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Theorem cdleme7ga 29126
Description: Part of proof of Lemma E in [Crawley] p. 113. See cdleme7 29127. (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 1071 . . 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 1073 . . . 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 1075 . . . 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 2253 . . . . 5  |-  ( Base `  K )  =  (
Base `  K )
6 cdleme4.j . . . . 5  |-  .\/  =  ( join `  K )
7 cdleme4.a . . . . 5  |-  A  =  ( Atoms `  K )
85, 6, 7hlatjcl 28245 . . . 4  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
92, 3, 4, 8syl3anc 1187 . . 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 990 . . . . 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 991 . . . . 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 992 . . . . 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 987 . . . . 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 996 . . . . 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 998 . . . . 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 29114 . . . . 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 1205 . . . 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 986 . . . . 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 1029 . . . . 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 997 . . . . 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 2253 . . . . . 6  |-  ( ( R  .\/  S ) 
./\  W )  =  ( ( R  .\/  S )  ./\  W )
2716, 6, 17, 7, 18, 19, 20, 1, 26cdleme7b 29122 . . . . 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 1199 . . . 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 28245 . . . 4  |-  ( ( K  e.  HL  /\  F  e.  A  /\  ( ( R  .\/  S )  ./\  W )  e.  A )  ->  ( F  .\/  ( ( R 
.\/  S )  ./\  W ) )  e.  (
Base `  K )
)
302, 22, 28, 29syl3anc 1187 . . 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 28242 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
322, 31syl 17 . . . 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 2253 . . . . 5  |-  ( Lines `  K )  =  (
Lines `  K )
34 eqid 2253 . . . . 5  |-  ( pmap `  K )  =  (
pmap `  K )
356, 7, 33, 34linepmap 28653 . . . 4  |-  ( ( ( K  e.  Lat  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  ( ( pmap `  K ) `  ( P  .\/  Q ) )  e.  ( Lines `  K ) )
3632, 3, 4, 14, 35syl31anc 1190 . . 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 1027 . . . . . . 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 28245 . . . . . . 7  |-  ( ( K  e.  HL  /\  R  e.  A  /\  S  e.  A )  ->  ( R  .\/  S
)  e.  ( Base `  K ) )
392, 37, 24, 38syl3anc 1187 . . . . . 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 1072 . . . . . . 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 28876 . . . . . . 7  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
4240, 41syl 17 . . . . . 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 14027 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( R  .\/  S )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( R  .\/  S )  ./\  W )  .<_  W )
4432, 39, 42, 43syl3anc 1187 . . . . 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 29115 . . . . . 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 1205 . . . . 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 3938 . . . . . 6  |-  ( ( ( ( R  .\/  S )  ./\  W )  .<_  W  /\  -.  F  .<_  W )  ->  (
( R  .\/  S
)  ./\  W )  =/=  F )
4847necomd 2495 . . . . 5  |-  ( ( ( ( R  .\/  S )  ./\  W )  .<_  W  /\  -.  F  .<_  W )  ->  F  =/=  ( ( R  .\/  S )  ./\  W )
)
4944, 46, 48syl2anc 645 . . . 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 28653 . . . 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 1190 . . 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 28168 . . . . . 6  |-  ( F  e.  A  ->  F  e.  ( Base `  K
) )
5322, 52syl 17 . . . . 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 14001 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( R  .\/  S )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( R  .\/  S )  ./\  W )  e.  ( Base `  K ) )
5532, 39, 42, 54syl3anc 1187 . . . . 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 14011 . . . . 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 1187 . . . 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 29123 . . . . . . . 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 1217 . . . . . . 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 2495 . . . . . 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 28239 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  AtLat )
622, 61syl 17 . . . . . . 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 28923 . . . . . . . 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 1191 . . . . . . 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 28191 . . . . . . 7  |-  ( ( K  e.  AtLat  /\  (
( R  .\/  S
)  ./\  W )  e.  A  /\  U  e.  A )  ->  ( -.  ( ( R  .\/  S )  ./\  W )  .<_  U  <->  ( ( R 
.\/  S )  ./\  W )  =/=  U ) )
6662, 28, 64, 65syl3anc 1187 . . . . . 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 225 . . . . 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 14028 . . . . . . . . 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 1189 . . . . . . . 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 200 . . . . . . 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 658 . . . . . 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 3928 . . . . . 6  |-  ( ( ( R  .\/  S
)  ./\  W )  .<_  U  <->  ( ( R 
.\/  S )  ./\  W )  .<_  ( ( P  .\/  Q )  ./\  W ) )
7371, 72syl6ibr 220 . . . . 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 170 . . . 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 3937 . . . . 5  |-  ( ( ( ( R  .\/  S )  ./\  W )  .<_  ( F  .\/  (
( R  .\/  S
)  ./\  W )
)  /\  -.  (
( R  .\/  S
)  ./\  W )  .<_  ( P  .\/  Q
) )  ->  ( F  .\/  ( ( R 
.\/  S )  ./\  W ) )  =/=  ( P  .\/  Q ) )
7675necomd 2495 . . . 4  |-  ( ( ( ( R  .\/  S )  ./\  W )  .<_  ( F  .\/  (
( R  .\/  S
)  ./\  W )
)  /\  -.  (
( R  .\/  S
)  ./\  W )  .<_  ( P  .\/  Q
) )  ->  ( P  .\/  Q )  =/=  ( F  .\/  (
( R  .\/  S
)  ./\  W )
) )
7757, 74, 76syl2anc 645 . . 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 29125 . . . 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 2437 . . 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 2253 . . . 4  |-  ( 0.
`  K )  =  ( 0. `  K
)
815, 17, 80, 7, 33, 342lnat 28662 . . 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 1215 . 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 2337 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 set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2412   class class class wbr 3920   ` cfv 4592  (class class class)co 5710   Basecbs 13022   lecple 13089   joincjn 13922   meetcmee 13923   0.cp0 13987   Latclat 13995   Atomscatm 28142   AtLatcal 28143   HLchlt 28229   Linesclines 28372   pmapcpmap 28375   LHypclh 28862
This theorem is referenced by:  cdleme7  29127  cdleme18c  29171  cdleme22f2  29225  cdlemefs32sn1aw  29292
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-iin 3806  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-iota 6143  df-undef 6182  df-riota 6190  df-poset 13924  df-plt 13936  df-lub 13952  df-glb 13953  df-join 13954  df-meet 13955  df-p0 13989  df-p1 13990  df-lat 13996  df-clat 14058  df-oposet 28055  df-ol 28057  df-oml 28058  df-covers 28145  df-ats 28146  df-atl 28177  df-cvlat 28201  df-hlat 28230  df-lines 28379  df-psubsp 28381  df-pmap 28382  df-padd 28674  df-lhyp 28866
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