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Theorem cdleme22cN 33879
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 1116 . . . . 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 32899 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
31, 2syl 17 . . . 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 1118 . . . . 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 1037 . . . . 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 2422 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
7 cdleme22.j . . . . . 6  |-  .\/  =  ( join `  K )
8 cdleme22.a . . . . . 6  |-  A  =  ( Atoms `  K )
96, 7, 8hlatjcl 32902 . . . . 5  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
101, 4, 5, 9syl3anc 1264 . . . 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 1117 . . . . 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 33533 . . . . 5  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
1411, 13syl 17 . . . 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 16323 . . . 4  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  ./\  W )  .<_  W )
183, 10, 14, 17syl3anc 1264 . . 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 1123 . . 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 4442 . . 3  |-  ( ( ( ( P  .\/  Q )  ./\  W )  .<_  W  /\  -.  S  .<_  W )  ->  (
( P  .\/  Q
)  ./\  W )  =/=  S )
2118, 19, 20syl2anc 665 . 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 1130 . . . . . . . . . 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 466 . . . . . . . . 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 466 . . . . . . . . . . 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 466 . . . . . . . . . . 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 1081 . . . . . . . . . . 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 1082 . . . . . . . . . . 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 1128 . . . . . . . . . . . 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 466 . . . . . . . . . . 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 1126 . . . . . . . . . . . 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 466 . . . . . . . . . . 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 1127 . . . . . . . . . . . 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 466 . . . . . . . . . . 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 462 . . . . . . . . . . 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 2422 . . . . . . . . . . . 12  |-  ( ( P  .\/  Q ) 
./\  W )  =  ( ( P  .\/  Q )  ./\  W )
3615, 7, 16, 8, 12, 35cdleme22aa 33876 . . . . . . . . . . 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 1293 . . . . . . . . . 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 6322 . . . . . . . . 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 4448 . . . . . . . 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 1131 . . . . . . . . 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 466 . . . . . . . 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 1122 . . . . . . . . . . 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 32825 . . . . . . . . . . 11  |-  ( S  e.  A  ->  S  e.  ( Base `  K
) )
4442, 43syl 17 . . . . . . . . . 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 1039 . . . . . . . . . . 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 1119 . . . . . . . . . . . 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 33578 . . . . . . . . . . . 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 1280 . . . . . . . . . . 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 32902 . . . . . . . . . . 11  |-  ( ( K  e.  HL  /\  T  e.  A  /\  ( ( P  .\/  Q )  ./\  W )  e.  A )  ->  ( T  .\/  ( ( P 
.\/  Q )  ./\  W ) )  e.  (
Base `  K )
)
501, 45, 48, 49syl3anc 1264 . . . . . . . . . 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 16324 . . . . . . . . . 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 1266 . . . . . . . . 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 466 . . . . . . . 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 929 . . . . . . 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 1129 . . . . . . . . . . . . 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 1185 . . . . . . . . . . . 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 1043 . . . . . . . . . . . . 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 1185 . . . . . . . . . . . 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 33878 . . . . . . . . . . . 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 1291 . . . . . . . . . . 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 32896 . . . . . . . . . . . . 13  |-  ( K  e.  HL  ->  K  e.  AtLat )
621, 61syl 17 . . . . . . . . . . . 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 2422 . . . . . . . . . . . . 13  |-  ( 0.
`  K )  =  ( 0. `  K
)
646, 15, 16, 63, 8atnle 32853 . . . . . . . . . . . 12  |-  ( ( K  e.  AtLat  /\  T  e.  A  /\  ( P  .\/  Q )  e.  ( Base `  K
) )  ->  ( -.  T  .<_  ( P 
.\/  Q )  <->  ( T  ./\  ( P  .\/  Q
) )  =  ( 0. `  K ) ) )
6562, 45, 10, 64syl3anc 1264 . . . . . . . . . . 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 213 . . . . . . . . . 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 6321 . . . . . . . . 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 32825 . . . . . . . . . . 11  |-  ( T  e.  A  ->  T  e.  ( Base `  K
) )
6945, 68syl 17 . . . . . . . . . 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 16322 . . . . . . . . . . 11  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  ./\  W )  .<_  ( P  .\/  Q ) )
713, 10, 14, 70syl3anc 1264 . . . . . . . . . 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 33401 . . . . . . . . . 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 1277 . . . . . . . . 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 32897 . . . . . . . . . . 11  |-  ( K  e.  HL  ->  K  e.  OL )
751, 74syl 17 . . . . . . . . . 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 16298 . . . . . . . . . . 11  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  ./\  W )  e.  ( Base `  K ) )
773, 10, 14, 76syl3anc 1264 . . . . . . . . . 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 32762 . . . . . . . . . 10  |-  ( ( K  e.  OL  /\  ( ( P  .\/  Q )  ./\  W )  e.  ( Base `  K
) )  ->  (
( 0. `  K
)  .\/  ( ( P  .\/  Q )  ./\  W ) )  =  ( ( P  .\/  Q
)  ./\  W )
)
7975, 77, 78syl2anc 665 . . . . . . . . 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 2471 . . . . . . . 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 466 . . . . . . 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 4448 . . . . . 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 32847 . . . . . . . 8  |-  ( ( K  e.  AtLat  /\  S  e.  A  /\  (
( P  .\/  Q
)  ./\  W )  e.  A )  ->  ( S  .<_  ( ( P 
.\/  Q )  ./\  W )  <->  S  =  (
( P  .\/  Q
)  ./\  W )
) )
8462, 42, 48, 83syl3anc 1264 . . . . . . 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 466 . . . . . 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 213 . . . . 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 2430 . . . 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 435 . . 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 2630 . 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872    =/= wne 2614   class class class wbr 4423   ` cfv 5601  (class class class)co 6306   Basecbs 15121   lecple 15197   joincjn 16189   meetcmee 16190   0.cp0 16283   Latclat 16291   OLcol 32710   Atomscatm 32799   AtLatcal 32800   HLchlt 32886   LHypclh 33519
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6598
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-reu 2778  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-iun 4301  df-iin 4302  df-br 4424  df-opab 4483  df-mpt 4484  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6268  df-ov 6309  df-oprab 6310  df-mpt2 6311  df-1st 6808  df-2nd 6809  df-preset 16173  df-poset 16191  df-plt 16204  df-lub 16220  df-glb 16221  df-join 16222  df-meet 16223  df-p0 16285  df-p1 16286  df-lat 16292  df-clat 16354  df-oposet 32712  df-ol 32714  df-oml 32715  df-covers 32802  df-ats 32803  df-atl 32834  df-cvlat 32858  df-hlat 32887  df-llines 33033  df-psubsp 33038  df-pmap 33039  df-padd 33331  df-lhyp 33523
This theorem is referenced by: (None)
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