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Theorem cdleme3g 30716
Description: Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme3fa 30718 and cdleme3 30719. (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 30713 . . 3  |-  F  =  ( ( R  .\/  U )  ./\  ( Q  .\/  V ) )
10 simp1l 981 . . . . 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 29846 . . . . 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 1078 . . . . 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 957 . . . . . 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 990 . . . . . 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 1076 . . . . . 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 985 . . . . . 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 30527 . . . . . 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 1188 . . . . 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 2404 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
2120, 2, 4hlatjcl 29849 . . . . 5  |-  ( ( K  e.  HL  /\  R  e.  A  /\  U  e.  A )  ->  ( R  .\/  U
)  e.  ( Base `  K ) )
2210, 13, 19, 21syl3anc 1184 . . . 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 986 . . . . . . 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 519 . . . . . 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 30714 . . . . . 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 1186 . . . . 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 29849 . . . . 5  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  V  e.  A )  ->  ( Q  .\/  V
)  e.  ( Base `  K ) )
2810, 16, 26, 27syl3anc 1184 . . . 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 14461 . . . 4  |-  ( ( K  e.  Lat  /\  ( R  .\/  U )  e.  ( Base `  K
)  /\  ( Q  .\/  V )  e.  (
Base `  K )
)  ->  ( ( R  .\/  U )  ./\  ( Q  .\/  V ) )  .<_  ( Q  .\/  V ) )
3012, 22, 28, 29syl3anc 1184 . . 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 4205 . 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 1077 . . 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 992 . . . . . 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 959 . . . . . 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 30699 . . . . . 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 1197 . . . . 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 29878 . . . . 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 1197 . . . 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 1074 . . . . . . . . 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 29849 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
4110, 39, 16, 40syl3anc 1184 . . . . . . . 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 982 . . . . . . . . 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 30480 . . . . . . . . 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 14461 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  ./\  W )  .<_  W )
4612, 41, 44, 45syl3anc 1184 . . . . . . 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 4205 . . . . . 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 29849 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  P  e.  A  /\  R  e.  A )  ->  ( P  .\/  R
)  e.  ( Base `  K ) )
4910, 39, 13, 48syl3anc 1184 . . . . . . . 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 14461 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( P  .\/  R )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  R )  ./\  W )  .<_  W )
5112, 49, 44, 50syl3anc 1184 . . . . . . 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 4205 . . . . . 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 29772 . . . . . . . 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 29772 . . . . . . . 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 14446 . . . . . . 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 1186 . . . . . 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 888 . . . . 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 29772 . . . . . . 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 29849 . . . . . . 7  |-  ( ( K  e.  HL  /\  U  e.  A  /\  V  e.  A )  ->  ( U  .\/  V
)  e.  ( Base `  K ) )
6310, 19, 26, 62syl3anc 1184 . . . . . 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 14440 . . . . . 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 1186 . . . . 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 656 . . . 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 42 . . 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 170 . 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 4190 . 2  |-  ( ( F  .<_  ( Q  .\/  V )  /\  -.  U  .<_  ( Q  .\/  V ) )  ->  F  =/=  U )
7031, 68, 69syl2anc 643 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 set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   Basecbs 13424   lecple 13491   joincjn 14356   meetcmee 14357   Latclat 14429   Atomscatm 29746   HLchlt 29833   LHypclh 30466
This theorem is referenced by:  cdleme3  30719  cdleme16b  30761  cdleme35a  30930
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-undef 6502  df-riota 6508  df-poset 14358  df-plt 14370  df-lub 14386  df-glb 14387  df-join 14388  df-meet 14389  df-p0 14423  df-p1 14424  df-lat 14430  df-clat 14492  df-oposet 29659  df-ol 29661  df-oml 29662  df-covers 29749  df-ats 29750  df-atl 29781  df-cvlat 29805  df-hlat 29834  df-lhyp 30470
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