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Theorem cdleme22cN 29661
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 1071 . . . . 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 28683 . . . . 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 1073 . . . . 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 992 . . . . 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 2256 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
7 cdleme22.j . . . . . 6  |-  .\/  =  ( join `  K )
8 cdleme22.a . . . . . 6  |-  A  =  ( Atoms `  K )
96, 7, 8hlatjcl 28686 . . . . 5  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
101, 4, 5, 9syl3anc 1187 . . . 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 1072 . . . . 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 29317 . . . . 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 14110 . . . 4  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  ./\  W )  .<_  W )
183, 10, 14, 17syl3anc 1187 . . 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 1078 . . 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 3981 . . 3  |-  ( ( ( ( P  .\/  Q )  ./\  W )  .<_  W  /\  -.  S  .<_  W )  ->  (
( P  .\/  Q
)  ./\  W )  =/=  S )
2118, 19, 20syl2anc 645 . 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 1085 . . . . . . . . . 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 453 . . . . . . . . 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 453 . . . . . . . . . . 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 453 . . . . . . . . . . 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 1036 . . . . . . . . . . 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 1037 . . . . . . . . . . 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 1083 . . . . . . . . . . . 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 453 . . . . . . . . . . 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 1081 . . . . . . . . . . . 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 453 . . . . . . . . . . 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 1082 . . . . . . . . . . . 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 453 . . . . . . . . . . 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 449 . . . . . . . . . . 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 2256 . . . . . . . . . . . 12  |-  ( ( P  .\/  Q ) 
./\  W )  =  ( ( P  .\/  Q )  ./\  W )
3615, 7, 16, 8, 12, 35cdleme22aa 29658 . . . . . . . . . . 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 1216 . . . . . . . . . 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 5773 . . . . . . . . 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 3987 . . . . . . . 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 1086 . . . . . . . . 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 453 . . . . . . . 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 1077 . . . . . . . . . . 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 28609 . . . . . . . . . . 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 994 . . . . . . . . . . 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 1074 . . . . . . . . . . . 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 29362 . . . . . . . . . . . 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 1203 . . . . . . . . . . 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 28686 . . . . . . . . . . 11  |-  ( ( K  e.  HL  /\  T  e.  A  /\  ( ( P  .\/  Q )  ./\  W )  e.  A )  ->  ( T  .\/  ( ( P 
.\/  Q )  ./\  W ) )  e.  (
Base `  K )
)
501, 45, 48, 49syl3anc 1187 . . . . . . . . . 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 14111 . . . . . . . . . 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 1189 . . . . . . . . 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 453 . . . . . . . 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 892 . . . . . . 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 1084 . . . . . . . . . . . . 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 1137 . . . . . . . . . . . 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 998 . . . . . . . . . . . . 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 1137 . . . . . . . . . . . 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 29660 . . . . . . . . . . . 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 1214 . . . . . . . . . . 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 28680 . . . . . . . . . . . . 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 2256 . . . . . . . . . . . . 13  |-  ( 0.
`  K )  =  ( 0. `  K
)
646, 15, 16, 63, 8atnle 28637 . . . . . . . . . . . 12  |-  ( ( K  e.  AtLat  /\  T  e.  A  /\  ( P  .\/  Q )  e.  ( Base `  K
) )  ->  ( -.  T  .<_  ( P 
.\/  Q )  <->  ( T  ./\  ( P  .\/  Q
) )  =  ( 0. `  K ) ) )
6562, 45, 10, 64syl3anc 1187 . . . . . . . . . . 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 203 . . . . . . . . . 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 5772 . . . . . . . . 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 28609 . . . . . . . . . . 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 14109 . . . . . . . . . . 11  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  ./\  W )  .<_  ( P  .\/  Q ) )
713, 10, 14, 70syl3anc 1187 . . . . . . . . . 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 29185 . . . . . . . . . 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 1200 . . . . . . . . 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 28681 . . . . . . . . . . 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 14084 . . . . . . . . . . 11  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  ./\  W )  e.  ( Base `  K ) )
773, 10, 14, 76syl3anc 1187 . . . . . . . . . 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 28546 . . . . . . . . . 10  |-  ( ( K  e.  OL  /\  ( ( P  .\/  Q )  ./\  W )  e.  ( Base `  K
) )  ->  (
( 0. `  K
)  .\/  ( ( P  .\/  Q )  ./\  W ) )  =  ( ( P  .\/  Q
)  ./\  W )
)
7975, 77, 78syl2anc 645 . . . . . . . . 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 2296 . . . . . . . 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 453 . . . . . . 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 3987 . . . . . 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 28631 . . . . . . . 8  |-  ( ( K  e.  AtLat  /\  S  e.  A  /\  (
( P  .\/  Q
)  ./\  W )  e.  A )  ->  ( S  .<_  ( ( P 
.\/  Q )  ./\  W )  <->  S  =  (
( P  .\/  Q
)  ./\  W )
) )
8462, 42, 48, 83syl3anc 1187 . . . . . . 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 453 . . . . . 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 203 . . . . 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 2261 . . . 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 425 . . 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 2455 . 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 16 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 5    -> wi 6    <-> wb 178    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2419   class class class wbr 3963   ` cfv 4638  (class class class)co 5757   Basecbs 13075   lecple 13142   joincjn 14005   meetcmee 14006   0.cp0 14070   Latclat 14078   OLcol 28494   Atomscatm 28583   AtLatcal 28584   HLchlt 28670   LHypclh 29303
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4071  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3769  df-iun 3848  df-iin 3849  df-br 3964  df-opab 4018  df-mpt 4019  df-id 4246  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-1st 6021  df-2nd 6022  df-iota 6190  df-undef 6229  df-riota 6237  df-poset 14007  df-plt 14019  df-lub 14035  df-glb 14036  df-join 14037  df-meet 14038  df-p0 14072  df-p1 14073  df-lat 14079  df-clat 14141  df-oposet 28496  df-ol 28498  df-oml 28499  df-covers 28586  df-ats 28587  df-atl 28618  df-cvlat 28642  df-hlat 28671  df-llines 28817  df-psubsp 28822  df-pmap 28823  df-padd 29115  df-lhyp 29307
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