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Theorem cdleme7ga 35675
Description: Part of proof of Lemma E in [Crawley] p. 113. See cdleme7 35676. (Contributed by NM, 8-Jun-2012.)
Hypotheses
Ref Expression
cdleme4.l  |-  .<_  =  ( le `  K )
cdleme4.j  |-  .\/  =  ( join `  K )
cdleme4.m  |-  ./\  =  ( meet `  K )
cdleme4.a  |-  A  =  ( Atoms `  K )
cdleme4.h  |-  H  =  ( LHyp `  K
)
cdleme4.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
cdleme4.f  |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
) )
cdleme4.g  |-  G  =  ( ( P  .\/  Q )  ./\  ( F  .\/  ( ( R  .\/  S )  ./\  W )
) )
Assertion
Ref Expression
cdleme7ga  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( P  .\/  Q
) ) )  ->  G  e.  A )

Proof of Theorem cdleme7ga
StepHypRef Expression
1 cdleme4.g . 2  |-  G  =  ( ( P  .\/  Q )  ./\  ( F  .\/  ( ( R  .\/  S )  ./\  W )
) )
2 simp11l 1106 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( P  .\/  Q
) ) )  ->  K  e.  HL )
3 simp12l 1108 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( P  .\/  Q
) ) )  ->  P  e.  A )
4 simp13l 1110 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( P  .\/  Q
) ) )  ->  Q  e.  A )
5 eqid 2441 . . . . 5  |-  ( Base `  K )  =  (
Base `  K )
6 cdleme4.j . . . . 5  |-  .\/  =  ( join `  K )
7 cdleme4.a . . . . 5  |-  A  =  ( Atoms `  K )
85, 6, 7hlatjcl 34793 . . . 4  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
92, 3, 4, 8syl3anc 1227 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( P  .\/  Q
) ) )  -> 
( P  .\/  Q
)  e.  ( Base `  K ) )
10 simp11 1025 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( P  .\/  Q
) ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
11 simp12 1026 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( P  .\/  Q
) ) )  -> 
( P  e.  A  /\  -.  P  .<_  W ) )
12 simp13 1027 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( P  .\/  Q
) ) )  -> 
( Q  e.  A  /\  -.  Q  .<_  W ) )
13 simp2r 1022 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( P  .\/  Q
) ) )  -> 
( S  e.  A  /\  -.  S  .<_  W ) )
14 simp31 1031 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( P  .\/  Q
) ) )  ->  P  =/=  Q )
15 simp33 1033 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( P  .\/  Q
) ) )  ->  -.  S  .<_  ( P 
.\/  Q ) )
16 cdleme4.l . . . . . 6  |-  .<_  =  ( le `  K )
17 cdleme4.m . . . . . 6  |-  ./\  =  ( meet `  K )
18 cdleme4.h . . . . . 6  |-  H  =  ( LHyp `  K
)
19 cdleme4.u . . . . . 6  |-  U  =  ( ( P  .\/  Q )  ./\  W )
20 cdleme4.f . . . . . 6  |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
) )
2116, 6, 17, 7, 18, 19, 20cdleme3fa 35663 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  F  e.  A )
2210, 11, 12, 13, 14, 15, 21syl132anc 1245 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( P  .\/  Q
) ) )  ->  F  e.  A )
23 simp2l 1021 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( P  .\/  Q
) ) )  -> 
( R  e.  A  /\  -.  R  .<_  W ) )
24 simp2rl 1064 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( P  .\/  Q
) ) )  ->  S  e.  A )
25 simp32 1032 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( P  .\/  Q
) ) )  ->  R  .<_  ( P  .\/  Q ) )
26 eqid 2441 . . . . . 6  |-  ( ( R  .\/  S ) 
./\  W )  =  ( ( R  .\/  S )  ./\  W )
2716, 6, 17, 7, 18, 19, 20, 1, 26cdleme7b 35671 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  ( P 
.\/  Q )  /\  R  .<_  ( P  .\/  Q ) ) )  -> 
( ( R  .\/  S )  ./\  W )  e.  A )
2810, 23, 24, 15, 25, 27syl113anc 1239 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( P  .\/  Q
) ) )  -> 
( ( R  .\/  S )  ./\  W )  e.  A )
295, 6, 7hlatjcl 34793 . . . 4  |-  ( ( K  e.  HL  /\  F  e.  A  /\  ( ( R  .\/  S )  ./\  W )  e.  A )  ->  ( F  .\/  ( ( R 
.\/  S )  ./\  W ) )  e.  (
Base `  K )
)
302, 22, 28, 29syl3anc 1227 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( P  .\/  Q
) ) )  -> 
( F  .\/  (
( R  .\/  S
)  ./\  W )
)  e.  ( Base `  K ) )
31 hllat 34790 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
322, 31syl 16 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( P  .\/  Q
) ) )  ->  K  e.  Lat )
33 eqid 2441 . . . . 5  |-  ( Lines `  K )  =  (
Lines `  K )
34 eqid 2441 . . . . 5  |-  ( pmap `  K )  =  (
pmap `  K )
356, 7, 33, 34linepmap 35201 . . . 4  |-  ( ( ( K  e.  Lat  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  ( ( pmap `  K ) `  ( P  .\/  Q ) )  e.  ( Lines `  K ) )
3632, 3, 4, 14, 35syl31anc 1230 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( P  .\/  Q
) ) )  -> 
( ( pmap `  K
) `  ( P  .\/  Q ) )  e.  ( Lines `  K )
)
37 simp2ll 1062 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( P  .\/  Q
) ) )  ->  R  e.  A )
385, 6, 7hlatjcl 34793 . . . . . . 7  |-  ( ( K  e.  HL  /\  R  e.  A  /\  S  e.  A )  ->  ( R  .\/  S
)  e.  ( Base `  K ) )
392, 37, 24, 38syl3anc 1227 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( P  .\/  Q
) ) )  -> 
( R  .\/  S
)  e.  ( Base `  K ) )
40 simp11r 1107 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( P  .\/  Q
) ) )  ->  W  e.  H )
415, 18lhpbase 35424 . . . . . . 7  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
4240, 41syl 16 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( P  .\/  Q
) ) )  ->  W  e.  ( Base `  K ) )
435, 16, 17latmle2 15576 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( R  .\/  S )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( R  .\/  S )  ./\  W )  .<_  W )
4432, 39, 42, 43syl3anc 1227 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( P  .\/  Q
) ) )  -> 
( ( R  .\/  S )  ./\  W )  .<_  W )
4516, 6, 17, 7, 18, 19, 20cdleme3 35664 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  -.  F  .<_  W )
4610, 11, 12, 13, 14, 15, 45syl132anc 1245 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( P  .\/  Q
) ) )  ->  -.  F  .<_  W )
47 nbrne2 4451 . . . . . 6  |-  ( ( ( ( R  .\/  S )  ./\  W )  .<_  W  /\  -.  F  .<_  W )  ->  (
( R  .\/  S
)  ./\  W )  =/=  F )
4847necomd 2712 . . . . 5  |-  ( ( ( ( R  .\/  S )  ./\  W )  .<_  W  /\  -.  F  .<_  W )  ->  F  =/=  ( ( R  .\/  S )  ./\  W )
)
4944, 46, 48syl2anc 661 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( P  .\/  Q
) ) )  ->  F  =/=  ( ( R 
.\/  S )  ./\  W ) )
506, 7, 33, 34linepmap 35201 . . . 4  |-  ( ( ( K  e.  Lat  /\  F  e.  A  /\  ( ( R  .\/  S )  ./\  W )  e.  A )  /\  F  =/=  ( ( R  .\/  S )  ./\  W )
)  ->  ( ( pmap `  K ) `  ( F  .\/  ( ( R  .\/  S ) 
./\  W ) ) )  e.  ( Lines `  K ) )
5132, 22, 28, 49, 50syl31anc 1230 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( P  .\/  Q
) ) )  -> 
( ( pmap `  K
) `  ( F  .\/  ( ( R  .\/  S )  ./\  W )
) )  e.  (
Lines `  K ) )
525, 7atbase 34716 . . . . . 6  |-  ( F  e.  A  ->  F  e.  ( Base `  K
) )
5322, 52syl 16 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( P  .\/  Q
) ) )  ->  F  e.  ( Base `  K ) )
545, 17latmcl 15551 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( R  .\/  S )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( R  .\/  S )  ./\  W )  e.  ( Base `  K ) )
5532, 39, 42, 54syl3anc 1227 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( P  .\/  Q
) ) )  -> 
( ( R  .\/  S )  ./\  W )  e.  ( Base `  K
) )
565, 16, 6latlej2 15560 . . . . 5  |-  ( ( K  e.  Lat  /\  F  e.  ( Base `  K )  /\  (
( R  .\/  S
)  ./\  W )  e.  ( Base `  K
) )  ->  (
( R  .\/  S
)  ./\  W )  .<_  ( F  .\/  (
( R  .\/  S
)  ./\  W )
) )
5732, 53, 55, 56syl3anc 1227 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( P  .\/  Q
) ) )  -> 
( ( R  .\/  S )  ./\  W )  .<_  ( F  .\/  (
( R  .\/  S
)  ./\  W )
) )
5816, 6, 17, 7, 18, 19, 20, 1, 26cdleme7c 35672 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( P  =/=  Q  /\  R  .<_  ( P 
.\/  Q )  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  U  =/=  ( ( R  .\/  S )  ./\  W )
)
5910, 11, 4, 23, 13, 14, 25, 15, 58syl323anc 1257 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( P  .\/  Q
) ) )  ->  U  =/=  ( ( R 
.\/  S )  ./\  W ) )
6059necomd 2712 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( P  .\/  Q
) ) )  -> 
( ( R  .\/  S )  ./\  W )  =/=  U )
61 hlatl 34787 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  AtLat )
622, 61syl 16 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( P  .\/  Q
) ) )  ->  K  e.  AtLat )
6316, 6, 17, 7, 18, 19lhpat2 35471 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q ) )  ->  U  e.  A
)
6410, 11, 4, 14, 63syl112anc 1231 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( P  .\/  Q
) ) )  ->  U  e.  A )
6516, 7atncmp 34739 . . . . . . 7  |-  ( ( K  e.  AtLat  /\  (
( R  .\/  S
)  ./\  W )  e.  A  /\  U  e.  A )  ->  ( -.  ( ( R  .\/  S )  ./\  W )  .<_  U  <->  ( ( R 
.\/  S )  ./\  W )  =/=  U ) )
6662, 28, 64, 65syl3anc 1227 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( P  .\/  Q
) ) )  -> 
( -.  ( ( R  .\/  S ) 
./\  W )  .<_  U 
<->  ( ( R  .\/  S )  ./\  W )  =/=  U ) )
6760, 66mpbird 232 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( P  .\/  Q
) ) )  ->  -.  ( ( R  .\/  S )  ./\  W )  .<_  U )
685, 16, 17latlem12 15577 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  ( ( ( R 
.\/  S )  ./\  W )  e.  ( Base `  K )  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
) )  ->  (
( ( ( R 
.\/  S )  ./\  W )  .<_  ( P  .\/  Q )  /\  (
( R  .\/  S
)  ./\  W )  .<_  W )  <->  ( ( R  .\/  S )  ./\  W )  .<_  ( ( P  .\/  Q )  ./\  W ) ) )
6932, 55, 9, 42, 68syl13anc 1229 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( P  .\/  Q
) ) )  -> 
( ( ( ( R  .\/  S ) 
./\  W )  .<_  ( P  .\/  Q )  /\  ( ( R 
.\/  S )  ./\  W )  .<_  W )  <->  ( ( R  .\/  S
)  ./\  W )  .<_  ( ( P  .\/  Q )  ./\  W )
) )
7069biimpd 207 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( P  .\/  Q
) ) )  -> 
( ( ( ( R  .\/  S ) 
./\  W )  .<_  ( P  .\/  Q )  /\  ( ( R 
.\/  S )  ./\  W )  .<_  W )  ->  ( ( R  .\/  S )  ./\  W )  .<_  ( ( P  .\/  Q )  ./\  W )
) )
7144, 70mpan2d 674 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( P  .\/  Q
) ) )  -> 
( ( ( R 
.\/  S )  ./\  W )  .<_  ( P  .\/  Q )  ->  (
( R  .\/  S
)  ./\  W )  .<_  ( ( P  .\/  Q )  ./\  W )
) )
7219breq2i 4441 . . . . . 6  |-  ( ( ( R  .\/  S
)  ./\  W )  .<_  U  <->  ( ( R 
.\/  S )  ./\  W )  .<_  ( ( P  .\/  Q )  ./\  W ) )
7371, 72syl6ibr 227 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( P  .\/  Q
) ) )  -> 
( ( ( R 
.\/  S )  ./\  W )  .<_  ( P  .\/  Q )  ->  (
( R  .\/  S
)  ./\  W )  .<_  U ) )
7467, 73mtod 177 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( P  .\/  Q
) ) )  ->  -.  ( ( R  .\/  S )  ./\  W )  .<_  ( P  .\/  Q
) )
75 nbrne1 4450 . . . . 5  |-  ( ( ( ( R  .\/  S )  ./\  W )  .<_  ( F  .\/  (
( R  .\/  S
)  ./\  W )
)  /\  -.  (
( R  .\/  S
)  ./\  W )  .<_  ( P  .\/  Q
) )  ->  ( F  .\/  ( ( R 
.\/  S )  ./\  W ) )  =/=  ( P  .\/  Q ) )
7675necomd 2712 . . . 4  |-  ( ( ( ( R  .\/  S )  ./\  W )  .<_  ( F  .\/  (
( R  .\/  S
)  ./\  W )
)  /\  -.  (
( R  .\/  S
)  ./\  W )  .<_  ( P  .\/  Q
) )  ->  ( P  .\/  Q )  =/=  ( F  .\/  (
( R  .\/  S
)  ./\  W )
) )
7757, 74, 76syl2anc 661 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( P  .\/  Q
) ) )  -> 
( P  .\/  Q
)  =/=  ( F 
.\/  ( ( R 
.\/  S )  ./\  W ) ) )
7816, 6, 17, 7, 18, 19, 20, 1, 26cdleme7e 35674 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( P  .\/  Q
) ) )  ->  G  =/=  ( 0. `  K ) )
791, 78syl5eqner 2742 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( P  .\/  Q
) ) )  -> 
( ( P  .\/  Q )  ./\  ( F  .\/  ( ( R  .\/  S )  ./\  W )
) )  =/=  ( 0. `  K ) )
80 eqid 2441 . . . 4  |-  ( 0.
`  K )  =  ( 0. `  K
)
815, 17, 80, 7, 33, 342lnat 35210 . . 3  |-  ( ( ( K  e.  HL  /\  ( P  .\/  Q
)  e.  ( Base `  K )  /\  ( F  .\/  ( ( R 
.\/  S )  ./\  W ) )  e.  (
Base `  K )
)  /\  ( (
( pmap `  K ) `  ( P  .\/  Q
) )  e.  (
Lines `  K )  /\  ( ( pmap `  K
) `  ( F  .\/  ( ( R  .\/  S )  ./\  W )
) )  e.  (
Lines `  K ) )  /\  ( ( P 
.\/  Q )  =/=  ( F  .\/  (
( R  .\/  S
)  ./\  W )
)  /\  ( ( P  .\/  Q )  ./\  ( F  .\/  ( ( R  .\/  S ) 
./\  W ) ) )  =/=  ( 0.
`  K ) ) )  ->  ( ( P  .\/  Q )  ./\  ( F  .\/  ( ( R  .\/  S ) 
./\  W ) ) )  e.  A )
822, 9, 30, 36, 51, 77, 79, 81syl322anc 1255 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( P  .\/  Q
) ) )  -> 
( ( P  .\/  Q )  ./\  ( F  .\/  ( ( R  .\/  S )  ./\  W )
) )  e.  A
)
831, 82syl5eqel 2533 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( P  .\/  Q
) ) )  ->  G  e.  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 972    = wceq 1381    e. wcel 1802    =/= wne 2636   class class class wbr 4433   ` cfv 5574  (class class class)co 6277   Basecbs 14504   lecple 14576   joincjn 15442   meetcmee 15443   0.cp0 15536   Latclat 15544   Atomscatm 34690   AtLatcal 34691   HLchlt 34777   Linesclines 34920   pmapcpmap 34923   LHypclh 35410
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-reu 2798  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-iun 4313  df-iin 4314  df-br 4434  df-opab 4492  df-mpt 4493  df-id 4781  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6781  df-2nd 6782  df-preset 15426  df-poset 15444  df-plt 15457  df-lub 15473  df-glb 15474  df-join 15475  df-meet 15476  df-p0 15538  df-p1 15539  df-lat 15545  df-clat 15607  df-oposet 34603  df-ol 34605  df-oml 34606  df-covers 34693  df-ats 34694  df-atl 34725  df-cvlat 34749  df-hlat 34778  df-lines 34927  df-psubsp 34929  df-pmap 34930  df-padd 35222  df-lhyp 35414
This theorem is referenced by:  cdleme7  35676  cdleme18c  35720  cdleme22f2  35775  cdlemefs32sn1aw  35842
  Copyright terms: Public domain W3C validator