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Theorem cdlemg18b 34164
Description: Lemma for cdlemg18c 34165. 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 1043 . 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 1034 . . . . . . . . . 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 1029 . . . . . . . . . . 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 1030 . . . . . . . . . . . 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 1038 . . . . . . . . . . . 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 1124 . . . . . . . . . . . 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 1110 . . . . . . . . . . . 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 33695 . . . . . . . . . . . 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 1274 . . . . . . . . . . 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 1005 . . . . . . . . . . . 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 1040 . . . . . . . . . . . 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 33623 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  Q  e.  A
)  ->  ( F `  Q )  e.  A
)
2016, 17, 6, 19syl3anc 1264 . . . . . . . . . . 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 32858 . . . . . . . . . . 11  |-  ( ( K  e.  HL  /\  U  e.  A  /\  ( F `  Q )  e.  A )  ->  U  .<_  ( U  .\/  ( F `  Q ) ) )
223, 15, 20, 21syl3anc 1264 . . . . . . . . . 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 32847 . . . . . . . . . . . 12  |-  ( K  e.  HL  ->  K  e.  Lat )
243, 23syl 17 . . . . . . . . . . 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 1122 . . . . . . . . . . . 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 2422 . . . . . . . . . . . . 13  |-  ( Base `  K )  =  (
Base `  K )
2726, 11atbase 32773 . . . . . . . . . . . 12  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
2825, 27syl 17 . . . . . . . . . . 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 32773 . . . . . . . . . . . 12  |-  ( U  e.  A  ->  U  e.  ( Base `  K
) )
3015, 29syl 17 . . . . . . . . . . 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 32850 . . . . . . . . . . . 12  |-  ( ( K  e.  HL  /\  U  e.  A  /\  ( F `  Q )  e.  A )  -> 
( U  .\/  ( F `  Q )
)  e.  ( Base `  K ) )
323, 15, 20, 31syl3anc 1264 . . . . . . . . . . 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 16295 . . . . . . . . . . 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 1266 . . . . . . . . . 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 929 . . . . . . . . 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 33698 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )
)  ->  ( P  .\/  U )  =  ( P  .\/  Q ) )
373, 4, 5, 6, 36syl22anc 1265 . . . . . . . . 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 1039 . . . . . . . . . . 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 34087 . . . . . . . . . . 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 1269 . . . . . . . . . 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 32851 . . . . . . . . . . 11  |-  ( ( K  e.  HL  /\  ( F `  Q )  e.  A  /\  U  e.  A )  ->  (
( F `  Q
)  .\/  U )  =  ( U  .\/  ( F `  Q ) ) )
423, 20, 15, 41syl3anc 1264 . . . . . . . . . 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 2464 . . . . . . . . 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 4450 . . . . . . . 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 33623 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  A
)  ->  ( F `  P )  e.  A
)
4616, 17, 25, 45syl3anc 1264 . . . . . . . . 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 32960 . . . . . . . . 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 1282 . . . . . . . 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 213 . . . . . . 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 32851 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( F `  P )  e.  A  /\  ( F `  Q )  e.  A )  ->  (
( F `  P
)  .\/  ( F `  Q ) )  =  ( ( F `  Q )  .\/  ( F `  P )
) )
513, 46, 20, 50syl3anc 1264 . . . . . . 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 2464 . . . . . 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 1204 . . . . 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 609 . . . 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 1199 . . 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 2634 . 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 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1868    =/= wne 2618   class class class wbr 4420   ` cfv 5597  (class class class)co 6301   Basecbs 15108   lecple 15184   joincjn 16176   meetcmee 16177   Latclat 16278   Atomscatm 32747   HLchlt 32834   LHypclh 33467   LTrncltrn 33584   trLctrl 33642
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656  ax-un 6593  ax-riotaBAD 32443
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-op 4003  df-uni 4217  df-iun 4298  df-iin 4299  df-br 4421  df-opab 4480  df-mpt 4481  df-id 4764  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-res 4861  df-ima 4862  df-iota 5561  df-fun 5599  df-fn 5600  df-f 5601  df-f1 5602  df-fo 5603  df-f1o 5604  df-fv 5605  df-riota 6263  df-ov 6304  df-oprab 6305  df-mpt2 6306  df-1st 6803  df-2nd 6804  df-undef 7024  df-map 7478  df-preset 16160  df-poset 16178  df-plt 16191  df-lub 16207  df-glb 16208  df-join 16209  df-meet 16210  df-p0 16272  df-p1 16273  df-lat 16279  df-clat 16341  df-oposet 32660  df-ol 32662  df-oml 32663  df-covers 32750  df-ats 32751  df-atl 32782  df-cvlat 32806  df-hlat 32835  df-llines 32981  df-lplanes 32982  df-lvols 32983  df-lines 32984  df-psubsp 32986  df-pmap 32987  df-padd 33279  df-lhyp 33471  df-laut 33472  df-ldil 33587  df-ltrn 33588  df-trl 33643
This theorem is referenced by:  cdlemg18c  34165
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