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Theorem cdleme22b 36464
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 1018 . . . . 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 1090 . . . . . 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 1091 . . . . . 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 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  =/=  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 35633 . . . . . 6  |-  ( ( ( K  e.  HL  /\  S  e.  A  /\  T  e.  A )  /\  S  =/=  T
)  ->  ( S  .\/  T )  e.  (
LLines `  K ) )
91, 2, 3, 4, 8syl31anc 1229 . . . . 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 35635 . . . . 5  |-  ( ( K  e.  HL  /\  ( S  .\/  T )  e.  ( LLines `  K
) )  ->  -.  ( S  .\/  T )  e.  A )
111, 9, 10syl2anc 659 . . . 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 35637 . . . . 5  |-  ( ( K  e.  HL  /\  ( S  .\/  T )  e.  ( LLines `  K
) )  ->  ( S  .\/  T )  =/=  ( 0. `  K
) )
141, 9, 13syl2anc 659 . . . 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 530 . . 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 2651 . . . . 5  |-  ( ( S  .\/  T )  =/=  ( 0. `  K )  <->  -.  ( S  .\/  T )  =  ( 0. `  K
) )
1716anbi2i 692 . . . 4  |-  ( ( -.  ( S  .\/  T )  e.  A  /\  ( S  .\/  T )  =/=  ( 0. `  K ) )  <->  ( -.  ( S  .\/  T )  e.  A  /\  -.  ( S  .\/  T )  =  ( 0. `  K ) ) )
18 pm4.56 493 . . . 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 1103 . . . . . . 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 1022 . . . . . . . 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 35496 . . . . . . . 8  |-  ( ( K  e.  HL  /\  T  e.  A  /\  V  e.  A )  ->  T  .<_  ( T  .\/  V ) )
251, 3, 22, 24syl3anc 1226 . . . . . . 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 35485 . . . . . . . . 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 35411 . . . . . . . . 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 35411 . . . . . . . . 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 35488 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  T  e.  A  /\  V  e.  A )  ->  ( T  .\/  V
)  e.  ( Base `  K ) )
341, 3, 22, 33syl3anc 1226 . . . . . . . 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 15891 . . . . . . . 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 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  .\/  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 463 . . . . 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 1104 . . . . . . 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 463 . . . . . 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 459 . . . . . 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 1027 . . . . . . . . 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 1028 . . . . . . . . 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 35488 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
451, 42, 43, 44syl3anc 1226 . . . . . . . 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 15891 . . . . . . . 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 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  .<_  ( P  .\/  Q )  /\  T  .<_  ( P 
.\/  Q ) )  <-> 
( S  .\/  T
)  .<_  ( P  .\/  Q ) ) )
4847adantr 463 . . . . . 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 35488 . . . . . . . 8  |-  ( ( K  e.  HL  /\  S  e.  A  /\  T  e.  A )  ->  ( S  .\/  T
)  e.  ( Base `  K ) )
511, 2, 3, 50syl3anc 1226 . . . . . . 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 15907 . . . . . . 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 1228 . . . . . 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 463 . . . . 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 432 . . 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 35484 . . . . . . . 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 463 . . . . . 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 463 . . . . . 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 754 . . . . . 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 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 ) ) ) )  ->  ( S  .\/  T )  .<_  ( ( T  .\/  V ) 
./\  ( P  .\/  Q ) ) )
6428, 23, 12, 6leat3 35417 . . . . . 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 1229 . . . . 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 603 . . . 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 4443 . . . . . . . . 9  |-  ( ( ( T  .\/  V
)  ./\  ( P  .\/  Q ) )  =  ( 0. `  K
)  ->  ( ( S  .\/  T )  .<_  ( ( T  .\/  V )  ./\  ( P  .\/  Q ) )  <->  ( S  .\/  T )  .<_  ( 0.
`  K ) ) )
6867biimpa 482 . . . . . . . 8  |-  ( ( ( ( T  .\/  V )  ./\  ( P  .\/  Q ) )  =  ( 0. `  K
)  /\  ( S  .\/  T )  .<_  ( ( T  .\/  V ) 
./\  ( P  .\/  Q ) ) )  -> 
( S  .\/  T
)  .<_  ( 0. `  K ) )
6928, 23, 12ople0 35309 . . . . . . . . 9  |-  ( ( K  e.  OP  /\  ( S  .\/  T )  e.  ( Base `  K
) )  ->  (
( S  .\/  T
)  .<_  ( 0. `  K )  <->  ( S  .\/  T )  =  ( 0. `  K ) ) )
7059, 51, 69syl2anc 659 . . . . . . . 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 427 . . . . . 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 391 . . . . 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 603 . . . 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 1102 . . . . 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 35647 . . . . 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 1241 . . . 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 379 . . 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 366    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   Basecbs 14716   lecple 14791   joincjn 15772   meetcmee 15773   0.cp0 15866   Latclat 15874   OPcops 35294   Atomscatm 35385   HLchlt 35472   LLinesclln 35612   LHypclh 36105
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-preset 15756  df-poset 15774  df-plt 15787  df-lub 15803  df-glb 15804  df-join 15805  df-meet 15806  df-p0 15868  df-p1 15869  df-lat 15875  df-clat 15937  df-oposet 35298  df-ol 35300  df-oml 35301  df-covers 35388  df-ats 35389  df-atl 35420  df-cvlat 35444  df-hlat 35473  df-llines 35619
This theorem is referenced by:  cdleme22cN  36465  cdleme27a  36490
  Copyright terms: Public domain W3C validator