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Theorem cdleme22b 29797
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 984 . . . . 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 1056 . . . . . 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 1057 . . . . . 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 1058 . . . . . 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 2284 . . . . . . 7  |-  ( LLines `  K )  =  (
LLines `  K )
85, 6, 7llni2 28968 . . . . . 6  |-  ( ( ( K  e.  HL  /\  S  e.  A  /\  T  e.  A )  /\  S  =/=  T
)  ->  ( S  .\/  T )  e.  (
LLines `  K ) )
91, 2, 3, 4, 8syl31anc 1190 . . . . 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 28970 . . . . 5  |-  ( ( K  e.  HL  /\  ( S  .\/  T )  e.  ( LLines `  K
) )  ->  -.  ( S  .\/  T )  e.  A )
111, 9, 10syl2anc 645 . . . 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 2284 . . . . . 6  |-  ( 0.
`  K )  =  ( 0. `  K
)
1312, 7llnn0 28972 . . . . 5  |-  ( ( K  e.  HL  /\  ( S  .\/  T )  e.  ( LLines `  K
) )  ->  ( S  .\/  T )  =/=  ( 0. `  K
) )
141, 9, 13syl2anc 645 . . . 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 520 . . 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 2449 . . . . 5  |-  ( ( S  .\/  T )  =/=  ( 0. `  K )  <->  -.  ( S  .\/  T )  =  ( 0. `  K
) )
1716anbi2i 678 . . . 4  |-  ( ( -.  ( S  .\/  T )  e.  A  /\  ( S  .\/  T )  =/=  ( 0. `  K ) )  <->  ( -.  ( S  .\/  T )  e.  A  /\  -.  ( S  .\/  T )  =  ( 0. `  K ) ) )
18 pm4.56 483 . . . 4  |-  ( ( -.  ( S  .\/  T )  e.  A  /\  -.  ( S  .\/  T
)  =  ( 0.
`  K ) )  <->  -.  ( ( S  .\/  T )  e.  A  \/  ( S  .\/  T )  =  ( 0. `  K ) ) )
1917, 18bitri 242 . . 3  |-  ( ( -.  ( S  .\/  T )  e.  A  /\  ( S  .\/  T )  =/=  ( 0. `  K ) )  <->  -.  (
( S  .\/  T
)  e.  A  \/  ( S  .\/  T )  =  ( 0. `  K ) ) )
2015, 19sylib 190 . 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 1069 . . . . . . 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 988 . . . . . . . 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 28831 . . . . . . . 8  |-  ( ( K  e.  HL  /\  T  e.  A  /\  V  e.  A )  ->  T  .<_  ( T  .\/  V ) )
251, 3, 22, 24syl3anc 1187 . . . . . . 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 28820 . . . . . . . . 9  |-  ( K  e.  HL  ->  K  e.  Lat )
271, 26syl 17 . . . . . . . 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 2284 . . . . . . . . . 10  |-  ( Base `  K )  =  (
Base `  K )
2928, 6atbase 28746 . . . . . . . . 9  |-  ( S  e.  A  ->  S  e.  ( Base `  K
) )
302, 29syl 17 . . . . . . . 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 28746 . . . . . . . . 9  |-  ( T  e.  A  ->  T  e.  ( Base `  K
) )
323, 31syl 17 . . . . . . . 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 28823 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  T  e.  A  /\  V  e.  A )  ->  ( T  .\/  V
)  e.  ( Base `  K ) )
341, 3, 22, 33syl3anc 1187 . . . . . . . 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 14162 . . . . . . . 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 1189 . . . . . . 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 892 . . . . . 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 453 . . . . 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 1070 . . . . . . 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 453 . . . . . 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 449 . . . . . 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 993 . . . . . . . . 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 994 . . . . . . . . 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 28823 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
451, 42, 43, 44syl3anc 1187 . . . . . . . 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 14162 . . . . . . . 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 1189 . . . . . . 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 453 . . . . . 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 892 . . . . 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 28823 . . . . . . . 8  |-  ( ( K  e.  HL  /\  S  e.  A  /\  T  e.  A )  ->  ( S  .\/  T
)  e.  ( Base `  K ) )
511, 2, 3, 50syl3anc 1187 . . . . . . 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 14178 . . . . . . 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 1189 . . . . . 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 453 . . . . 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 892 . . . 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 425 . . 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 28819 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  OP )
591, 58syl 17 . . . . . . 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 453 . . . . . 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 453 . . . . . 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 735 . . . . . 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 736 . . . . . 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 28752 . . . . . 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 1190 . . . . 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 591 . . . 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 4028 . . . . . . . . 9  |-  ( ( ( T  .\/  V
)  ./\  ( P  .\/  Q ) )  =  ( 0. `  K
)  ->  ( ( S  .\/  T )  .<_  ( ( T  .\/  V )  ./\  ( P  .\/  Q ) )  <->  ( S  .\/  T )  .<_  ( 0.
`  K ) ) )
6867biimpa 472 . . . . . . . 8  |-  ( ( ( ( T  .\/  V )  ./\  ( P  .\/  Q ) )  =  ( 0. `  K
)  /\  ( S  .\/  T )  .<_  ( ( T  .\/  V ) 
./\  ( P  .\/  Q ) ) )  -> 
( S  .\/  T
)  .<_  ( 0. `  K ) )
6928, 23, 12ople0 28644 . . . . . . . . 9  |-  ( ( K  e.  OP  /\  ( S  .\/  T )  e.  ( Base `  K
) )  ->  (
( S  .\/  T
)  .<_  ( 0. `  K )  <->  ( S  .\/  T )  =  ( 0. `  K ) ) )
7059, 51, 69syl2anc 645 . . . . . . . 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 212 . . . . . . 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 420 . . . . . 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 384 . . . . 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 591 . . . 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 1068 . . . . 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 28982 . . . . 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 1202 . . . 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 372 . . 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 42 . 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 170 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 5    -> wi 6    <-> wb 178    \/ wo 359    /\ wa 360    /\ w3a 939    = wceq 1628    e. wcel 1688    =/= wne 2447   class class class wbr 4024   ` cfv 5221  (class class class)co 5819   Basecbs 13142   lecple 13209   joincjn 14072   meetcmee 14073   0.cp0 14137   Latclat 14145   OPcops 28629   Atomscatm 28720   HLchlt 28807   LLinesclln 28947   LHypclh 29440
This theorem is referenced by:  cdleme22cN  29798  cdleme27a  29823
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-13 1690  ax-14 1692  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1534  df-nf 1537  df-sb 1636  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-1st 6083  df-2nd 6084  df-iota 6252  df-undef 6291  df-riota 6299  df-poset 14074  df-plt 14086  df-lub 14102  df-glb 14103  df-join 14104  df-meet 14105  df-p0 14139  df-lat 14146  df-clat 14208  df-oposet 28633  df-ol 28635  df-oml 28636  df-covers 28723  df-ats 28724  df-atl 28755  df-cvlat 28779  df-hlat 28808  df-llines 28954
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