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Theorem cdlemg18b 31476
Description: Lemma for cdlemg18c 31477. 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 995 . 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 986 . . . . . . . . . 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 981 . . . . . . . . . . 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 982 . . . . . . . . . . . 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 990 . . . . . . . . . . . 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 1076 . . . . . . . . . . . 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 1062 . . . . . . . . . . . 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 31008 . . . . . . . . . . . 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 1194 . . . . . . . . . . 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 957 . . . . . . . . . . . 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 992 . . . . . . . . . . . 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 30937 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  Q  e.  A
)  ->  ( F `  Q )  e.  A
)
2016, 17, 6, 19syl3anc 1184 . . . . . . . . . . 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 30172 . . . . . . . . . . 11  |-  ( ( K  e.  HL  /\  U  e.  A  /\  ( F `  Q )  e.  A )  ->  U  .<_  ( U  .\/  ( F `  Q ) ) )
223, 15, 20, 21syl3anc 1184 . . . . . . . . . 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 30161 . . . . . . . . . . . 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 1074 . . . . . . . . . . . 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 2436 . . . . . . . . . . . . 13  |-  ( Base `  K )  =  (
Base `  K )
2726, 11atbase 30087 . . . . . . . . . . . 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 30087 . . . . . . . . . . . 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 30164 . . . . . . . . . . . 12  |-  ( ( K  e.  HL  /\  U  e.  A  /\  ( F `  Q )  e.  A )  -> 
( U  .\/  ( F `  Q )
)  e.  ( Base `  K ) )
323, 15, 20, 31syl3anc 1184 . . . . . . . . . . 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 14491 . . . . . . . . . . 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 1186 . . . . . . . . . 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 888 . . . . . . . . 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 31011 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )
)  ->  ( P  .\/  U )  =  ( P  .\/  Q ) )
373, 4, 5, 6, 36syl22anc 1185 . . . . . . . . 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 991 . . . . . . . . . . 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 31399 . . . . . . . . . . 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 1189 . . . . . . . . . 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 30165 . . . . . . . . . . 11  |-  ( ( K  e.  HL  /\  ( F `  Q )  e.  A  /\  U  e.  A )  ->  (
( F `  Q
)  .\/  U )  =  ( U  .\/  ( F `  Q ) ) )
423, 20, 15, 41syl3anc 1184 . . . . . . . . . 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 2469 . . . . . . . . 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 4241 . . . . . . . 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 30937 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  A
)  ->  ( F `  P )  e.  A
)
4616, 17, 25, 45syl3anc 1184 . . . . . . . . 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 30274 . . . . . . . . 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 1202 . . . . . . . 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 202 . . . . . . 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 30165 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( F `  P )  e.  A  /\  ( F `  Q )  e.  A )  ->  (
( F `  P
)  .\/  ( F `  Q ) )  =  ( ( F `  Q )  .\/  ( F `  P )
) )
513, 46, 20, 50syl3anc 1184 . . . . . . 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 2469 . . . . . 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 1152 . . . . 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 590 . . . 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 1147 . . 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 2637 . 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 set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2599   class class class wbr 4212   ` cfv 5454  (class class class)co 6081   Basecbs 13469   lecple 13536   joincjn 14401   meetcmee 14402   Latclat 14474   Atomscatm 30061   HLchlt 30148   LHypclh 30781   LTrncltrn 30898   trLctrl 30955
This theorem is referenced by:  cdlemg18c  31477
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-iin 4096  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-undef 6543  df-riota 6549  df-map 7020  df-poset 14403  df-plt 14415  df-lub 14431  df-glb 14432  df-join 14433  df-meet 14434  df-p0 14468  df-p1 14469  df-lat 14475  df-clat 14537  df-oposet 29974  df-ol 29976  df-oml 29977  df-covers 30064  df-ats 30065  df-atl 30096  df-cvlat 30120  df-hlat 30149  df-llines 30295  df-lplanes 30296  df-lvols 30297  df-lines 30298  df-psubsp 30300  df-pmap 30301  df-padd 30593  df-lhyp 30785  df-laut 30786  df-ldil 30901  df-ltrn 30902  df-trl 30956
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