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Theorem cdleme7c 35059
Description: Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme7ga 35062 and cdleme7 35063. (Contributed by NM, 7-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 )
) )
cdleme7.v  |-  V  =  ( ( R  .\/  S )  ./\  W )
Assertion
Ref Expression
cdleme7c  |-  ( ( ( ( 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  =/=  V )

Proof of Theorem cdleme7c
StepHypRef Expression
1 cdleme4.u . . . 4  |-  U  =  ( ( P  .\/  Q )  ./\  W )
2 cdleme7.v . . . 4  |-  V  =  ( ( R  .\/  S )  ./\  W )
31, 2oveq12i 6296 . . 3  |-  ( U 
./\  V )  =  ( ( ( P 
.\/  Q )  ./\  W )  ./\  ( ( R  .\/  S )  ./\  W ) )
4 simp11 1026 . . . . . . . 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 ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
5 simp12l 1109 . . . . . . . 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 ) ) )  ->  P  e.  A )
6 simp13 1028 . . . . . . . 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 ) ) )  ->  Q  e.  A )
7 simp2l 1022 . . . . . . . 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 ) ) )  ->  ( R  e.  A  /\  -.  R  .<_  W ) )
8 simp32 1033 . . . . . . . 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 ) ) )  ->  R  .<_  ( P  .\/  Q ) )
9 cdleme4.l . . . . . . . . 9  |-  .<_  =  ( le `  K )
10 cdleme4.j . . . . . . . . 9  |-  .\/  =  ( join `  K )
11 cdleme4.m . . . . . . . . 9  |-  ./\  =  ( meet `  K )
12 cdleme4.a . . . . . . . . 9  |-  A  =  ( Atoms `  K )
13 cdleme4.h . . . . . . . . 9  |-  H  =  ( LHyp `  K
)
149, 10, 11, 12, 13, 1cdleme4 35052 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  R  .<_  ( P 
.\/  Q ) )  ->  ( P  .\/  Q )  =  ( R 
.\/  U ) )
154, 5, 6, 7, 8, 14syl131anc 1241 . . . . . . 7  |-  ( ( ( ( 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 ) ) )  ->  ( P  .\/  Q )  =  ( R  .\/  U ) )
1615oveq1d 6299 . . . . . 6  |-  ( ( ( ( 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 ) ) )  ->  ( ( P  .\/  Q )  ./\  ( R  .\/  S ) )  =  ( ( R  .\/  U ) 
./\  ( R  .\/  S ) ) )
17 simp11l 1107 . . . . . . 7  |-  ( ( ( ( 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 ) ) )  ->  K  e.  HL )
18 simp12 1027 . . . . . . . 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 ) ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
19 simp31 1032 . . . . . . . 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 ) ) )  ->  P  =/=  Q )
209, 10, 11, 12, 13, 1lhpat2 34859 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q ) )  ->  U  e.  A
)
214, 18, 6, 19, 20syl112anc 1232 . . . . . . 7  |-  ( ( ( ( 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  e.  A )
22 simp2rl 1065 . . . . . . 7  |-  ( ( ( ( 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 ) ) )  ->  S  e.  A )
23 simp2ll 1063 . . . . . . 7  |-  ( ( ( ( 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 ) ) )  ->  R  e.  A )
24 hllat 34178 . . . . . . . . . . 11  |-  ( K  e.  HL  ->  K  e.  Lat )
2517, 24syl 16 . . . . . . . . . 10  |-  ( ( ( ( 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 ) ) )  ->  K  e.  Lat )
26 eqid 2467 . . . . . . . . . . . 12  |-  ( Base `  K )  =  (
Base `  K )
2726, 10, 12hlatjcl 34181 . . . . . . . . . . 11  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
2817, 5, 6, 27syl3anc 1228 . . . . . . . . . 10  |-  ( ( ( ( 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 ) ) )  ->  ( P  .\/  Q )  e.  (
Base `  K )
)
29 simp11r 1108 . . . . . . . . . . 11  |-  ( ( ( ( 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 ) ) )  ->  W  e.  H )
3026, 13lhpbase 34812 . . . . . . . . . . 11  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
3129, 30syl 16 . . . . . . . . . 10  |-  ( ( ( ( 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 ) ) )  ->  W  e.  ( Base `  K )
)
3226, 9, 11latmle2 15564 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  ./\  W )  .<_  W )
3325, 28, 31, 32syl3anc 1228 . . . . . . . . 9  |-  ( ( ( ( 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 ) ) )  ->  ( ( P  .\/  Q )  ./\  W )  .<_  W )
341, 33syl5eqbr 4480 . . . . . . . 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  .<_  W )
35 simp2rr 1066 . . . . . . . 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 ) ) )  ->  -.  S  .<_  W )
36 nbrne2 4465 . . . . . . . 8  |-  ( ( U  .<_  W  /\  -.  S  .<_  W )  ->  U  =/=  S
)
3734, 35, 36syl2anc 661 . . . . . . 7  |-  ( ( ( ( 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  =/=  S )
38 cdleme4.f . . . . . . . 8  |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
) )
39 cdleme4.g . . . . . . . 8  |-  G  =  ( ( P  .\/  Q )  ./\  ( F  .\/  ( ( R  .\/  S )  ./\  W )
) )
409, 10, 11, 12, 13, 1, 38, 39cdleme7aa 35056 . . . . . . 7  |-  ( ( ( ( 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 ) ) )  ->  -.  R  .<_  ( U  .\/  S
) )
419, 10, 11, 122llnma2 34603 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( U  e.  A  /\  S  e.  A  /\  R  e.  A
)  /\  ( U  =/=  S  /\  -.  R  .<_  ( U  .\/  S
) ) )  -> 
( ( R  .\/  U )  ./\  ( R  .\/  S ) )  =  R )
4217, 21, 22, 23, 37, 40, 41syl132anc 1246 . . . . . 6  |-  ( ( ( ( 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 ) ) )  ->  ( ( R  .\/  U )  ./\  ( R  .\/  S ) )  =  R )
4316, 42eqtrd 2508 . . . . 5  |-  ( ( ( ( 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 ) ) )  ->  ( ( P  .\/  Q )  ./\  ( R  .\/  S ) )  =  R )
4443oveq1d 6299 . . . 4  |-  ( ( ( ( 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 ) ) )  ->  ( (
( P  .\/  Q
)  ./\  ( R  .\/  S ) )  ./\  W )  =  ( R 
./\  W ) )
45 hlol 34176 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  OL )
4617, 45syl 16 . . . . 5  |-  ( ( ( ( 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 ) ) )  ->  K  e.  OL )
4726, 10, 12hlatjcl 34181 . . . . . 6  |-  ( ( K  e.  HL  /\  R  e.  A  /\  S  e.  A )  ->  ( R  .\/  S
)  e.  ( Base `  K ) )
4817, 23, 22, 47syl3anc 1228 . . . . 5  |-  ( ( ( ( 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 ) ) )  ->  ( R  .\/  S )  e.  (
Base `  K )
)
4926, 11latmmdir 34050 . . . . 5  |-  ( ( K  e.  OL  /\  ( ( P  .\/  Q )  e.  ( Base `  K )  /\  ( R  .\/  S )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
) )  ->  (
( ( P  .\/  Q )  ./\  ( R  .\/  S ) )  ./\  W )  =  ( ( ( P  .\/  Q
)  ./\  W )  ./\  ( ( R  .\/  S )  ./\  W )
) )
5046, 28, 48, 31, 49syl13anc 1230 . . . 4  |-  ( ( ( ( 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 ) ) )  ->  ( (
( P  .\/  Q
)  ./\  ( R  .\/  S ) )  ./\  W )  =  ( ( ( P  .\/  Q
)  ./\  W )  ./\  ( ( R  .\/  S )  ./\  W )
) )
51 eqid 2467 . . . . . 6  |-  ( 0.
`  K )  =  ( 0. `  K
)
529, 11, 51, 12, 13lhpmat 34844 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  -> 
( R  ./\  W
)  =  ( 0.
`  K ) )
534, 7, 52syl2anc 661 . . . 4  |-  ( ( ( ( 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 ) ) )  ->  ( R  ./\ 
W )  =  ( 0. `  K ) )
5444, 50, 533eqtr3d 2516 . . 3  |-  ( ( ( ( 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 ) ) )  ->  ( (
( P  .\/  Q
)  ./\  W )  ./\  ( ( R  .\/  S )  ./\  W )
)  =  ( 0.
`  K ) )
553, 54syl5eq 2520 . 2  |-  ( ( ( ( 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  ./\ 
V )  =  ( 0. `  K ) )
56 hlatl 34175 . . . 4  |-  ( K  e.  HL  ->  K  e.  AtLat )
5717, 56syl 16 . . 3  |-  ( ( ( ( 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 ) ) )  ->  K  e.  AtLat
)
58 simp33 1034 . . . 4  |-  ( ( ( ( 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 ) ) )  ->  -.  S  .<_  ( P  .\/  Q
) )
599, 10, 11, 12, 13, 1, 38, 39, 2cdleme7b 35058 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  ( P 
.\/  Q )  /\  R  .<_  ( P  .\/  Q ) ) )  ->  V  e.  A )
604, 7, 22, 58, 8, 59syl113anc 1240 . . 3  |-  ( ( ( ( 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 ) ) )  ->  V  e.  A )
6111, 51, 12atnem0 34133 . . 3  |-  ( ( K  e.  AtLat  /\  U  e.  A  /\  V  e.  A )  ->  ( U  =/=  V  <->  ( U  ./\ 
V )  =  ( 0. `  K ) ) )
6257, 21, 60, 61syl3anc 1228 . 2  |-  ( ( ( ( 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  =/=  V  <->  ( U  ./\  V )  =  ( 0.
`  K ) ) )
6355, 62mpbird 232 1  |-  ( ( ( ( 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  =/=  V )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   class class class wbr 4447   ` cfv 5588  (class class class)co 6284   Basecbs 14490   lecple 14562   joincjn 15431   meetcmee 15432   0.cp0 15524   Latclat 15532   OLcol 33989   Atomscatm 34078   AtLatcal 34079   HLchlt 34165   LHypclh 34798
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-1st 6784  df-2nd 6785  df-poset 15433  df-plt 15445  df-lub 15461  df-glb 15462  df-join 15463  df-meet 15464  df-p0 15526  df-p1 15527  df-lat 15533  df-clat 15595  df-oposet 33991  df-ol 33993  df-oml 33994  df-covers 34081  df-ats 34082  df-atl 34113  df-cvlat 34137  df-hlat 34166  df-psubsp 34317  df-pmap 34318  df-padd 34610  df-lhyp 34802
This theorem is referenced by:  cdleme7d  35060  cdleme7ga  35062
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