  Q  /\  S  .<_  ( ( F `  P ) 
.\/  ( F `  Q ) ) ) )  /\  ( ( G `  P )  =/=  P  /\  ( R `  G )  .<_  ( P  .\/  Q
)  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  Q  e.  (
Base `  K )
)
3212, 17, 8, 9ltrnj 30321 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  (
Base `  K )  /\  Q  e.  ( Base `  K ) ) )  ->  ( F `  ( P  .\/  Q
) )  =  ( ( F `  P
)  .\/  ( F `  Q ) ) )
333, 4, 29, 31, 32syl112anc 1186 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( P  =/=  Q  /\  S  .<_  ( ( F `  P ) 
.\/  ( F `  Q ) ) ) )  /\  ( ( G `  P )  =/=  P  /\  ( R `  G )  .<_  ( P  .\/  Q
)  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  ( F `  ( P  .\/  Q ) )  =  ( ( F `  P ) 
.\/  ( F `  Q ) ) )
3427, 33breq12d 4036 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( P  =/=  Q  /\  S  .<_  ( ( F `  P ) 
.\/  ( F `  Q ) ) ) )  /\  ( ( G `  P )  =/=  P  /\  ( R `  G )  .<_  ( P  .\/  Q
)  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  ( ( F `
 ( `' F `  S ) )  .<_  ( F `  ( P 
.\/  Q ) )  <-> 
S  .<_  ( ( F `
 P )  .\/  ( F `  Q ) ) ) )
3521, 34bitr2d 245 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( P  =/=  Q  /\  S  .<_  ( ( F `  P ) 
.\/  ( F `  Q ) ) ) )  /\  ( ( G `  P )  =/=  P  /\  ( R `  G )  .<_  ( P  .\/  Q
)  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  ( S  .<_  ( ( F `  P
)  .\/  ( F `  Q ) )  <->  ( `' F `  S )  .<_  ( P  .\/  Q
) ) )
362, 35mpbid 201 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( P  =/=  Q  /\  S  .<_  ( ( F `  P ) 
.\/  ( F `  Q ) ) ) )  /\  ( ( G `  P )  =/=  P  /\  ( R `  G )  .<_  ( P  .\/  Q
)  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  ( `' F `  S )  .<_  ( P 
.\/  Q ) )
37 simp33 993 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( P  =/=  Q  /\  S  .<_  ( ( F `  P ) 
.\/  ( F `  Q ) ) ) )  /\  ( ( G `  P )  =/=  P  /\  ( R `  G )  .<_  ( P  .\/  Q
)  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) )
38 simp23l 1076 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( P  =/=  Q  /\  S  .<_  ( ( F `  P ) 
.\/  ( F `  Q ) ) ) )  /\  ( ( G `  P )  =/=  P  /\  ( R `  G )  .<_  ( P  .\/  Q
)  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  P  =/=  Q
)
39 simp21 988 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( P  =/=  Q  /\  S  .<_  ( ( F `  P ) 
.\/  ( F `  Q ) ) ) )  /\  ( ( G `  P )  =/=  P  /\  ( R `  G )  .<_  ( P  .\/  Q
)  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  ( S  e.  A  /\  -.  S  .<_  W ) )
406, 7, 8, 9ltrncnvel 30331 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  ( ( `' F `  S )  e.  A  /\  -.  ( `' F `  S ) 
.<_  W ) )
413, 4, 39, 40syl3anc 1182 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( P  =/=  Q  /\  S  .<_  ( ( F `  P ) 
.\/  ( F `  Q ) ) ) )  /\  ( ( G `  P )  =/=  P  /\  ( R `  G )  .<_  ( P  .\/  Q
)  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  ( ( `' F `  S )  e.  A  /\  -.  ( `' F `  S ) 
.<_  W ) )
426, 17, 7cdleme0nex 30479 . . 3  |-  ( ( ( K  e.  HL  /\  ( `' F `  S )  .<_  ( P 
.\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q )  /\  ( ( `' F `  S )  e.  A  /\  -.  ( `' F `  S )  .<_  W ) )  ->  ( ( `' F `  S )  =  P  \/  ( `' F `  S )  =  Q ) )
431, 36, 37, 15, 16, 38, 41, 42syl331anc 1207 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( P  =/=  Q  /\  S  .<_  ( ( F `  P ) 
.\/  ( F `  Q ) ) ) )  /\  ( ( G `  P )  =/=  P  /\  ( R `  G )  .<_  ( P  .\/  Q
)  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  ( ( `' F `  S )  =  P  \/  ( `' F `  S )  =  Q ) )
44 eqcom 2285 . . . 4  |-  ( ( F `  P )  =  S  <->  S  =  ( F `  P ) )
45 f1ocnvfvb 5795 . . . . 5  |-  ( ( F : ( Base `  K ) -1-1-onto-> ( Base `  K
)  /\  P  e.  ( Base `  K )  /\  S  e.  ( Base `  K ) )  ->  ( ( F `
 P )  =  S  <->  ( `' F `  S )  =  P ) )
4623, 29, 25, 45syl3anc 1182 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( P  =/=  Q  /\  S  .<_  ( ( F `  P ) 
.\/  ( F `  Q ) ) ) )  /\  ( ( G `  P )  =/=  P  /\  ( R `  G )  .<_  ( P  .\/  Q
)  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  ( ( F `
 P )  =  S  <->  ( `' F `  S )  =  P ) )
4744, 46syl5rbbr 251 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( P  =/=  Q  /\  S  .<_  ( ( F `  P ) 
.\/  ( F `  Q ) ) ) )  /\  ( ( G `  P )  =/=  P  /\  ( R `  G )  .<_  ( P  .\/  Q
)  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  ( ( `' F `  S )  =  P  <->  S  =  ( F `  P ) ) )
48 eqcom 2285 . . . 4  |-  ( ( F `  Q )  =  S  <->  S  =  ( F `  Q ) )
49 f1ocnvfvb 5795 . . . . 5  |-  ( ( F : ( Base `  K ) -1-1-onto-> ( Base `  K
)  /\  Q  e.  ( Base `  K )  /\  S  e.  ( Base `  K ) )  ->  ( ( F `
 Q )  =  S  <->  ( `' F `  S )  =  Q ) )
5023, 31, 25, 49syl3anc 1182 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( P  =/=  Q  /\  S  .<_  ( ( F `  P ) 
.\/  ( F `  Q ) ) ) )  /\  ( ( G `  P )  =/=  P  /\  ( R `  G )  .<_  ( P  .\/  Q
)  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  ( ( F `
 Q )  =  S  <->  ( `' F `  S )  =  Q ) )
5148, 50syl5rbbr 251 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( P  =/=  Q  /\  S  .<_  ( ( F `  P ) 
.\/  ( F `  Q ) ) ) )  /\  ( ( G `  P )  =/=  P  /\  ( R `  G )  .<_  ( P  .\/  Q
)  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  ( ( `' F `  S )  =  Q  <->  S  =  ( F `  Q ) ) )
5247, 51orbi12d 690 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( P  =/=  Q  /\  S  .<_  ( ( F `  P ) 
.\/  ( F `  Q ) ) ) )  /\  ( ( G `  P )  =/=  P  /\  ( R `  G )  .<_  ( P  .\/  Q
)  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  ( ( ( `' F `  S )  =  P  \/  ( `' F `  S )  =  Q )  <->  ( S  =  ( F `  P )  \/  S  =  ( F `  Q ) ) ) )
5343, 52mpbid 201 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( P  =/=  Q  /\  S  .<_  ( ( F `  P ) 
.\/  ( F `  Q ) ) ) )  /\  ( ( G `  P )  =/=  P  /\  ( R `  G )  .<_  ( P  .\/  Q
)  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  ( S  =  ( F `  P
)  \/  S  =  ( F `  Q
) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544   class class class wbr 4023   `'ccnv 4688   -1-1-onto->wf1o 5254   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215   joincjn 14078   meetcmee 14079   Atomscatm 29453   HLchlt 29540   LHypclh 30173   LTrncltrn 30290   trLctrl 30347
This theorem is referenced by:  cdlemg17i  30858
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-map 6774  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-p0 14145  df-lat 14152  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-lhyp 30177  df-laut 30178  df-ldil 30293  df-ltrn 30294
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