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Theorem cdlemg17h 31465
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 1068 . . 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 1079 . . . 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 987 . . . . . 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 1076 . . . . . 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 1074 . . . . . . . 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 30938 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  S  e.  A
)  ->  ( `' F `  S )  e.  A )
113, 4, 5, 10syl3anc 1184 . . . . . . 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 2436 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  K )
1312, 7atbase 30087 . . . . . . 7  |-  ( ( `' F `  S )  e.  A  ->  ( `' F `  S )  e.  ( Base `  K
) )
1411, 13syl 16 . . . . . 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 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
) ) ) )  ->  P  e.  A
)
16 simp13l 1072 . . . . . . 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 30164 . . . . . . 7  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
191, 15, 16, 18syl3anc 1184 . . . . . 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 30926 . . . . . 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 1188 . . . . 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 30921 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  F :
( Base `  K ) -1-1-onto-> ( Base `  K ) )
233, 4, 22syl2anc 643 . . . . . . 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 30087 . . . . . . . 8  |-  ( S  e.  A  ->  S  e.  ( Base `  K
) )
255, 24syl 16 . . . . . . 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 6015 . . . . . . 7  |-  ( ( F : ( Base `  K ) -1-1-onto-> ( Base `  K
)  /\  S  e.  ( Base `  K )
)  ->  ( F `  ( `' F `  S ) )  =  S )
2723, 25, 26syl2anc 643 . . . . . 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 30087 . . . . . . . 8  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
2915, 28syl 16 . . . . . . 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 30087 . . . . . . . 8  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
3116, 30syl 16 . . . . . . 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 30929 . . . . . . 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 1188 . . . . . 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 4225 . . . . 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 246 . . . 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 202 . . 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 995 . . 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 1078 . . 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 990 . . . 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 30939 . . . 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 1184 . . 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 31087 . . 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 1209 . 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 2438 . . . 4  |-  ( ( F `  P )  =  S  <->  S  =  ( F `  P ) )
45 f1ocnvfvb 6017 . . . . 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 1184 . . . 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 252 . . 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 2438 . . . 4  |-  ( ( F `  Q )  =  S  <->  S  =  ( F `  Q ) )
49 f1ocnvfvb 6017 . . . . 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 1184 . . . 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 252 . . 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 691 . 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 202 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 177    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2599   E.wrex 2706   class class class wbr 4212   `'ccnv 4877   -1-1-onto->wf1o 5453   ` cfv 5454  (class class class)co 6081   Basecbs 13469   lecple 13536   joincjn 14401   meetcmee 14402   Atomscatm 30061   HLchlt 30148   LHypclh 30781   LTrncltrn 30898   trLctrl 30955
This theorem is referenced by:  cdlemg17i  31466
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-undef 6543  df-riota 6549  df-map 7020  df-poset 14403  df-plt 14415  df-lub 14431  df-glb 14432  df-join 14433  df-p0 14468  df-lat 14475  df-oposet 29974  df-ol 29976  df-oml 29977  df-covers 30064  df-ats 30065  df-atl 30096  df-cvlat 30120  df-hlat 30149  df-lhyp 30785  df-laut 30786  df-ldil 30901  df-ltrn 30902
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