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Theorem cdleme22cN 30824
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  -. v  <_ p  \/ q. (Contributed by NM, 3-Dec-2012.) (New usage is discouraged.)
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
cdleme22cN  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( ( P  =/=  Q  /\  S  =/=  T )  /\  ( S  .<_  ( T 
.\/  V )  /\  S  .<_  ( P  .\/  Q ) )  /\  ( T  .\/  V )  =/=  ( P  .\/  Q
) ) )  ->  -.  V  .<_  ( P 
.\/  Q ) )

Proof of Theorem cdleme22cN
StepHypRef Expression
1 simp11l 1068 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( ( P  =/=  Q  /\  S  =/=  T )  /\  ( S  .<_  ( T 
.\/  V )  /\  S  .<_  ( P  .\/  Q ) )  /\  ( T  .\/  V )  =/=  ( P  .\/  Q
) ) )  ->  K  e.  HL )
2 hllat 29846 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
31, 2syl 16 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( ( P  =/=  Q  /\  S  =/=  T )  /\  ( S  .<_  ( T 
.\/  V )  /\  S  .<_  ( P  .\/  Q ) )  /\  ( T  .\/  V )  =/=  ( P  .\/  Q
) ) )  ->  K  e.  Lat )
4 simp12l 1070 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( ( P  =/=  Q  /\  S  =/=  T )  /\  ( S  .<_  ( T 
.\/  V )  /\  S  .<_  ( P  .\/  Q ) )  /\  ( T  .\/  V )  =/=  ( P  .\/  Q
) ) )  ->  P  e.  A )
5 simp13 989 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( ( P  =/=  Q  /\  S  =/=  T )  /\  ( S  .<_  ( T 
.\/  V )  /\  S  .<_  ( P  .\/  Q ) )  /\  ( T  .\/  V )  =/=  ( P  .\/  Q
) ) )  ->  Q  e.  A )
6 eqid 2404 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
7 cdleme22.j . . . . . 6  |-  .\/  =  ( join `  K )
8 cdleme22.a . . . . . 6  |-  A  =  ( Atoms `  K )
96, 7, 8hlatjcl 29849 . . . . 5  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
101, 4, 5, 9syl3anc 1184 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( ( P  =/=  Q  /\  S  =/=  T )  /\  ( S  .<_  ( T 
.\/  V )  /\  S  .<_  ( P  .\/  Q ) )  /\  ( T  .\/  V )  =/=  ( P  .\/  Q
) ) )  -> 
( P  .\/  Q
)  e.  ( Base `  K ) )
11 simp11r 1069 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( ( P  =/=  Q  /\  S  =/=  T )  /\  ( S  .<_  ( T 
.\/  V )  /\  S  .<_  ( P  .\/  Q ) )  /\  ( T  .\/  V )  =/=  ( P  .\/  Q
) ) )  ->  W  e.  H )
12 cdleme22.h . . . . . 6  |-  H  =  ( LHyp `  K
)
136, 12lhpbase 30480 . . . . 5  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
1411, 13syl 16 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( ( P  =/=  Q  /\  S  =/=  T )  /\  ( S  .<_  ( T 
.\/  V )  /\  S  .<_  ( P  .\/  Q ) )  /\  ( T  .\/  V )  =/=  ( P  .\/  Q
) ) )  ->  W  e.  ( Base `  K ) )
15 cdleme22.l . . . . 5  |-  .<_  =  ( le `  K )
16 cdleme22.m . . . . 5  |-  ./\  =  ( meet `  K )
176, 15, 16latmle2 14461 . . . 4  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  ./\  W )  .<_  W )
183, 10, 14, 17syl3anc 1184 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( ( P  =/=  Q  /\  S  =/=  T )  /\  ( S  .<_  ( T 
.\/  V )  /\  S  .<_  ( P  .\/  Q ) )  /\  ( T  .\/  V )  =/=  ( P  .\/  Q
) ) )  -> 
( ( P  .\/  Q )  ./\  W )  .<_  W )
19 simp21r 1075 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( ( P  =/=  Q  /\  S  =/=  T )  /\  ( S  .<_  ( T 
.\/  V )  /\  S  .<_  ( P  .\/  Q ) )  /\  ( T  .\/  V )  =/=  ( P  .\/  Q
) ) )  ->  -.  S  .<_  W )
20 nbrne2 4190 . . 3  |-  ( ( ( ( P  .\/  Q )  ./\  W )  .<_  W  /\  -.  S  .<_  W )  ->  (
( P  .\/  Q
)  ./\  W )  =/=  S )
2118, 19, 20syl2anc 643 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( ( P  =/=  Q  /\  S  =/=  T )  /\  ( S  .<_  ( T 
.\/  V )  /\  S  .<_  ( P  .\/  Q ) )  /\  ( T  .\/  V )  =/=  ( P  .\/  Q
) ) )  -> 
( ( P  .\/  Q )  ./\  W )  =/=  S )
22 simp32l 1082 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( ( P  =/=  Q  /\  S  =/=  T )  /\  ( S  .<_  ( T 
.\/  V )  /\  S  .<_  ( P  .\/  Q ) )  /\  ( T  .\/  V )  =/=  ( P  .\/  Q
) ) )  ->  S  .<_  ( T  .\/  V ) )
2322adantr 452 . . . . . . . . 9  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( ( P  =/=  Q  /\  S  =/=  T )  /\  ( S  .<_  ( T 
.\/  V )  /\  S  .<_  ( P  .\/  Q ) )  /\  ( T  .\/  V )  =/=  ( P  .\/  Q
) ) )  /\  V  .<_  ( P  .\/  Q ) )  ->  S  .<_  ( T  .\/  V
) )
241adantr 452 . . . . . . . . . . 11  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( ( P  =/=  Q  /\  S  =/=  T )  /\  ( S  .<_  ( T 
.\/  V )  /\  S  .<_  ( P  .\/  Q ) )  /\  ( T  .\/  V )  =/=  ( P  .\/  Q
) ) )  /\  V  .<_  ( P  .\/  Q ) )  ->  K  e.  HL )
2511adantr 452 . . . . . . . . . . 11  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( ( P  =/=  Q  /\  S  =/=  T )  /\  ( S  .<_  ( T 
.\/  V )  /\  S  .<_  ( P  .\/  Q ) )  /\  ( T  .\/  V )  =/=  ( P  .\/  Q
) ) )  /\  V  .<_  ( P  .\/  Q ) )  ->  W  e.  H )
26 simpl12 1033 . . . . . . . . . . 11  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( ( P  =/=  Q  /\  S  =/=  T )  /\  ( S  .<_  ( T 
.\/  V )  /\  S  .<_  ( P  .\/  Q ) )  /\  ( T  .\/  V )  =/=  ( P  .\/  Q
) ) )  /\  V  .<_  ( P  .\/  Q ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
27 simpl13 1034 . . . . . . . . . . 11  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( ( P  =/=  Q  /\  S  =/=  T )  /\  ( S  .<_  ( T 
.\/  V )  /\  S  .<_  ( P  .\/  Q ) )  /\  ( T  .\/  V )  =/=  ( P  .\/  Q
) ) )  /\  V  .<_  ( P  .\/  Q ) )  ->  Q  e.  A )
28 simp31l 1080 . . . . . . . . . . . 12  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( ( P  =/=  Q  /\  S  =/=  T )  /\  ( S  .<_  ( T 
.\/  V )  /\  S  .<_  ( P  .\/  Q ) )  /\  ( T  .\/  V )  =/=  ( P  .\/  Q
) ) )  ->  P  =/=  Q )
2928adantr 452 . . . . . . . . . . 11  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( ( P  =/=  Q  /\  S  =/=  T )  /\  ( S  .<_  ( T 
.\/  V )  /\  S  .<_  ( P  .\/  Q ) )  /\  ( T  .\/  V )  =/=  ( P  .\/  Q
) ) )  /\  V  .<_  ( P  .\/  Q ) )  ->  P  =/=  Q )
30 simp23l 1078 . . . . . . . . . . . 12  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( ( P  =/=  Q  /\  S  =/=  T )  /\  ( S  .<_  ( T 
.\/  V )  /\  S  .<_  ( P  .\/  Q ) )  /\  ( T  .\/  V )  =/=  ( P  .\/  Q
) ) )  ->  V  e.  A )
3130adantr 452 . . . . . . . . . . 11  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( ( P  =/=  Q  /\  S  =/=  T )  /\  ( S  .<_  ( T 
.\/  V )  /\  S  .<_  ( P  .\/  Q ) )  /\  ( T  .\/  V )  =/=  ( P  .\/  Q
) ) )  /\  V  .<_  ( P  .\/  Q ) )  ->  V  e.  A )
32 simp23r 1079 . . . . . . . . . . . 12  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( ( P  =/=  Q  /\  S  =/=  T )  /\  ( S  .<_  ( T 
.\/  V )  /\  S  .<_  ( P  .\/  Q ) )  /\  ( T  .\/  V )  =/=  ( P  .\/  Q
) ) )  ->  V  .<_  W )
3332adantr 452 . . . . . . . . . . 11  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( ( P  =/=  Q  /\  S  =/=  T )  /\  ( S  .<_  ( T 
.\/  V )  /\  S  .<_  ( P  .\/  Q ) )  /\  ( T  .\/  V )  =/=  ( P  .\/  Q
) ) )  /\  V  .<_  ( P  .\/  Q ) )  ->  V  .<_  W )
34 simpr 448 . . . . . . . . . . 11  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( ( P  =/=  Q  /\  S  =/=  T )  /\  ( S  .<_  ( T 
.\/  V )  /\  S  .<_  ( P  .\/  Q ) )  /\  ( T  .\/  V )  =/=  ( P  .\/  Q
) ) )  /\  V  .<_  ( P  .\/  Q ) )  ->  V  .<_  ( P  .\/  Q
) )
35 eqid 2404 . . . . . . . . . . . 12  |-  ( ( P  .\/  Q ) 
./\  W )  =  ( ( P  .\/  Q )  ./\  W )
3615, 7, 16, 8, 12, 35cdleme22aa 30821 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  P  =/=  Q )  /\  ( V  e.  A  /\  V  .<_  W  /\  V  .<_  ( P  .\/  Q ) ) )  ->  V  =  ( ( P  .\/  Q )  ./\  W ) )
3724, 25, 26, 27, 29, 31, 33, 34, 36syl233anc 1213 . . . . . . . . . 10  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( ( P  =/=  Q  /\  S  =/=  T )  /\  ( S  .<_  ( T 
.\/  V )  /\  S  .<_  ( P  .\/  Q ) )  /\  ( T  .\/  V )  =/=  ( P  .\/  Q
) ) )  /\  V  .<_  ( P  .\/  Q ) )  ->  V  =  ( ( P 
.\/  Q )  ./\  W ) )
3837oveq2d 6056 . . . . . . . . 9  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( ( P  =/=  Q  /\  S  =/=  T )  /\  ( S  .<_  ( T 
.\/  V )  /\  S  .<_  ( P  .\/  Q ) )  /\  ( T  .\/  V )  =/=  ( P  .\/  Q
) ) )  /\  V  .<_  ( P  .\/  Q ) )  ->  ( T  .\/  V )  =  ( T  .\/  (
( P  .\/  Q
)  ./\  W )
) )
3923, 38breqtrd 4196 . . . . . . . 8  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( ( P  =/=  Q  /\  S  =/=  T )  /\  ( S  .<_  ( T 
.\/  V )  /\  S  .<_  ( P  .\/  Q ) )  /\  ( T  .\/  V )  =/=  ( P  .\/  Q
) ) )  /\  V  .<_  ( P  .\/  Q ) )  ->  S  .<_  ( T  .\/  (
( P  .\/  Q
)  ./\  W )
) )
40 simp32r 1083 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( ( P  =/=  Q  /\  S  =/=  T )  /\  ( S  .<_  ( T 
.\/  V )  /\  S  .<_  ( P  .\/  Q ) )  /\  ( T  .\/  V )  =/=  ( P  .\/  Q
) ) )  ->  S  .<_  ( P  .\/  Q ) )
4140adantr 452 . . . . . . . 8  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( ( P  =/=  Q  /\  S  =/=  T )  /\  ( S  .<_  ( T 
.\/  V )  /\  S  .<_  ( P  .\/  Q ) )  /\  ( T  .\/  V )  =/=  ( P  .\/  Q
) ) )  /\  V  .<_  ( P  .\/  Q ) )  ->  S  .<_  ( P  .\/  Q
) )
42 simp21l 1074 . . . . . . . . . . 11  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( ( P  =/=  Q  /\  S  =/=  T )  /\  ( S  .<_  ( T 
.\/  V )  /\  S  .<_  ( P  .\/  Q ) )  /\  ( T  .\/  V )  =/=  ( P  .\/  Q
) ) )  ->  S  e.  A )
436, 8atbase 29772 . . . . . . . . . . 11  |-  ( S  e.  A  ->  S  e.  ( Base `  K
) )
4442, 43syl 16 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( ( P  =/=  Q  /\  S  =/=  T )  /\  ( S  .<_  ( T 
.\/  V )  /\  S  .<_  ( P  .\/  Q ) )  /\  ( T  .\/  V )  =/=  ( P  .\/  Q
) ) )  ->  S  e.  ( Base `  K ) )
45 simp22 991 . . . . . . . . . . 11  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( ( P  =/=  Q  /\  S  =/=  T )  /\  ( S  .<_  ( T 
.\/  V )  /\  S  .<_  ( P  .\/  Q ) )  /\  ( T  .\/  V )  =/=  ( P  .\/  Q
) ) )  ->  T  e.  A )
46 simp12r 1071 . . . . . . . . . . . 12  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( ( P  =/=  Q  /\  S  =/=  T )  /\  ( S  .<_  ( T 
.\/  V )  /\  S  .<_  ( P  .\/  Q ) )  /\  ( T  .\/  V )  =/=  ( P  .\/  Q
) ) )  ->  -.  P  .<_  W )
4715, 7, 16, 8, 12lhpat 30525 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q ) )  ->  ( ( P 
.\/  Q )  ./\  W )  e.  A )
481, 11, 4, 46, 5, 28, 47syl222anc 1200 . . . . . . . . . . 11  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( ( P  =/=  Q  /\  S  =/=  T )  /\  ( S  .<_  ( T 
.\/  V )  /\  S  .<_  ( P  .\/  Q ) )  /\  ( T  .\/  V )  =/=  ( P  .\/  Q
) ) )  -> 
( ( P  .\/  Q )  ./\  W )  e.  A )
496, 7, 8hlatjcl 29849 . . . . . . . . . . 11  |-  ( ( K  e.  HL  /\  T  e.  A  /\  ( ( P  .\/  Q )  ./\  W )  e.  A )  ->  ( T  .\/  ( ( P 
.\/  Q )  ./\  W ) )  e.  (
Base `  K )
)
501, 45, 48, 49syl3anc 1184 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( ( P  =/=  Q  /\  S  =/=  T )  /\  ( S  .<_  ( T 
.\/  V )  /\  S  .<_  ( P  .\/  Q ) )  /\  ( T  .\/  V )  =/=  ( P  .\/  Q
) ) )  -> 
( T  .\/  (
( P  .\/  Q
)  ./\  W )
)  e.  ( Base `  K ) )
516, 15, 16latlem12 14462 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  ( S  e.  ( Base `  K )  /\  ( T  .\/  ( ( P  .\/  Q ) 
./\  W ) )  e.  ( Base `  K
)  /\  ( P  .\/  Q )  e.  (
Base `  K )
) )  ->  (
( S  .<_  ( T 
.\/  ( ( P 
.\/  Q )  ./\  W ) )  /\  S  .<_  ( P  .\/  Q
) )  <->  S  .<_  ( ( T  .\/  (
( P  .\/  Q
)  ./\  W )
)  ./\  ( P  .\/  Q ) ) ) )
523, 44, 50, 10, 51syl13anc 1186 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( ( P  =/=  Q  /\  S  =/=  T )  /\  ( S  .<_  ( T 
.\/  V )  /\  S  .<_  ( P  .\/  Q ) )  /\  ( T  .\/  V )  =/=  ( P  .\/  Q
) ) )  -> 
( ( S  .<_  ( T  .\/  ( ( P  .\/  Q ) 
./\  W ) )  /\  S  .<_  ( P 
.\/  Q ) )  <-> 
S  .<_  ( ( T 
.\/  ( ( P 
.\/  Q )  ./\  W ) )  ./\  ( P  .\/  Q ) ) ) )
5352adantr 452 . . . . . . . 8  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( ( P  =/=  Q  /\  S  =/=  T )  /\  ( S  .<_  ( T 
.\/  V )  /\  S  .<_  ( P  .\/  Q ) )  /\  ( T  .\/  V )  =/=  ( P  .\/  Q
) ) )  /\  V  .<_  ( P  .\/  Q ) )  ->  (
( S  .<_  ( T 
.\/  ( ( P 
.\/  Q )  ./\  W ) )  /\  S  .<_  ( P  .\/  Q
) )  <->  S  .<_  ( ( T  .\/  (
( P  .\/  Q
)  ./\  W )
)  ./\  ( P  .\/  Q ) ) ) )
5439, 41, 53mpbi2and 888 . . . . . . 7  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( ( P  =/=  Q  /\  S  =/=  T )  /\  ( S  .<_  ( T 
.\/  V )  /\  S  .<_  ( P  .\/  Q ) )  /\  ( T  .\/  V )  =/=  ( P  .\/  Q
) ) )  /\  V  .<_  ( P  .\/  Q ) )  ->  S  .<_  ( ( T  .\/  ( ( P  .\/  Q )  ./\  W )
)  ./\  ( P  .\/  Q ) ) )
55 simp31r 1081 . . . . . . . . . . . . 13  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( ( P  =/=  Q  /\  S  =/=  T )  /\  ( S  .<_  ( T 
.\/  V )  /\  S  .<_  ( P  .\/  Q ) )  /\  ( T  .\/  V )  =/=  ( P  .\/  Q
) ) )  ->  S  =/=  T )
5642, 45, 553jca 1134 . . . . . . . . . . . 12  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( ( P  =/=  Q  /\  S  =/=  T )  /\  ( S  .<_  ( T 
.\/  V )  /\  S  .<_  ( P  .\/  Q ) )  /\  ( T  .\/  V )  =/=  ( P  .\/  Q
) ) )  -> 
( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )
57 simp33 995 . . . . . . . . . . . . 13  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( ( P  =/=  Q  /\  S  =/=  T )  /\  ( S  .<_  ( T 
.\/  V )  /\  S  .<_  ( P  .\/  Q ) )  /\  ( T  .\/  V )  =/=  ( P  .\/  Q
) ) )  -> 
( T  .\/  V
)  =/=  ( P 
.\/  Q ) )
5857, 22, 403jca 1134 . . . . . . . . . . . 12  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( ( P  =/=  Q  /\  S  =/=  T )  /\  ( S  .<_  ( T 
.\/  V )  /\  S  .<_  ( P  .\/  Q ) )  /\  ( T  .\/  V )  =/=  ( P  .\/  Q
) ) )  -> 
( ( T  .\/  V )  =/=  ( P 
.\/  Q )  /\  S  .<_  ( T  .\/  V )  /\  S  .<_  ( P  .\/  Q ) ) )
5915, 7, 16, 8, 12cdleme22b 30823 . . . . . . . . . . . 12  |-  ( ( ( 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 ) )
601, 56, 4, 5, 28, 30, 58, 59syl232anc 1211 . . . . . . . . . . 11  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( ( P  =/=  Q  /\  S  =/=  T )  /\  ( S  .<_  ( T 
.\/  V )  /\  S  .<_  ( P  .\/  Q ) )  /\  ( T  .\/  V )  =/=  ( P  .\/  Q
) ) )  ->  -.  T  .<_  ( P 
.\/  Q ) )
61 hlatl 29843 . . . . . . . . . . . . 13  |-  ( K  e.  HL  ->  K  e.  AtLat )
621, 61syl 16 . . . . . . . . . . . 12  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( ( P  =/=  Q  /\  S  =/=  T )  /\  ( S  .<_  ( T 
.\/  V )  /\  S  .<_  ( P  .\/  Q ) )  /\  ( T  .\/  V )  =/=  ( P  .\/  Q
) ) )  ->  K  e.  AtLat )
63 eqid 2404 . . . . . . . . . . . . 13  |-  ( 0.
`  K )  =  ( 0. `  K
)
646, 15, 16, 63, 8atnle 29800 . . . . . . . . . . . 12  |-  ( ( K  e.  AtLat  /\  T  e.  A  /\  ( P  .\/  Q )  e.  ( Base `  K
) )  ->  ( -.  T  .<_  ( P 
.\/  Q )  <->  ( T  ./\  ( P  .\/  Q
) )  =  ( 0. `  K ) ) )
6562, 45, 10, 64syl3anc 1184 . . . . . . . . . . 11  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( ( P  =/=  Q  /\  S  =/=  T )  /\  ( S  .<_  ( T 
.\/  V )  /\  S  .<_  ( P  .\/  Q ) )  /\  ( T  .\/  V )  =/=  ( P  .\/  Q
) ) )  -> 
( -.  T  .<_  ( P  .\/  Q )  <-> 
( T  ./\  ( P  .\/  Q ) )  =  ( 0. `  K ) ) )
6660, 65mpbid 202 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( ( P  =/=  Q  /\  S  =/=  T )  /\  ( S  .<_  ( T 
.\/  V )  /\  S  .<_  ( P  .\/  Q ) )  /\  ( T  .\/  V )  =/=  ( P  .\/  Q
) ) )  -> 
( T  ./\  ( P  .\/  Q ) )  =  ( 0. `  K ) )
6766oveq1d 6055 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( ( P  =/=  Q  /\  S  =/=  T )  /\  ( S  .<_  ( T 
.\/  V )  /\  S  .<_  ( P  .\/  Q ) )  /\  ( T  .\/  V )  =/=  ( P  .\/  Q
) ) )  -> 
( ( T  ./\  ( P  .\/  Q ) )  .\/  ( ( P  .\/  Q ) 
./\  W ) )  =  ( ( 0.
`  K )  .\/  ( ( P  .\/  Q )  ./\  W )
) )
686, 8atbase 29772 . . . . . . . . . . 11  |-  ( T  e.  A  ->  T  e.  ( Base `  K
) )
6945, 68syl 16 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( ( P  =/=  Q  /\  S  =/=  T )  /\  ( S  .<_  ( T 
.\/  V )  /\  S  .<_  ( P  .\/  Q ) )  /\  ( T  .\/  V )  =/=  ( P  .\/  Q
) ) )  ->  T  e.  ( Base `  K ) )
706, 15, 16latmle1 14460 . . . . . . . . . . 11  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  ./\  W )  .<_  ( P  .\/  Q ) )
713, 10, 14, 70syl3anc 1184 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( ( P  =/=  Q  /\  S  =/=  T )  /\  ( S  .<_  ( T 
.\/  V )  /\  S  .<_  ( P  .\/  Q ) )  /\  ( T  .\/  V )  =/=  ( P  .\/  Q
) ) )  -> 
( ( P  .\/  Q )  ./\  W )  .<_  ( P  .\/  Q
) )
726, 15, 7, 16, 8atmod4i1 30348 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  ( ( ( P 
.\/  Q )  ./\  W )  e.  A  /\  T  e.  ( Base `  K )  /\  ( P  .\/  Q )  e.  ( Base `  K
) )  /\  (
( P  .\/  Q
)  ./\  W )  .<_  ( P  .\/  Q
) )  ->  (
( T  ./\  ( P  .\/  Q ) ) 
.\/  ( ( P 
.\/  Q )  ./\  W ) )  =  ( ( T  .\/  (
( P  .\/  Q
)  ./\  W )
)  ./\  ( P  .\/  Q ) ) )
731, 48, 69, 10, 71, 72syl131anc 1197 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( ( P  =/=  Q  /\  S  =/=  T )  /\  ( S  .<_  ( T 
.\/  V )  /\  S  .<_  ( P  .\/  Q ) )  /\  ( T  .\/  V )  =/=  ( P  .\/  Q
) ) )  -> 
( ( T  ./\  ( P  .\/  Q ) )  .\/  ( ( P  .\/  Q ) 
./\  W ) )  =  ( ( T 
.\/  ( ( P 
.\/  Q )  ./\  W ) )  ./\  ( P  .\/  Q ) ) )
74 hlol 29844 . . . . . . . . . . 11  |-  ( K  e.  HL  ->  K  e.  OL )
751, 74syl 16 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( ( P  =/=  Q  /\  S  =/=  T )  /\  ( S  .<_  ( T 
.\/  V )  /\  S  .<_  ( P  .\/  Q ) )  /\  ( T  .\/  V )  =/=  ( P  .\/  Q
) ) )  ->  K  e.  OL )
766, 16latmcl 14435 . . . . . . . . . . 11  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  ./\  W )  e.  ( Base `  K ) )
773, 10, 14, 76syl3anc 1184 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( ( P  =/=  Q  /\  S  =/=  T )  /\  ( S  .<_  ( T 
.\/  V )  /\  S  .<_  ( P  .\/  Q ) )  /\  ( T  .\/  V )  =/=  ( P  .\/  Q
) ) )  -> 
( ( P  .\/  Q )  ./\  W )  e.  ( Base `  K
) )
786, 7, 63olj02 29709 . . . . . . . . . 10  |-  ( ( K  e.  OL  /\  ( ( P  .\/  Q )  ./\  W )  e.  ( Base `  K
) )  ->  (
( 0. `  K
)  .\/  ( ( P  .\/  Q )  ./\  W ) )  =  ( ( P  .\/  Q
)  ./\  W )
)
7975, 77, 78syl2anc 643 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( ( P  =/=  Q  /\  S  =/=  T )  /\  ( S  .<_  ( T 
.\/  V )  /\  S  .<_  ( P  .\/  Q ) )  /\  ( T  .\/  V )  =/=  ( P  .\/  Q
) ) )  -> 
( ( 0. `  K )  .\/  (
( P  .\/  Q
)  ./\  W )
)  =  ( ( P  .\/  Q ) 
./\  W ) )
8067, 73, 793eqtr3d 2444 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( ( P  =/=  Q  /\  S  =/=  T )  /\  ( S  .<_  ( T 
.\/  V )  /\  S  .<_  ( P  .\/  Q ) )  /\  ( T  .\/  V )  =/=  ( P  .\/  Q
) ) )  -> 
( ( T  .\/  ( ( P  .\/  Q )  ./\  W )
)  ./\  ( P  .\/  Q ) )  =  ( ( P  .\/  Q )  ./\  W )
)
8180adantr 452 . . . . . . 7  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( ( P  =/=  Q  /\  S  =/=  T )  /\  ( S  .<_  ( T 
.\/  V )  /\  S  .<_  ( P  .\/  Q ) )  /\  ( T  .\/  V )  =/=  ( P  .\/  Q
) ) )  /\  V  .<_  ( P  .\/  Q ) )  ->  (
( T  .\/  (
( P  .\/  Q
)  ./\  W )
)  ./\  ( P  .\/  Q ) )  =  ( ( P  .\/  Q )  ./\  W )
)
8254, 81breqtrd 4196 . . . . . 6  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( ( P  =/=  Q  /\  S  =/=  T )  /\  ( S  .<_  ( T 
.\/  V )  /\  S  .<_  ( P  .\/  Q ) )  /\  ( T  .\/  V )  =/=  ( P  .\/  Q
) ) )  /\  V  .<_  ( P  .\/  Q ) )  ->  S  .<_  ( ( P  .\/  Q )  ./\  W )
)
8315, 8atcmp 29794 . . . . . . . 8  |-  ( ( K  e.  AtLat  /\  S  e.  A  /\  (
( P  .\/  Q
)  ./\  W )  e.  A )  ->  ( S  .<_  ( ( P 
.\/  Q )  ./\  W )  <->  S  =  (
( P  .\/  Q
)  ./\  W )
) )
8462, 42, 48, 83syl3anc 1184 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( ( P  =/=  Q  /\  S  =/=  T )  /\  ( S  .<_  ( T 
.\/  V )  /\  S  .<_  ( P  .\/  Q ) )  /\  ( T  .\/  V )  =/=  ( P  .\/  Q
) ) )  -> 
( S  .<_  ( ( P  .\/  Q ) 
./\  W )  <->  S  =  ( ( P  .\/  Q )  ./\  W )
) )
8584adantr 452 . . . . . 6  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( ( P  =/=  Q  /\  S  =/=  T )  /\  ( S  .<_  ( T 
.\/  V )  /\  S  .<_  ( P  .\/  Q ) )  /\  ( T  .\/  V )  =/=  ( P  .\/  Q
) ) )  /\  V  .<_  ( P  .\/  Q ) )  ->  ( S  .<_  ( ( P 
.\/  Q )  ./\  W )  <->  S  =  (
( P  .\/  Q
)  ./\  W )
) )
8682, 85mpbid 202 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( ( P  =/=  Q  /\  S  =/=  T )  /\  ( S  .<_  ( T 
.\/  V )  /\  S  .<_  ( P  .\/  Q ) )  /\  ( T  .\/  V )  =/=  ( P  .\/  Q
) ) )  /\  V  .<_  ( P  .\/  Q ) )  ->  S  =  ( ( P 
.\/  Q )  ./\  W ) )
8786eqcomd 2409 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( ( P  =/=  Q  /\  S  =/=  T )  /\  ( S  .<_  ( T 
.\/  V )  /\  S  .<_  ( P  .\/  Q ) )  /\  ( T  .\/  V )  =/=  ( P  .\/  Q
) ) )  /\  V  .<_  ( P  .\/  Q ) )  ->  (
( P  .\/  Q
)  ./\  W )  =  S )
8887ex 424 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( ( P  =/=  Q  /\  S  =/=  T )  /\  ( S  .<_  ( T 
.\/  V )  /\  S  .<_  ( P  .\/  Q ) )  /\  ( T  .\/  V )  =/=  ( P  .\/  Q
) ) )  -> 
( V  .<_  ( P 
.\/  Q )  -> 
( ( P  .\/  Q )  ./\  W )  =  S ) )
8988necon3ad 2603 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( ( P  =/=  Q  /\  S  =/=  T )  /\  ( S  .<_  ( T 
.\/  V )  /\  S  .<_  ( P  .\/  Q ) )  /\  ( T  .\/  V )  =/=  ( P  .\/  Q
) ) )  -> 
( ( ( P 
.\/  Q )  ./\  W )  =/=  S  ->  -.  V  .<_  ( P 
.\/  Q ) ) )
9021, 89mpd 15 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( ( P  =/=  Q  /\  S  =/=  T )  /\  ( S  .<_  ( T 
.\/  V )  /\  S  .<_  ( P  .\/  Q ) )  /\  ( T  .\/  V )  =/=  ( P  .\/  Q
) ) )  ->  -.  V  .<_  ( P 
.\/  Q ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   Basecbs 13424   lecple 13491   joincjn 14356   meetcmee 14357   0.cp0 14421   Latclat 14429   OLcol 29657   Atomscatm 29746   AtLatcal 29747   HLchlt 29833   LHypclh 30466
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-undef 6502  df-riota 6508  df-poset 14358  df-plt 14370  df-lub 14386  df-glb 14387  df-join 14388  df-meet 14389  df-p0 14423  df-p1 14424  df-lat 14430  df-clat 14492  df-oposet 29659  df-ol 29661  df-oml 29662  df-covers 29749  df-ats 29750  df-atl 29781  df-cvlat 29805  df-hlat 29834  df-llines 29980  df-psubsp 29985  df-pmap 29986  df-padd 30278  df-lhyp 30470
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