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Theorem cdleme22b 34308
Description: Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph, 5th line on p. 115. Show that t  \/ v =/= p  \/ q and s  <_ p  \/ q implies  -. t  <_ p  \/ q. (Contributed by NM, 2-Dec-2012.)
Hypotheses
Ref Expression
cdleme22.l  |-  .<_  =  ( le `  K )
cdleme22.j  |-  .\/  =  ( join `  K )
cdleme22.m  |-  ./\  =  ( meet `  K )
cdleme22.a  |-  A  =  ( Atoms `  K )
cdleme22.h  |-  H  =  ( LHyp `  K
)
Assertion
Ref Expression
cdleme22b  |-  ( ( ( K  e.  HL  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q )  /\  ( V  e.  A  /\  ( ( T  .\/  V )  =/=  ( P 
.\/  Q )  /\  S  .<_  ( T  .\/  V )  /\  S  .<_  ( P  .\/  Q ) ) ) )  ->  -.  T  .<_  ( P 
.\/  Q ) )

Proof of Theorem cdleme22b
StepHypRef Expression
1 simp1l 1012 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q )  /\  ( V  e.  A  /\  ( ( T  .\/  V )  =/=  ( P 
.\/  Q )  /\  S  .<_  ( T  .\/  V )  /\  S  .<_  ( P  .\/  Q ) ) ) )  ->  K  e.  HL )
2 simp1r1 1084 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q )  /\  ( V  e.  A  /\  ( ( T  .\/  V )  =/=  ( P 
.\/  Q )  /\  S  .<_  ( T  .\/  V )  /\  S  .<_  ( P  .\/  Q ) ) ) )  ->  S  e.  A )
3 simp1r2 1085 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q )  /\  ( V  e.  A  /\  ( ( T  .\/  V )  =/=  ( P 
.\/  Q )  /\  S  .<_  ( T  .\/  V )  /\  S  .<_  ( P  .\/  Q ) ) ) )  ->  T  e.  A )
4 simp1r3 1086 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q )  /\  ( V  e.  A  /\  ( ( T  .\/  V )  =/=  ( P 
.\/  Q )  /\  S  .<_  ( T  .\/  V )  /\  S  .<_  ( P  .\/  Q ) ) ) )  ->  S  =/=  T )
5 cdleme22.j . . . . . . 7  |-  .\/  =  ( join `  K )
6 cdleme22.a . . . . . . 7  |-  A  =  ( Atoms `  K )
7 eqid 2454 . . . . . . 7  |-  ( LLines `  K )  =  (
LLines `  K )
85, 6, 7llni2 33479 . . . . . 6  |-  ( ( ( K  e.  HL  /\  S  e.  A  /\  T  e.  A )  /\  S  =/=  T
)  ->  ( S  .\/  T )  e.  (
LLines `  K ) )
91, 2, 3, 4, 8syl31anc 1222 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q )  /\  ( V  e.  A  /\  ( ( T  .\/  V )  =/=  ( P 
.\/  Q )  /\  S  .<_  ( T  .\/  V )  /\  S  .<_  ( P  .\/  Q ) ) ) )  -> 
( S  .\/  T
)  e.  ( LLines `  K ) )
106, 7llnneat 33481 . . . . 5  |-  ( ( K  e.  HL  /\  ( S  .\/  T )  e.  ( LLines `  K
) )  ->  -.  ( S  .\/  T )  e.  A )
111, 9, 10syl2anc 661 . . . 4  |-  ( ( ( K  e.  HL  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q )  /\  ( V  e.  A  /\  ( ( T  .\/  V )  =/=  ( P 
.\/  Q )  /\  S  .<_  ( T  .\/  V )  /\  S  .<_  ( P  .\/  Q ) ) ) )  ->  -.  ( S  .\/  T
)  e.  A )
12 eqid 2454 . . . . . 6  |-  ( 0.
`  K )  =  ( 0. `  K
)
1312, 7llnn0 33483 . . . . 5  |-  ( ( K  e.  HL  /\  ( S  .\/  T )  e.  ( LLines `  K
) )  ->  ( S  .\/  T )  =/=  ( 0. `  K
) )
141, 9, 13syl2anc 661 . . . 4  |-  ( ( ( K  e.  HL  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q )  /\  ( V  e.  A  /\  ( ( T  .\/  V )  =/=  ( P 
.\/  Q )  /\  S  .<_  ( T  .\/  V )  /\  S  .<_  ( P  .\/  Q ) ) ) )  -> 
( S  .\/  T
)  =/=  ( 0.
`  K ) )
1511, 14jca 532 . . 3  |-  ( ( ( K  e.  HL  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q )  /\  ( V  e.  A  /\  ( ( T  .\/  V )  =/=  ( P 
.\/  Q )  /\  S  .<_  ( T  .\/  V )  /\  S  .<_  ( P  .\/  Q ) ) ) )  -> 
( -.  ( S 
.\/  T )  e.  A  /\  ( S 
.\/  T )  =/=  ( 0. `  K
) ) )
16 df-ne 2649 . . . . 5  |-  ( ( S  .\/  T )  =/=  ( 0. `  K )  <->  -.  ( S  .\/  T )  =  ( 0. `  K
) )
1716anbi2i 694 . . . 4  |-  ( ( -.  ( S  .\/  T )  e.  A  /\  ( S  .\/  T )  =/=  ( 0. `  K ) )  <->  ( -.  ( S  .\/  T )  e.  A  /\  -.  ( S  .\/  T )  =  ( 0. `  K ) ) )
18 pm4.56 495 . . . 4  |-  ( ( -.  ( S  .\/  T )  e.  A  /\  -.  ( S  .\/  T
)  =  ( 0.
`  K ) )  <->  -.  ( ( S  .\/  T )  e.  A  \/  ( S  .\/  T )  =  ( 0. `  K ) ) )
1917, 18bitri 249 . . 3  |-  ( ( -.  ( S  .\/  T )  e.  A  /\  ( S  .\/  T )  =/=  ( 0. `  K ) )  <->  -.  (
( S  .\/  T
)  e.  A  \/  ( S  .\/  T )  =  ( 0. `  K ) ) )
2015, 19sylib 196 . 2  |-  ( ( ( K  e.  HL  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q )  /\  ( V  e.  A  /\  ( ( T  .\/  V )  =/=  ( P 
.\/  Q )  /\  S  .<_  ( T  .\/  V )  /\  S  .<_  ( P  .\/  Q ) ) ) )  ->  -.  ( ( S  .\/  T )  e.  A  \/  ( S  .\/  T )  =  ( 0. `  K ) ) )
21 simp3r2 1097 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q )  /\  ( V  e.  A  /\  ( ( T  .\/  V )  =/=  ( P 
.\/  Q )  /\  S  .<_  ( T  .\/  V )  /\  S  .<_  ( P  .\/  Q ) ) ) )  ->  S  .<_  ( T  .\/  V ) )
22 simp3l 1016 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q )  /\  ( V  e.  A  /\  ( ( T  .\/  V )  =/=  ( P 
.\/  Q )  /\  S  .<_  ( T  .\/  V )  /\  S  .<_  ( P  .\/  Q ) ) ) )  ->  V  e.  A )
23 cdleme22.l . . . . . . . . 9  |-  .<_  =  ( le `  K )
2423, 5, 6hlatlej1 33342 . . . . . . . 8  |-  ( ( K  e.  HL  /\  T  e.  A  /\  V  e.  A )  ->  T  .<_  ( T  .\/  V ) )
251, 3, 22, 24syl3anc 1219 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q )  /\  ( V  e.  A  /\  ( ( T  .\/  V )  =/=  ( P 
.\/  Q )  /\  S  .<_  ( T  .\/  V )  /\  S  .<_  ( P  .\/  Q ) ) ) )  ->  T  .<_  ( T  .\/  V ) )
26 hllat 33331 . . . . . . . . 9  |-  ( K  e.  HL  ->  K  e.  Lat )
271, 26syl 16 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q )  /\  ( V  e.  A  /\  ( ( T  .\/  V )  =/=  ( P 
.\/  Q )  /\  S  .<_  ( T  .\/  V )  /\  S  .<_  ( P  .\/  Q ) ) ) )  ->  K  e.  Lat )
28 eqid 2454 . . . . . . . . . 10  |-  ( Base `  K )  =  (
Base `  K )
2928, 6atbase 33257 . . . . . . . . 9  |-  ( S  e.  A  ->  S  e.  ( Base `  K
) )
302, 29syl 16 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q )  /\  ( V  e.  A  /\  ( ( T  .\/  V )  =/=  ( P 
.\/  Q )  /\  S  .<_  ( T  .\/  V )  /\  S  .<_  ( P  .\/  Q ) ) ) )  ->  S  e.  ( Base `  K ) )
3128, 6atbase 33257 . . . . . . . . 9  |-  ( T  e.  A  ->  T  e.  ( Base `  K
) )
323, 31syl 16 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q )  /\  ( V  e.  A  /\  ( ( T  .\/  V )  =/=  ( P 
.\/  Q )  /\  S  .<_  ( T  .\/  V )  /\  S  .<_  ( P  .\/  Q ) ) ) )  ->  T  e.  ( Base `  K ) )
3328, 5, 6hlatjcl 33334 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  T  e.  A  /\  V  e.  A )  ->  ( T  .\/  V
)  e.  ( Base `  K ) )
341, 3, 22, 33syl3anc 1219 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q )  /\  ( V  e.  A  /\  ( ( T  .\/  V )  =/=  ( P 
.\/  Q )  /\  S  .<_  ( T  .\/  V )  /\  S  .<_  ( P  .\/  Q ) ) ) )  -> 
( T  .\/  V
)  e.  ( Base `  K ) )
3528, 23, 5latjle12 15350 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( S  e.  ( Base `  K )  /\  T  e.  ( Base `  K )  /\  ( T  .\/  V )  e.  ( Base `  K
) ) )  -> 
( ( S  .<_  ( T  .\/  V )  /\  T  .<_  ( T 
.\/  V ) )  <-> 
( S  .\/  T
)  .<_  ( T  .\/  V ) ) )
3627, 30, 32, 34, 35syl13anc 1221 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q )  /\  ( V  e.  A  /\  ( ( T  .\/  V )  =/=  ( P 
.\/  Q )  /\  S  .<_  ( T  .\/  V )  /\  S  .<_  ( P  .\/  Q ) ) ) )  -> 
( ( S  .<_  ( T  .\/  V )  /\  T  .<_  ( T 
.\/  V ) )  <-> 
( S  .\/  T
)  .<_  ( T  .\/  V ) ) )
3721, 25, 36mpbi2and 912 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q )  /\  ( V  e.  A  /\  ( ( T  .\/  V )  =/=  ( P 
.\/  Q )  /\  S  .<_  ( T  .\/  V )  /\  S  .<_  ( P  .\/  Q ) ) ) )  -> 
( S  .\/  T
)  .<_  ( T  .\/  V ) )
3837adantr 465 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/= 
T ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( V  e.  A  /\  (
( T  .\/  V
)  =/=  ( P 
.\/  Q )  /\  S  .<_  ( T  .\/  V )  /\  S  .<_  ( P  .\/  Q ) ) ) )  /\  T  .<_  ( P  .\/  Q ) )  ->  ( S  .\/  T )  .<_  ( T  .\/  V ) )
39 simp3r3 1098 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q )  /\  ( V  e.  A  /\  ( ( T  .\/  V )  =/=  ( P 
.\/  Q )  /\  S  .<_  ( T  .\/  V )  /\  S  .<_  ( P  .\/  Q ) ) ) )  ->  S  .<_  ( P  .\/  Q ) )
4039adantr 465 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/= 
T ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( V  e.  A  /\  (
( T  .\/  V
)  =/=  ( P 
.\/  Q )  /\  S  .<_  ( T  .\/  V )  /\  S  .<_  ( P  .\/  Q ) ) ) )  /\  T  .<_  ( P  .\/  Q ) )  ->  S  .<_  ( P  .\/  Q
) )
41 simpr 461 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/= 
T ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( V  e.  A  /\  (
( T  .\/  V
)  =/=  ( P 
.\/  Q )  /\  S  .<_  ( T  .\/  V )  /\  S  .<_  ( P  .\/  Q ) ) ) )  /\  T  .<_  ( P  .\/  Q ) )  ->  T  .<_  ( P  .\/  Q
) )
42 simp21 1021 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q )  /\  ( V  e.  A  /\  ( ( T  .\/  V )  =/=  ( P 
.\/  Q )  /\  S  .<_  ( T  .\/  V )  /\  S  .<_  ( P  .\/  Q ) ) ) )  ->  P  e.  A )
43 simp22 1022 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q )  /\  ( V  e.  A  /\  ( ( T  .\/  V )  =/=  ( P 
.\/  Q )  /\  S  .<_  ( T  .\/  V )  /\  S  .<_  ( P  .\/  Q ) ) ) )  ->  Q  e.  A )
4428, 5, 6hlatjcl 33334 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
451, 42, 43, 44syl3anc 1219 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q )  /\  ( V  e.  A  /\  ( ( T  .\/  V )  =/=  ( P 
.\/  Q )  /\  S  .<_  ( T  .\/  V )  /\  S  .<_  ( P  .\/  Q ) ) ) )  -> 
( P  .\/  Q
)  e.  ( Base `  K ) )
4628, 23, 5latjle12 15350 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( S  e.  ( Base `  K )  /\  T  e.  ( Base `  K )  /\  ( P  .\/  Q )  e.  ( Base `  K
) ) )  -> 
( ( S  .<_  ( P  .\/  Q )  /\  T  .<_  ( P 
.\/  Q ) )  <-> 
( S  .\/  T
)  .<_  ( P  .\/  Q ) ) )
4727, 30, 32, 45, 46syl13anc 1221 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q )  /\  ( V  e.  A  /\  ( ( T  .\/  V )  =/=  ( P 
.\/  Q )  /\  S  .<_  ( T  .\/  V )  /\  S  .<_  ( P  .\/  Q ) ) ) )  -> 
( ( S  .<_  ( P  .\/  Q )  /\  T  .<_  ( P 
.\/  Q ) )  <-> 
( S  .\/  T
)  .<_  ( P  .\/  Q ) ) )
4847adantr 465 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/= 
T ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( V  e.  A  /\  (
( T  .\/  V
)  =/=  ( P 
.\/  Q )  /\  S  .<_  ( T  .\/  V )  /\  S  .<_  ( P  .\/  Q ) ) ) )  /\  T  .<_  ( P  .\/  Q ) )  ->  (
( S  .<_  ( P 
.\/  Q )  /\  T  .<_  ( P  .\/  Q ) )  <->  ( S  .\/  T )  .<_  ( P 
.\/  Q ) ) )
4940, 41, 48mpbi2and 912 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/= 
T ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( V  e.  A  /\  (
( T  .\/  V
)  =/=  ( P 
.\/  Q )  /\  S  .<_  ( T  .\/  V )  /\  S  .<_  ( P  .\/  Q ) ) ) )  /\  T  .<_  ( P  .\/  Q ) )  ->  ( S  .\/  T )  .<_  ( P  .\/  Q ) )
5028, 5, 6hlatjcl 33334 . . . . . . . 8  |-  ( ( K  e.  HL  /\  S  e.  A  /\  T  e.  A )  ->  ( S  .\/  T
)  e.  ( Base `  K ) )
511, 2, 3, 50syl3anc 1219 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q )  /\  ( V  e.  A  /\  ( ( T  .\/  V )  =/=  ( P 
.\/  Q )  /\  S  .<_  ( T  .\/  V )  /\  S  .<_  ( P  .\/  Q ) ) ) )  -> 
( S  .\/  T
)  e.  ( Base `  K ) )
52 cdleme22.m . . . . . . . 8  |-  ./\  =  ( meet `  K )
5328, 23, 52latlem12 15366 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( ( S  .\/  T )  e.  ( Base `  K )  /\  ( T  .\/  V )  e.  ( Base `  K
)  /\  ( P  .\/  Q )  e.  (
Base `  K )
) )  ->  (
( ( S  .\/  T )  .<_  ( T  .\/  V )  /\  ( S  .\/  T )  .<_  ( P  .\/  Q ) )  <->  ( S  .\/  T )  .<_  ( ( T  .\/  V )  ./\  ( P  .\/  Q ) ) ) )
5427, 51, 34, 45, 53syl13anc 1221 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q )  /\  ( V  e.  A  /\  ( ( T  .\/  V )  =/=  ( P 
.\/  Q )  /\  S  .<_  ( T  .\/  V )  /\  S  .<_  ( P  .\/  Q ) ) ) )  -> 
( ( ( S 
.\/  T )  .<_  ( T  .\/  V )  /\  ( S  .\/  T )  .<_  ( P  .\/  Q ) )  <->  ( S  .\/  T )  .<_  ( ( T  .\/  V ) 
./\  ( P  .\/  Q ) ) ) )
5554adantr 465 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/= 
T ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( V  e.  A  /\  (
( T  .\/  V
)  =/=  ( P 
.\/  Q )  /\  S  .<_  ( T  .\/  V )  /\  S  .<_  ( P  .\/  Q ) ) ) )  /\  T  .<_  ( P  .\/  Q ) )  ->  (
( ( S  .\/  T )  .<_  ( T  .\/  V )  /\  ( S  .\/  T )  .<_  ( P  .\/  Q ) )  <->  ( S  .\/  T )  .<_  ( ( T  .\/  V )  ./\  ( P  .\/  Q ) ) ) )
5638, 49, 55mpbi2and 912 . . . 4  |-  ( ( ( ( K  e.  HL  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/= 
T ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( V  e.  A  /\  (
( T  .\/  V
)  =/=  ( P 
.\/  Q )  /\  S  .<_  ( T  .\/  V )  /\  S  .<_  ( P  .\/  Q ) ) ) )  /\  T  .<_  ( P  .\/  Q ) )  ->  ( S  .\/  T )  .<_  ( ( T  .\/  V )  ./\  ( P  .\/  Q ) ) )
5756ex 434 . . 3  |-  ( ( ( K  e.  HL  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q )  /\  ( V  e.  A  /\  ( ( T  .\/  V )  =/=  ( P 
.\/  Q )  /\  S  .<_  ( T  .\/  V )  /\  S  .<_  ( P  .\/  Q ) ) ) )  -> 
( T  .<_  ( P 
.\/  Q )  -> 
( S  .\/  T
)  .<_  ( ( T 
.\/  V )  ./\  ( P  .\/  Q ) ) ) )
58 hlop 33330 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  OP )
591, 58syl 16 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q )  /\  ( V  e.  A  /\  ( ( T  .\/  V )  =/=  ( P 
.\/  Q )  /\  S  .<_  ( T  .\/  V )  /\  S  .<_  ( P  .\/  Q ) ) ) )  ->  K  e.  OP )
6059adantr 465 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/= 
T ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( V  e.  A  /\  (
( T  .\/  V
)  =/=  ( P 
.\/  Q )  /\  S  .<_  ( T  .\/  V )  /\  S  .<_  ( P  .\/  Q ) ) ) )  /\  ( ( ( T 
.\/  V )  ./\  ( P  .\/  Q ) )  e.  A  /\  ( S  .\/  T ) 
.<_  ( ( T  .\/  V )  ./\  ( P  .\/  Q ) ) ) )  ->  K  e.  OP )
6151adantr 465 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/= 
T ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( V  e.  A  /\  (
( T  .\/  V
)  =/=  ( P 
.\/  Q )  /\  S  .<_  ( T  .\/  V )  /\  S  .<_  ( P  .\/  Q ) ) ) )  /\  ( ( ( T 
.\/  V )  ./\  ( P  .\/  Q ) )  e.  A  /\  ( S  .\/  T ) 
.<_  ( ( T  .\/  V )  ./\  ( P  .\/  Q ) ) ) )  ->  ( S  .\/  T )  e.  (
Base `  K )
)
62 simprl 755 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/= 
T ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( V  e.  A  /\  (
( T  .\/  V
)  =/=  ( P 
.\/  Q )  /\  S  .<_  ( T  .\/  V )  /\  S  .<_  ( P  .\/  Q ) ) ) )  /\  ( ( ( T 
.\/  V )  ./\  ( P  .\/  Q ) )  e.  A  /\  ( S  .\/  T ) 
.<_  ( ( T  .\/  V )  ./\  ( P  .\/  Q ) ) ) )  ->  ( ( T  .\/  V )  ./\  ( P  .\/  Q ) )  e.  A )
63 simprr 756 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/= 
T ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( V  e.  A  /\  (
( T  .\/  V
)  =/=  ( P 
.\/  Q )  /\  S  .<_  ( T  .\/  V )  /\  S  .<_  ( P  .\/  Q ) ) ) )  /\  ( ( ( T 
.\/  V )  ./\  ( P  .\/  Q ) )  e.  A  /\  ( S  .\/  T ) 
.<_  ( ( T  .\/  V )  ./\  ( P  .\/  Q ) ) ) )  ->  ( S  .\/  T )  .<_  ( ( T  .\/  V ) 
./\  ( P  .\/  Q ) ) )
6428, 23, 12, 6leat3 33263 . . . . . 6  |-  ( ( ( K  e.  OP  /\  ( S  .\/  T
)  e.  ( Base `  K )  /\  (
( T  .\/  V
)  ./\  ( P  .\/  Q ) )  e.  A )  /\  ( S  .\/  T )  .<_  ( ( T  .\/  V )  ./\  ( P  .\/  Q ) ) )  ->  ( ( S 
.\/  T )  e.  A  \/  ( S 
.\/  T )  =  ( 0. `  K
) ) )
6560, 61, 62, 63, 64syl31anc 1222 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/= 
T ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( V  e.  A  /\  (
( T  .\/  V
)  =/=  ( P 
.\/  Q )  /\  S  .<_  ( T  .\/  V )  /\  S  .<_  ( P  .\/  Q ) ) ) )  /\  ( ( ( T 
.\/  V )  ./\  ( P  .\/  Q ) )  e.  A  /\  ( S  .\/  T ) 
.<_  ( ( T  .\/  V )  ./\  ( P  .\/  Q ) ) ) )  ->  ( ( S  .\/  T )  e.  A  \/  ( S 
.\/  T )  =  ( 0. `  K
) ) )
6665exp32 605 . . . 4  |-  ( ( ( K  e.  HL  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q )  /\  ( V  e.  A  /\  ( ( T  .\/  V )  =/=  ( P 
.\/  Q )  /\  S  .<_  ( T  .\/  V )  /\  S  .<_  ( P  .\/  Q ) ) ) )  -> 
( ( ( T 
.\/  V )  ./\  ( P  .\/  Q ) )  e.  A  -> 
( ( S  .\/  T )  .<_  ( ( T  .\/  V )  ./\  ( P  .\/  Q ) )  ->  ( ( S  .\/  T )  e.  A  \/  ( S 
.\/  T )  =  ( 0. `  K
) ) ) ) )
67 breq2 4403 . . . . . . . . 9  |-  ( ( ( T  .\/  V
)  ./\  ( P  .\/  Q ) )  =  ( 0. `  K
)  ->  ( ( S  .\/  T )  .<_  ( ( T  .\/  V )  ./\  ( P  .\/  Q ) )  <->  ( S  .\/  T )  .<_  ( 0.
`  K ) ) )
6867biimpa 484 . . . . . . . 8  |-  ( ( ( ( T  .\/  V )  ./\  ( P  .\/  Q ) )  =  ( 0. `  K
)  /\  ( S  .\/  T )  .<_  ( ( T  .\/  V ) 
./\  ( P  .\/  Q ) ) )  -> 
( S  .\/  T
)  .<_  ( 0. `  K ) )
6928, 23, 12ople0 33155 . . . . . . . . 9  |-  ( ( K  e.  OP  /\  ( S  .\/  T )  e.  ( Base `  K
) )  ->  (
( S  .\/  T
)  .<_  ( 0. `  K )  <->  ( S  .\/  T )  =  ( 0. `  K ) ) )
7059, 51, 69syl2anc 661 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q )  /\  ( V  e.  A  /\  ( ( T  .\/  V )  =/=  ( P 
.\/  Q )  /\  S  .<_  ( T  .\/  V )  /\  S  .<_  ( P  .\/  Q ) ) ) )  -> 
( ( S  .\/  T )  .<_  ( 0. `  K )  <->  ( S  .\/  T )  =  ( 0. `  K ) ) )
7168, 70syl5ib 219 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q )  /\  ( V  e.  A  /\  ( ( T  .\/  V )  =/=  ( P 
.\/  Q )  /\  S  .<_  ( T  .\/  V )  /\  S  .<_  ( P  .\/  Q ) ) ) )  -> 
( ( ( ( T  .\/  V ) 
./\  ( P  .\/  Q ) )  =  ( 0. `  K )  /\  ( S  .\/  T )  .<_  ( ( T  .\/  V )  ./\  ( P  .\/  Q ) ) )  ->  ( S  .\/  T )  =  ( 0. `  K
) ) )
7271imp 429 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/= 
T ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( V  e.  A  /\  (
( T  .\/  V
)  =/=  ( P 
.\/  Q )  /\  S  .<_  ( T  .\/  V )  /\  S  .<_  ( P  .\/  Q ) ) ) )  /\  ( ( ( T 
.\/  V )  ./\  ( P  .\/  Q ) )  =  ( 0.
`  K )  /\  ( S  .\/  T ) 
.<_  ( ( T  .\/  V )  ./\  ( P  .\/  Q ) ) ) )  ->  ( S  .\/  T )  =  ( 0. `  K ) )
7372olcd 393 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/= 
T ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( V  e.  A  /\  (
( T  .\/  V
)  =/=  ( P 
.\/  Q )  /\  S  .<_  ( T  .\/  V )  /\  S  .<_  ( P  .\/  Q ) ) ) )  /\  ( ( ( T 
.\/  V )  ./\  ( P  .\/  Q ) )  =  ( 0.
`  K )  /\  ( S  .\/  T ) 
.<_  ( ( T  .\/  V )  ./\  ( P  .\/  Q ) ) ) )  ->  ( ( S  .\/  T )  e.  A  \/  ( S 
.\/  T )  =  ( 0. `  K
) ) )
7473exp32 605 . . . 4  |-  ( ( ( K  e.  HL  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q )  /\  ( V  e.  A  /\  ( ( T  .\/  V )  =/=  ( P 
.\/  Q )  /\  S  .<_  ( T  .\/  V )  /\  S  .<_  ( P  .\/  Q ) ) ) )  -> 
( ( ( T 
.\/  V )  ./\  ( P  .\/  Q ) )  =  ( 0.
`  K )  -> 
( ( S  .\/  T )  .<_  ( ( T  .\/  V )  ./\  ( P  .\/  Q ) )  ->  ( ( S  .\/  T )  e.  A  \/  ( S 
.\/  T )  =  ( 0. `  K
) ) ) ) )
75 simp3r1 1096 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q )  /\  ( V  e.  A  /\  ( ( T  .\/  V )  =/=  ( P 
.\/  Q )  /\  S  .<_  ( T  .\/  V )  /\  S  .<_  ( P  .\/  Q ) ) ) )  -> 
( T  .\/  V
)  =/=  ( P 
.\/  Q ) )
765, 52, 12, 62atmat0 33493 . . . . 5  |-  ( ( ( K  e.  HL  /\  T  e.  A  /\  V  e.  A )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( T  .\/  V
)  =/=  ( P 
.\/  Q ) ) )  ->  ( (
( T  .\/  V
)  ./\  ( P  .\/  Q ) )  e.  A  \/  ( ( T  .\/  V ) 
./\  ( P  .\/  Q ) )  =  ( 0. `  K ) ) )
771, 3, 22, 42, 43, 75, 76syl33anc 1234 . . . 4  |-  ( ( ( K  e.  HL  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q )  /\  ( V  e.  A  /\  ( ( T  .\/  V )  =/=  ( P 
.\/  Q )  /\  S  .<_  ( T  .\/  V )  /\  S  .<_  ( P  .\/  Q ) ) ) )  -> 
( ( ( T 
.\/  V )  ./\  ( P  .\/  Q ) )  e.  A  \/  ( ( T  .\/  V )  ./\  ( P  .\/  Q ) )  =  ( 0. `  K
) ) )
7866, 74, 77mpjaod 381 . . 3  |-  ( ( ( K  e.  HL  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q )  /\  ( V  e.  A  /\  ( ( T  .\/  V )  =/=  ( P 
.\/  Q )  /\  S  .<_  ( T  .\/  V )  /\  S  .<_  ( P  .\/  Q ) ) ) )  -> 
( ( S  .\/  T )  .<_  ( ( T  .\/  V )  ./\  ( P  .\/  Q ) )  ->  ( ( S  .\/  T )  e.  A  \/  ( S 
.\/  T )  =  ( 0. `  K
) ) ) )
7957, 78syld 44 . 2  |-  ( ( ( K  e.  HL  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q )  /\  ( V  e.  A  /\  ( ( T  .\/  V )  =/=  ( P 
.\/  Q )  /\  S  .<_  ( T  .\/  V )  /\  S  .<_  ( P  .\/  Q ) ) ) )  -> 
( T  .<_  ( P 
.\/  Q )  -> 
( ( S  .\/  T )  e.  A  \/  ( S  .\/  T )  =  ( 0. `  K ) ) ) )
8020, 79mtod 177 1  |-  ( ( ( K  e.  HL  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q )  /\  ( V  e.  A  /\  ( ( T  .\/  V )  =/=  ( P 
.\/  Q )  /\  S  .<_  ( T  .\/  V )  /\  S  .<_  ( P  .\/  Q ) ) ) )  ->  -.  T  .<_  ( P 
.\/  Q ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2647   class class class wbr 4399   ` cfv 5525  (class class class)co 6199   Basecbs 14291   lecple 14363   joincjn 15232   meetcmee 15233   0.cp0 15325   Latclat 15333   OPcops 33140   Atomscatm 33231   HLchlt 33318   LLinesclln 33458   LHypclh 33951
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-ral 2803  df-rex 2804  df-reu 2805  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-op 3991  df-uni 4199  df-iun 4280  df-br 4400  df-opab 4458  df-mpt 4459  df-id 4743  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6160  df-ov 6202  df-oprab 6203  df-poset 15234  df-plt 15246  df-lub 15262  df-glb 15263  df-join 15264  df-meet 15265  df-p0 15327  df-p1 15328  df-lat 15334  df-clat 15396  df-oposet 33144  df-ol 33146  df-oml 33147  df-covers 33234  df-ats 33235  df-atl 33266  df-cvlat 33290  df-hlat 33319  df-llines 33465
This theorem is referenced by:  cdleme22cN  34309  cdleme27a  34334
  Copyright terms: Public domain W3C validator