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Theorem cdleme3g 35661
Description: Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme3fa 35663 and cdleme3 35664. (Contributed by NM, 7-Jun-2012.)
Hypotheses
Ref Expression
cdleme1.l  |-  .<_  =  ( le `  K )
cdleme1.j  |-  .\/  =  ( join `  K )
cdleme1.m  |-  ./\  =  ( meet `  K )
cdleme1.a  |-  A  =  ( Atoms `  K )
cdleme1.h  |-  H  =  ( LHyp `  K
)
cdleme1.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
cdleme1.f  |-  F  =  ( ( R  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
) )
cdleme3.3  |-  V  =  ( ( P  .\/  R )  ./\  W )
Assertion
Ref Expression
cdleme3g  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P 
.\/  Q ) ) )  ->  F  =/=  U )

Proof of Theorem cdleme3g
StepHypRef Expression
1 cdleme1.l . . . 4  |-  .<_  =  ( le `  K )
2 cdleme1.j . . . 4  |-  .\/  =  ( join `  K )
3 cdleme1.m . . . 4  |-  ./\  =  ( meet `  K )
4 cdleme1.a . . . 4  |-  A  =  ( Atoms `  K )
5 cdleme1.h . . . 4  |-  H  =  ( LHyp `  K
)
6 cdleme1.u . . . 4  |-  U  =  ( ( P  .\/  Q )  ./\  W )
7 cdleme1.f . . . 4  |-  F  =  ( ( R  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
) )
8 cdleme3.3 . . . 4  |-  V  =  ( ( P  .\/  R )  ./\  W )
91, 2, 3, 4, 5, 6, 7, 8cdleme3d 35658 . . 3  |-  F  =  ( ( R  .\/  U )  ./\  ( Q  .\/  V ) )
10 simp1l 1019 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P 
.\/  Q ) ) )  ->  K  e.  HL )
11 hllat 34790 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
1210, 11syl 16 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P 
.\/  Q ) ) )  ->  K  e.  Lat )
13 simp23l 1116 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P 
.\/  Q ) ) )  ->  R  e.  A )
14 simp1 995 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P 
.\/  Q ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
15 simp21 1028 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P 
.\/  Q ) ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
16 simp22l 1114 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P 
.\/  Q ) ) )  ->  Q  e.  A )
17 simp3l 1023 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P 
.\/  Q ) ) )  ->  P  =/=  Q )
181, 2, 3, 4, 5, 6lhpat2 35471 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q ) )  ->  U  e.  A
)
1914, 15, 16, 17, 18syl112anc 1231 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P 
.\/  Q ) ) )  ->  U  e.  A )
20 eqid 2441 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
2120, 2, 4hlatjcl 34793 . . . . 5  |-  ( ( K  e.  HL  /\  R  e.  A  /\  U  e.  A )  ->  ( R  .\/  U
)  e.  ( Base `  K ) )
2210, 13, 19, 21syl3anc 1227 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P 
.\/  Q ) ) )  ->  ( R  .\/  U )  e.  (
Base `  K )
)
23 simp3r 1024 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P 
.\/  Q ) ) )  ->  -.  R  .<_  ( P  .\/  Q
) )
2413, 23jca 532 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P 
.\/  Q ) ) )  ->  ( R  e.  A  /\  -.  R  .<_  ( P  .\/  Q
) ) )
251, 2, 3, 4, 5, 6, 7, 8cdleme3e 35659 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  ( P 
.\/  Q ) ) ) )  ->  V  e.  A )
2614, 15, 16, 24, 25syl13anc 1229 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P 
.\/  Q ) ) )  ->  V  e.  A )
2720, 2, 4hlatjcl 34793 . . . . 5  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  V  e.  A )  ->  ( Q  .\/  V
)  e.  ( Base `  K ) )
2810, 16, 26, 27syl3anc 1227 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P 
.\/  Q ) ) )  ->  ( Q  .\/  V )  e.  (
Base `  K )
)
2920, 1, 3latmle2 15576 . . . 4  |-  ( ( K  e.  Lat  /\  ( R  .\/  U )  e.  ( Base `  K
)  /\  ( Q  .\/  V )  e.  (
Base `  K )
)  ->  ( ( R  .\/  U )  ./\  ( Q  .\/  V ) )  .<_  ( Q  .\/  V ) )
3012, 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 ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P 
.\/  Q ) ) )  ->  ( ( R  .\/  U )  ./\  ( Q  .\/  V ) )  .<_  ( Q  .\/  V ) )
319, 30syl5eqbr 4466 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P 
.\/  Q ) ) )  ->  F  .<_  ( Q  .\/  V ) )
32 simp22r 1115 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P 
.\/  Q ) ) )  ->  -.  Q  .<_  W )
33 simp23 1030 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P 
.\/  Q ) ) )  ->  ( R  e.  A  /\  -.  R  .<_  W ) )
34 simp3 997 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P 
.\/  Q ) ) )  ->  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q
) ) )
351, 2, 3, 4, 5, 6, 8cdleme0e 35644 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  ->  U  =/=  V )
3614, 15, 16, 33, 34, 35syl131anc 1240 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P 
.\/  Q ) ) )  ->  U  =/=  V )
371, 2, 4hlatexch2 34822 . . . . 5  |-  ( ( K  e.  HL  /\  ( U  e.  A  /\  Q  e.  A  /\  V  e.  A
)  /\  U  =/=  V )  ->  ( U  .<_  ( Q  .\/  V
)  ->  Q  .<_  ( U  .\/  V ) ) )
3810, 19, 16, 26, 36, 37syl131anc 1240 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P 
.\/  Q ) ) )  ->  ( U  .<_  ( Q  .\/  V
)  ->  Q  .<_  ( U  .\/  V ) ) )
39 simp21l 1112 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P 
.\/  Q ) ) )  ->  P  e.  A )
4020, 2, 4hlatjcl 34793 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
4110, 39, 16, 40syl3anc 1227 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P 
.\/  Q ) ) )  ->  ( P  .\/  Q )  e.  (
Base `  K )
)
42 simp1r 1020 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P 
.\/  Q ) ) )  ->  W  e.  H )
4320, 5lhpbase 35424 . . . . . . . . 9  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
4442, 43syl 16 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P 
.\/  Q ) ) )  ->  W  e.  ( Base `  K )
)
4520, 1, 3latmle2 15576 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  ./\  W )  .<_  W )
4612, 41, 44, 45syl3anc 1227 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P 
.\/  Q ) ) )  ->  ( ( P  .\/  Q )  ./\  W )  .<_  W )
476, 46syl5eqbr 4466 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P 
.\/  Q ) ) )  ->  U  .<_  W )
4820, 2, 4hlatjcl 34793 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  P  e.  A  /\  R  e.  A )  ->  ( P  .\/  R
)  e.  ( Base `  K ) )
4910, 39, 13, 48syl3anc 1227 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P 
.\/  Q ) ) )  ->  ( P  .\/  R )  e.  (
Base `  K )
)
5020, 1, 3latmle2 15576 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( P  .\/  R )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  R )  ./\  W )  .<_  W )
5112, 49, 44, 50syl3anc 1227 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P 
.\/  Q ) ) )  ->  ( ( P  .\/  R )  ./\  W )  .<_  W )
528, 51syl5eqbr 4466 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P 
.\/  Q ) ) )  ->  V  .<_  W )
5320, 4atbase 34716 . . . . . . . 8  |-  ( U  e.  A  ->  U  e.  ( Base `  K
) )
5419, 53syl 16 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P 
.\/  Q ) ) )  ->  U  e.  ( Base `  K )
)
5520, 4atbase 34716 . . . . . . . 8  |-  ( V  e.  A  ->  V  e.  ( Base `  K
) )
5626, 55syl 16 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P 
.\/  Q ) ) )  ->  V  e.  ( Base `  K )
)
5720, 1, 2latjle12 15561 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( U  e.  ( Base `  K )  /\  V  e.  ( Base `  K )  /\  W  e.  ( Base `  K
) ) )  -> 
( ( U  .<_  W  /\  V  .<_  W )  <-> 
( U  .\/  V
)  .<_  W ) )
5812, 54, 56, 44, 57syl13anc 1229 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P 
.\/  Q ) ) )  ->  ( ( U  .<_  W  /\  V  .<_  W )  <->  ( U  .\/  V )  .<_  W ) )
5947, 52, 58mpbi2and 919 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P 
.\/  Q ) ) )  ->  ( U  .\/  V )  .<_  W )
6020, 4atbase 34716 . . . . . . 7  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
6116, 60syl 16 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P 
.\/  Q ) ) )  ->  Q  e.  ( Base `  K )
)
6220, 2, 4hlatjcl 34793 . . . . . . 7  |-  ( ( K  e.  HL  /\  U  e.  A  /\  V  e.  A )  ->  ( U  .\/  V
)  e.  ( Base `  K ) )
6310, 19, 26, 62syl3anc 1227 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P 
.\/  Q ) ) )  ->  ( U  .\/  V )  e.  (
Base `  K )
)
6420, 1lattr 15555 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( Q  e.  ( Base `  K )  /\  ( U  .\/  V )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
) )  ->  (
( Q  .<_  ( U 
.\/  V )  /\  ( U  .\/  V ) 
.<_  W )  ->  Q  .<_  W ) )
6512, 61, 63, 44, 64syl13anc 1229 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P 
.\/  Q ) ) )  ->  ( ( Q  .<_  ( U  .\/  V )  /\  ( U 
.\/  V )  .<_  W )  ->  Q  .<_  W ) )
6659, 65mpan2d 674 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P 
.\/  Q ) ) )  ->  ( Q  .<_  ( U  .\/  V
)  ->  Q  .<_  W ) )
6738, 66syld 44 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P 
.\/  Q ) ) )  ->  ( U  .<_  ( Q  .\/  V
)  ->  Q  .<_  W ) )
6832, 67mtod 177 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P 
.\/  Q ) ) )  ->  -.  U  .<_  ( Q  .\/  V
) )
69 nbrne2 4451 . 2  |-  ( ( F  .<_  ( Q  .\/  V )  /\  -.  U  .<_  ( Q  .\/  V ) )  ->  F  =/=  U )
7031, 68, 69syl2anc 661 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P 
.\/  Q ) ) )  ->  F  =/=  U )
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   Latclat 15544   Atomscatm 34690   HLchlt 34777   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-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-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-lhyp 35414
This theorem is referenced by:  cdleme3  35664  cdleme16b  35706  cdleme35a  35876
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