Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdleme22b Structured version   Unicode version

Theorem cdleme22b 35155
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 1020 . . . . 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 1092 . . . . . 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 1093 . . . . . 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 1094 . . . . . 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 2467 . . . . . . 7  |-  ( LLines `  K )  =  (
LLines `  K )
85, 6, 7llni2 34326 . . . . . 6  |-  ( ( ( K  e.  HL  /\  S  e.  A  /\  T  e.  A )  /\  S  =/=  T
)  ->  ( S  .\/  T )  e.  (
LLines `  K ) )
91, 2, 3, 4, 8syl31anc 1231 . . . . 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 34328 . . . . 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 2467 . . . . . 6  |-  ( 0.
`  K )  =  ( 0. `  K
)
1312, 7llnn0 34330 . . . . 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 2664 . . . . 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 1105 . . . . . . 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 1024 . . . . . . . 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 34189 . . . . . . . 8  |-  ( ( K  e.  HL  /\  T  e.  A  /\  V  e.  A )  ->  T  .<_  ( T  .\/  V ) )
251, 3, 22, 24syl3anc 1228 . . . . . . 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 34178 . . . . . . . . 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 2467 . . . . . . . . . 10  |-  ( Base `  K )  =  (
Base `  K )
2928, 6atbase 34104 . . . . . . . . 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 34104 . . . . . . . . 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 34181 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  T  e.  A  /\  V  e.  A )  ->  ( T  .\/  V
)  e.  ( Base `  K ) )
341, 3, 22, 33syl3anc 1228 . . . . . . . 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 15549 . . . . . . . 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 1230 . . . . . . 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 919 . . . . . 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 1106 . . . . . . 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 1029 . . . . . . . . 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 1030 . . . . . . . . 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 34181 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
451, 42, 43, 44syl3anc 1228 . . . . . . . 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 15549 . . . . . . . 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 1230 . . . . . . 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 919 . . . . 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 34181 . . . . . . . 8  |-  ( ( K  e.  HL  /\  S  e.  A  /\  T  e.  A )  ->  ( S  .\/  T
)  e.  ( Base `  K ) )
511, 2, 3, 50syl3anc 1228 . . . . . . 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 15565 . . . . . . 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 1230 . . . . . 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 919 . . . 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 34177 . . . . . . . 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 34110 . . . . . 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 1231 . . . . 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 4451 . . . . . . . . 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 34002 . . . . . . . . 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 1104 . . . . 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 34340 . . . . 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 1243 . . . 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 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   OPcops 33987   Atomscatm 34078   HLchlt 34165   LLinesclln 34305   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-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-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-llines 34312
This theorem is referenced by:  cdleme22cN  35156  cdleme27a  35181
  Copyright terms: Public domain W3C validator