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Theorem cdlemg18b 36506
Description: Lemma for cdlemg18c 36507. TODO: fix comment. (Contributed by NM, 15-May-2013.)
Hypotheses
Ref Expression
cdlemg12.l  |-  .<_  =  ( le `  K )
cdlemg12.j  |-  .\/  =  ( join `  K )
cdlemg12.m  |-  ./\  =  ( meet `  K )
cdlemg12.a  |-  A  =  ( Atoms `  K )
cdlemg12.h  |-  H  =  ( LHyp `  K
)
cdlemg12.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemg12b.r  |-  R  =  ( ( trL `  K
) `  W )
cdlemg18b.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
Assertion
Ref Expression
cdlemg18b  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( P  =/=  Q  /\  ( F `
 P )  =/= 
Q  /\  ( ( F `  Q )  .\/  ( F `  P
) )  =/=  ( P  .\/  Q ) ) )  ->  -.  P  .<_  ( U  .\/  ( F `  Q )
) )

Proof of Theorem cdlemg18b
StepHypRef Expression
1 simp33 1034 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( P  =/=  Q  /\  ( F `
 P )  =/= 
Q  /\  ( ( F `  Q )  .\/  ( F `  P
) )  =/=  ( P  .\/  Q ) ) )  ->  ( ( F `  Q )  .\/  ( F `  P
) )  =/=  ( P  .\/  Q ) )
2 simp3r 1025 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( ( P  =/=  Q  /\  ( F `  P )  =/=  Q  /\  ( ( F `  Q ) 
.\/  ( F `  P ) )  =/=  ( P  .\/  Q
) )  /\  P  .<_  ( U  .\/  ( F `  Q )
) ) )  ->  P  .<_  ( U  .\/  ( F `  Q ) ) )
3 simp1l 1020 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( ( P  =/=  Q  /\  ( F `  P )  =/=  Q  /\  ( ( F `  Q ) 
.\/  ( F `  P ) )  =/=  ( P  .\/  Q
) )  /\  P  .<_  ( U  .\/  ( F `  Q )
) ) )  ->  K  e.  HL )
4 simp1r 1021 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( ( P  =/=  Q  /\  ( F `  P )  =/=  Q  /\  ( ( F `  Q ) 
.\/  ( F `  P ) )  =/=  ( P  .\/  Q
) )  /\  P  .<_  ( U  .\/  ( F `  Q )
) ) )  ->  W  e.  H )
5 simp21 1029 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( ( P  =/=  Q  /\  ( F `  P )  =/=  Q  /\  ( ( F `  Q ) 
.\/  ( F `  P ) )  =/=  ( P  .\/  Q
) )  /\  P  .<_  ( U  .\/  ( F `  Q )
) ) )  -> 
( P  e.  A  /\  -.  P  .<_  W ) )
6 simp22l 1115 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( ( P  =/=  Q  /\  ( F `  P )  =/=  Q  /\  ( ( F `  Q ) 
.\/  ( F `  P ) )  =/=  ( P  .\/  Q
) )  /\  P  .<_  ( U  .\/  ( F `  Q )
) ) )  ->  Q  e.  A )
7 simp3l1 1101 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( ( P  =/=  Q  /\  ( F `  P )  =/=  Q  /\  ( ( F `  Q ) 
.\/  ( F `  P ) )  =/=  ( P  .\/  Q
) )  /\  P  .<_  ( U  .\/  ( F `  Q )
) ) )  ->  P  =/=  Q )
8 cdlemg12.l . . . . . . . . . . . . 13  |-  .<_  =  ( le `  K )
9 cdlemg12.j . . . . . . . . . . . . 13  |-  .\/  =  ( join `  K )
10 cdlemg12.m . . . . . . . . . . . . 13  |-  ./\  =  ( meet `  K )
11 cdlemg12.a . . . . . . . . . . . . 13  |-  A  =  ( Atoms `  K )
12 cdlemg12.h . . . . . . . . . . . . 13  |-  H  =  ( LHyp `  K
)
13 cdlemg18b.u . . . . . . . . . . . . 13  |-  U  =  ( ( P  .\/  Q )  ./\  W )
148, 9, 10, 11, 12, 13cdleme0a 36037 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q ) )  ->  U  e.  A
)
153, 4, 5, 6, 7, 14syl212anc 1238 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( ( P  =/=  Q  /\  ( F `  P )  =/=  Q  /\  ( ( F `  Q ) 
.\/  ( F `  P ) )  =/=  ( P  .\/  Q
) )  /\  P  .<_  ( U  .\/  ( F `  Q )
) ) )  ->  U  e.  A )
16 simp1 996 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( ( P  =/=  Q  /\  ( F `  P )  =/=  Q  /\  ( ( F `  Q ) 
.\/  ( F `  P ) )  =/=  ( P  .\/  Q
) )  /\  P  .<_  ( U  .\/  ( F `  Q )
) ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
17 simp23 1031 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( ( P  =/=  Q  /\  ( F `  P )  =/=  Q  /\  ( ( F `  Q ) 
.\/  ( F `  P ) )  =/=  ( P  .\/  Q
) )  /\  P  .<_  ( U  .\/  ( F `  Q )
) ) )  ->  F  e.  T )
18 cdlemg12.t . . . . . . . . . . . . 13  |-  T  =  ( ( LTrn `  K
) `  W )
198, 11, 12, 18ltrnat 35965 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  Q  e.  A
)  ->  ( F `  Q )  e.  A
)
2016, 17, 6, 19syl3anc 1228 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( ( P  =/=  Q  /\  ( F `  P )  =/=  Q  /\  ( ( F `  Q ) 
.\/  ( F `  P ) )  =/=  ( P  .\/  Q
) )  /\  P  .<_  ( U  .\/  ( F `  Q )
) ) )  -> 
( F `  Q
)  e.  A )
218, 9, 11hlatlej1 35200 . . . . . . . . . . 11  |-  ( ( K  e.  HL  /\  U  e.  A  /\  ( F `  Q )  e.  A )  ->  U  .<_  ( U  .\/  ( F `  Q ) ) )
223, 15, 20, 21syl3anc 1228 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( ( P  =/=  Q  /\  ( F `  P )  =/=  Q  /\  ( ( F `  Q ) 
.\/  ( F `  P ) )  =/=  ( P  .\/  Q
) )  /\  P  .<_  ( U  .\/  ( F `  Q )
) ) )  ->  U  .<_  ( U  .\/  ( F `  Q ) ) )
23 hllat 35189 . . . . . . . . . . . 12  |-  ( K  e.  HL  ->  K  e.  Lat )
243, 23syl 16 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( ( P  =/=  Q  /\  ( F `  P )  =/=  Q  /\  ( ( F `  Q ) 
.\/  ( F `  P ) )  =/=  ( P  .\/  Q
) )  /\  P  .<_  ( U  .\/  ( F `  Q )
) ) )  ->  K  e.  Lat )
25 simp21l 1113 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( ( P  =/=  Q  /\  ( F `  P )  =/=  Q  /\  ( ( F `  Q ) 
.\/  ( F `  P ) )  =/=  ( P  .\/  Q
) )  /\  P  .<_  ( U  .\/  ( F `  Q )
) ) )  ->  P  e.  A )
26 eqid 2457 . . . . . . . . . . . . 13  |-  ( Base `  K )  =  (
Base `  K )
2726, 11atbase 35115 . . . . . . . . . . . 12  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
2825, 27syl 16 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( ( P  =/=  Q  /\  ( F `  P )  =/=  Q  /\  ( ( F `  Q ) 
.\/  ( F `  P ) )  =/=  ( P  .\/  Q
) )  /\  P  .<_  ( U  .\/  ( F `  Q )
) ) )  ->  P  e.  ( Base `  K ) )
2926, 11atbase 35115 . . . . . . . . . . . 12  |-  ( U  e.  A  ->  U  e.  ( Base `  K
) )
3015, 29syl 16 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( ( P  =/=  Q  /\  ( F `  P )  =/=  Q  /\  ( ( F `  Q ) 
.\/  ( F `  P ) )  =/=  ( P  .\/  Q
) )  /\  P  .<_  ( U  .\/  ( F `  Q )
) ) )  ->  U  e.  ( Base `  K ) )
3126, 9, 11hlatjcl 35192 . . . . . . . . . . . 12  |-  ( ( K  e.  HL  /\  U  e.  A  /\  ( F `  Q )  e.  A )  -> 
( U  .\/  ( F `  Q )
)  e.  ( Base `  K ) )
323, 15, 20, 31syl3anc 1228 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( ( P  =/=  Q  /\  ( F `  P )  =/=  Q  /\  ( ( F `  Q ) 
.\/  ( F `  P ) )  =/=  ( P  .\/  Q
) )  /\  P  .<_  ( U  .\/  ( F `  Q )
) ) )  -> 
( U  .\/  ( F `  Q )
)  e.  ( Base `  K ) )
3326, 8, 9latjle12 15818 . . . . . . . . . . 11  |-  ( ( K  e.  Lat  /\  ( P  e.  ( Base `  K )  /\  U  e.  ( Base `  K )  /\  ( U  .\/  ( F `  Q ) )  e.  ( Base `  K
) ) )  -> 
( ( P  .<_  ( U  .\/  ( F `
 Q ) )  /\  U  .<_  ( U 
.\/  ( F `  Q ) ) )  <-> 
( P  .\/  U
)  .<_  ( U  .\/  ( F `  Q ) ) ) )
3424, 28, 30, 32, 33syl13anc 1230 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( ( P  =/=  Q  /\  ( F `  P )  =/=  Q  /\  ( ( F `  Q ) 
.\/  ( F `  P ) )  =/=  ( P  .\/  Q
) )  /\  P  .<_  ( U  .\/  ( F `  Q )
) ) )  -> 
( ( P  .<_  ( U  .\/  ( F `
 Q ) )  /\  U  .<_  ( U 
.\/  ( F `  Q ) ) )  <-> 
( P  .\/  U
)  .<_  ( U  .\/  ( F `  Q ) ) ) )
352, 22, 34mpbi2and 921 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( ( P  =/=  Q  /\  ( F `  P )  =/=  Q  /\  ( ( F `  Q ) 
.\/  ( F `  P ) )  =/=  ( P  .\/  Q
) )  /\  P  .<_  ( U  .\/  ( F `  Q )
) ) )  -> 
( P  .\/  U
)  .<_  ( U  .\/  ( F `  Q ) ) )
368, 9, 10, 11, 12, 13cdleme0cp 36040 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )
)  ->  ( P  .\/  U )  =  ( P  .\/  Q ) )
373, 4, 5, 6, 36syl22anc 1229 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( ( P  =/=  Q  /\  ( F `  P )  =/=  Q  /\  ( ( F `  Q ) 
.\/  ( F `  P ) )  =/=  ( P  .\/  Q
) )  /\  P  .<_  ( U  .\/  ( F `  Q )
) ) )  -> 
( P  .\/  U
)  =  ( P 
.\/  Q ) )
38 simp22 1030 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( ( P  =/=  Q  /\  ( F `  P )  =/=  Q  /\  ( ( F `  Q ) 
.\/  ( F `  P ) )  =/=  ( P  .\/  Q
) )  /\  P  .<_  ( U  .\/  ( F `  Q )
) ) )  -> 
( Q  e.  A  /\  -.  Q  .<_  W ) )
3912, 18, 8, 9, 11, 10, 13cdlemg2kq 36429 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  F  e.  T )  ->  (
( F `  P
)  .\/  ( F `  Q ) )  =  ( ( F `  Q )  .\/  U
) )
4016, 5, 38, 17, 39syl121anc 1233 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( ( P  =/=  Q  /\  ( F `  P )  =/=  Q  /\  ( ( F `  Q ) 
.\/  ( F `  P ) )  =/=  ( P  .\/  Q
) )  /\  P  .<_  ( U  .\/  ( F `  Q )
) ) )  -> 
( ( F `  P )  .\/  ( F `  Q )
)  =  ( ( F `  Q ) 
.\/  U ) )
419, 11hlatjcom 35193 . . . . . . . . . . 11  |-  ( ( K  e.  HL  /\  ( F `  Q )  e.  A  /\  U  e.  A )  ->  (
( F `  Q
)  .\/  U )  =  ( U  .\/  ( F `  Q ) ) )
423, 20, 15, 41syl3anc 1228 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( ( P  =/=  Q  /\  ( F `  P )  =/=  Q  /\  ( ( F `  Q ) 
.\/  ( F `  P ) )  =/=  ( P  .\/  Q
) )  /\  P  .<_  ( U  .\/  ( F `  Q )
) ) )  -> 
( ( F `  Q )  .\/  U
)  =  ( U 
.\/  ( F `  Q ) ) )
4340, 42eqtr2d 2499 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( ( P  =/=  Q  /\  ( F `  P )  =/=  Q  /\  ( ( F `  Q ) 
.\/  ( F `  P ) )  =/=  ( P  .\/  Q
) )  /\  P  .<_  ( U  .\/  ( F `  Q )
) ) )  -> 
( U  .\/  ( F `  Q )
)  =  ( ( F `  P ) 
.\/  ( F `  Q ) ) )
4435, 37, 433brtr3d 4485 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( ( P  =/=  Q  /\  ( F `  P )  =/=  Q  /\  ( ( F `  Q ) 
.\/  ( F `  P ) )  =/=  ( P  .\/  Q
) )  /\  P  .<_  ( U  .\/  ( F `  Q )
) ) )  -> 
( P  .\/  Q
)  .<_  ( ( F `
 P )  .\/  ( F `  Q ) ) )
458, 11, 12, 18ltrnat 35965 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  A
)  ->  ( F `  P )  e.  A
)
4616, 17, 25, 45syl3anc 1228 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( ( P  =/=  Q  /\  ( F `  P )  =/=  Q  /\  ( ( F `  Q ) 
.\/  ( F `  P ) )  =/=  ( P  .\/  Q
) )  /\  P  .<_  ( U  .\/  ( F `  Q )
) ) )  -> 
( F `  P
)  e.  A )
478, 9, 11ps-1 35302 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( ( F `  P )  e.  A  /\  ( F `  Q )  e.  A ) )  -> 
( ( P  .\/  Q )  .<_  ( ( F `  P )  .\/  ( F `  Q
) )  <->  ( P  .\/  Q )  =  ( ( F `  P
)  .\/  ( F `  Q ) ) ) )
483, 25, 6, 7, 46, 20, 47syl132anc 1246 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( ( P  =/=  Q  /\  ( F `  P )  =/=  Q  /\  ( ( F `  Q ) 
.\/  ( F `  P ) )  =/=  ( P  .\/  Q
) )  /\  P  .<_  ( U  .\/  ( F `  Q )
) ) )  -> 
( ( P  .\/  Q )  .<_  ( ( F `  P )  .\/  ( F `  Q
) )  <->  ( P  .\/  Q )  =  ( ( F `  P
)  .\/  ( F `  Q ) ) ) )
4944, 48mpbid 210 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( ( P  =/=  Q  /\  ( F `  P )  =/=  Q  /\  ( ( F `  Q ) 
.\/  ( F `  P ) )  =/=  ( P  .\/  Q
) )  /\  P  .<_  ( U  .\/  ( F `  Q )
) ) )  -> 
( P  .\/  Q
)  =  ( ( F `  P ) 
.\/  ( F `  Q ) ) )
509, 11hlatjcom 35193 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( F `  P )  e.  A  /\  ( F `  Q )  e.  A )  ->  (
( F `  P
)  .\/  ( F `  Q ) )  =  ( ( F `  Q )  .\/  ( F `  P )
) )
513, 46, 20, 50syl3anc 1228 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( ( P  =/=  Q  /\  ( F `  P )  =/=  Q  /\  ( ( F `  Q ) 
.\/  ( F `  P ) )  =/=  ( P  .\/  Q
) )  /\  P  .<_  ( U  .\/  ( F `  Q )
) ) )  -> 
( ( F `  P )  .\/  ( F `  Q )
)  =  ( ( F `  Q ) 
.\/  ( F `  P ) ) )
5249, 51eqtr2d 2499 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( ( P  =/=  Q  /\  ( F `  P )  =/=  Q  /\  ( ( F `  Q ) 
.\/  ( F `  P ) )  =/=  ( P  .\/  Q
) )  /\  P  .<_  ( U  .\/  ( F `  Q )
) ) )  -> 
( ( F `  Q )  .\/  ( F `  P )
)  =  ( P 
.\/  Q ) )
53523exp 1195 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  ->  ( (
( P  =/=  Q  /\  ( F `  P
)  =/=  Q  /\  ( ( F `  Q )  .\/  ( F `  P )
)  =/=  ( P 
.\/  Q ) )  /\  P  .<_  ( U 
.\/  ( F `  Q ) ) )  ->  ( ( F `
 Q )  .\/  ( F `  P ) )  =  ( P 
.\/  Q ) ) ) )
5453exp4a 606 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  ->  ( ( P  =/=  Q  /\  ( F `  P )  =/=  Q  /\  ( ( F `  Q ) 
.\/  ( F `  P ) )  =/=  ( P  .\/  Q
) )  ->  ( P  .<_  ( U  .\/  ( F `  Q ) )  ->  ( ( F `  Q )  .\/  ( F `  P
) )  =  ( P  .\/  Q ) ) ) ) )
55543imp 1190 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( P  =/=  Q  /\  ( F `
 P )  =/= 
Q  /\  ( ( F `  Q )  .\/  ( F `  P
) )  =/=  ( P  .\/  Q ) ) )  ->  ( P  .<_  ( U  .\/  ( F `  Q )
)  ->  ( ( F `  Q )  .\/  ( F `  P
) )  =  ( P  .\/  Q ) ) )
5655necon3ad 2667 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( P  =/=  Q  /\  ( F `
 P )  =/= 
Q  /\  ( ( F `  Q )  .\/  ( F `  P
) )  =/=  ( P  .\/  Q ) ) )  ->  ( (
( F `  Q
)  .\/  ( F `  P ) )  =/=  ( P  .\/  Q
)  ->  -.  P  .<_  ( U  .\/  ( F `  Q )
) ) )
571, 56mpd 15 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( P  =/=  Q  /\  ( F `
 P )  =/= 
Q  /\  ( ( F `  Q )  .\/  ( F `  P
) )  =/=  ( P  .\/  Q ) ) )  ->  -.  P  .<_  ( U  .\/  ( F `  Q )
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819    =/= wne 2652   class class class wbr 4456   ` cfv 5594  (class class class)co 6296   Basecbs 14643   lecple 14718   joincjn 15699   meetcmee 15700   Latclat 15801   Atomscatm 35089   HLchlt 35176   LHypclh 35809   LTrncltrn 35926   trLctrl 35984
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-riotaBAD 34785
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-undef 7020  df-map 7440  df-preset 15683  df-poset 15701  df-plt 15714  df-lub 15730  df-glb 15731  df-join 15732  df-meet 15733  df-p0 15795  df-p1 15796  df-lat 15802  df-clat 15864  df-oposet 35002  df-ol 35004  df-oml 35005  df-covers 35092  df-ats 35093  df-atl 35124  df-cvlat 35148  df-hlat 35177  df-llines 35323  df-lplanes 35324  df-lvols 35325  df-lines 35326  df-psubsp 35328  df-pmap 35329  df-padd 35621  df-lhyp 35813  df-laut 35814  df-ldil 35929  df-ltrn 35930  df-trl 35985
This theorem is referenced by:  cdlemg18c  36507
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