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Theorem cdleme22b 31152
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 979 . . . . 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 1051 . . . . . 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 1052 . . . . . 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 1053 . . . . . 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 2296 . . . . . . 7  |-  ( LLines `  K )  =  (
LLines `  K )
85, 6, 7llni2 30323 . . . . . 6  |-  ( ( ( K  e.  HL  /\  S  e.  A  /\  T  e.  A )  /\  S  =/=  T
)  ->  ( S  .\/  T )  e.  (
LLines `  K ) )
91, 2, 3, 4, 8syl31anc 1185 . . . . 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 30325 . . . . 5  |-  ( ( K  e.  HL  /\  ( S  .\/  T )  e.  ( LLines `  K
) )  ->  -.  ( S  .\/  T )  e.  A )
111, 9, 10syl2anc 642 . . . 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 2296 . . . . . 6  |-  ( 0.
`  K )  =  ( 0. `  K
)
1312, 7llnn0 30327 . . . . 5  |-  ( ( K  e.  HL  /\  ( S  .\/  T )  e.  ( LLines `  K
) )  ->  ( S  .\/  T )  =/=  ( 0. `  K
) )
141, 9, 13syl2anc 642 . . . 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 518 . . 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 2461 . . . . 5  |-  ( ( S  .\/  T )  =/=  ( 0. `  K )  <->  -.  ( S  .\/  T )  =  ( 0. `  K
) )
1716anbi2i 675 . . . 4  |-  ( ( -.  ( S  .\/  T )  e.  A  /\  ( S  .\/  T )  =/=  ( 0. `  K ) )  <->  ( -.  ( S  .\/  T )  e.  A  /\  -.  ( S  .\/  T )  =  ( 0. `  K ) ) )
18 pm4.56 481 . . . 4  |-  ( ( -.  ( S  .\/  T )  e.  A  /\  -.  ( S  .\/  T
)  =  ( 0.
`  K ) )  <->  -.  ( ( S  .\/  T )  e.  A  \/  ( S  .\/  T )  =  ( 0. `  K ) ) )
1917, 18bitri 240 . . 3  |-  ( ( -.  ( S  .\/  T )  e.  A  /\  ( S  .\/  T )  =/=  ( 0. `  K ) )  <->  -.  (
( S  .\/  T
)  e.  A  \/  ( S  .\/  T )  =  ( 0. `  K ) ) )
2015, 19sylib 188 . 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 1064 . . . . . . 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 983 . . . . . . . 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 30186 . . . . . . . 8  |-  ( ( K  e.  HL  /\  T  e.  A  /\  V  e.  A )  ->  T  .<_  ( T  .\/  V ) )
251, 3, 22, 24syl3anc 1182 . . . . . . 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 30175 . . . . . . . . 9  |-  ( K  e.  HL  ->  K  e.  Lat )
271, 26syl 15 . . . . . . . 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 2296 . . . . . . . . . 10  |-  ( Base `  K )  =  (
Base `  K )
2928, 6atbase 30101 . . . . . . . . 9  |-  ( S  e.  A  ->  S  e.  ( Base `  K
) )
302, 29syl 15 . . . . . . . 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 30101 . . . . . . . . 9  |-  ( T  e.  A  ->  T  e.  ( Base `  K
) )
323, 31syl 15 . . . . . . . 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 30178 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  T  e.  A  /\  V  e.  A )  ->  ( T  .\/  V
)  e.  ( Base `  K ) )
341, 3, 22, 33syl3anc 1182 . . . . . . . 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 14184 . . . . . . . 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 1184 . . . . . . 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 887 . . . . . 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 451 . . . . 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 1065 . . . . . . 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 451 . . . . . 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 447 . . . . . 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 988 . . . . . . . . 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 989 . . . . . . . . 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 30178 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
451, 42, 43, 44syl3anc 1182 . . . . . . . 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 14184 . . . . . . . 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 1184 . . . . . . 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 451 . . . . . 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 887 . . . . 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 30178 . . . . . . . 8  |-  ( ( K  e.  HL  /\  S  e.  A  /\  T  e.  A )  ->  ( S  .\/  T
)  e.  ( Base `  K ) )
511, 2, 3, 50syl3anc 1182 . . . . . . 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 14200 . . . . . . 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 1184 . . . . . 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 451 . . . . 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 887 . . . 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 423 . . 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 30174 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  OP )
591, 58syl 15 . . . . . . 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 451 . . . . . 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 451 . . . . . 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 732 . . . . . 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 733 . . . . . 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 30107 . . . . . 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 1185 . . . . 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 588 . . . 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 4043 . . . . . . . . 9  |-  ( ( ( T  .\/  V
)  ./\  ( P  .\/  Q ) )  =  ( 0. `  K
)  ->  ( ( S  .\/  T )  .<_  ( ( T  .\/  V )  ./\  ( P  .\/  Q ) )  <->  ( S  .\/  T )  .<_  ( 0.
`  K ) ) )
6867biimpa 470 . . . . . . . 8  |-  ( ( ( ( T  .\/  V )  ./\  ( P  .\/  Q ) )  =  ( 0. `  K
)  /\  ( S  .\/  T )  .<_  ( ( T  .\/  V ) 
./\  ( P  .\/  Q ) ) )  -> 
( S  .\/  T
)  .<_  ( 0. `  K ) )
6928, 23, 12ople0 29999 . . . . . . . . 9  |-  ( ( K  e.  OP  /\  ( S  .\/  T )  e.  ( Base `  K
) )  ->  (
( S  .\/  T
)  .<_  ( 0. `  K )  <->  ( S  .\/  T )  =  ( 0. `  K ) ) )
7059, 51, 69syl2anc 642 . . . . . . . 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 210 . . . . . . 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 418 . . . . . 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 382 . . . . 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 588 . . . 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 1063 . . . . 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 30337 . . . . 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 1197 . . . 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 370 . . 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 40 . 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 168 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 set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Basecbs 13164   lecple 13231   joincjn 14094   meetcmee 14095   0.cp0 14159   Latclat 14167   OPcops 29984   Atomscatm 30075   HLchlt 30162   LLinesclln 30302   LHypclh 30795
This theorem is referenced by:  cdleme22cN  31153  cdleme27a  31178
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-poset 14096  df-plt 14108  df-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-p0 14161  df-lat 14168  df-clat 14230  df-oposet 29988  df-ol 29990  df-oml 29991  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163  df-llines 30309
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