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Theorem trlval4 33170
Description: The value of the trace of a lattice translation in terms of 2 atoms. (Contributed by NM, 3-May-2013.)
Hypotheses
Ref Expression
trlval3.l  |-  .<_  =  ( le `  K )
trlval3.j  |-  .\/  =  ( join `  K )
trlval3.m  |-  ./\  =  ( meet `  K )
trlval3.a  |-  A  =  ( Atoms `  K )
trlval3.h  |-  H  =  ( LHyp `  K
)
trlval3.t  |-  T  =  ( ( LTrn `  K
) `  W )
trlval3.r  |-  R  =  ( ( trL `  K
) `  W )
Assertion
Ref Expression
trlval4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  ( R `  F ) 
.<_  ( P  .\/  Q
) ) )  -> 
( R `  F
)  =  ( ( P  .\/  ( F `
 P ) ) 
./\  ( Q  .\/  ( F `  Q ) ) ) )

Proof of Theorem trlval4
StepHypRef Expression
1 simp1 995 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  ( R `  F ) 
.<_  ( P  .\/  Q
) ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
2 simp21 1028 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  ( R `  F ) 
.<_  ( P  .\/  Q
) ) )  ->  F  e.  T )
3 simp22 1029 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  ( R `  F ) 
.<_  ( P  .\/  Q
) ) )  -> 
( P  e.  A  /\  -.  P  .<_  W ) )
4 simp23 1030 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  ( R `  F ) 
.<_  ( P  .\/  Q
) ) )  -> 
( Q  e.  A  /\  -.  Q  .<_  W ) )
5 simp3r 1024 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  ( R `  F ) 
.<_  ( P  .\/  Q
) ) )  ->  -.  ( R `  F
)  .<_  ( P  .\/  Q ) )
6 simpl1l 1046 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  ( R `  F )  .<_  ( P 
.\/  Q ) ) )  /\  ( P 
.\/  ( F `  P ) )  =  ( Q  .\/  ( F `  Q )
) )  ->  K  e.  HL )
7 simp23l 1116 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  ( R `  F ) 
.<_  ( P  .\/  Q
) ) )  ->  Q  e.  A )
87adantr 463 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  ( R `  F )  .<_  ( P 
.\/  Q ) ) )  /\  ( P 
.\/  ( F `  P ) )  =  ( Q  .\/  ( F `  Q )
) )  ->  Q  e.  A )
9 simpl1 998 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  ( R `  F )  .<_  ( P 
.\/  Q ) ) )  /\  ( P 
.\/  ( F `  P ) )  =  ( Q  .\/  ( F `  Q )
) )  ->  ( K  e.  HL  /\  W  e.  H ) )
10 simpl21 1073 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  ( R `  F )  .<_  ( P 
.\/  Q ) ) )  /\  ( P 
.\/  ( F `  P ) )  =  ( Q  .\/  ( F `  Q )
) )  ->  F  e.  T )
11 trlval3.l . . . . . . . . . 10  |-  .<_  =  ( le `  K )
12 trlval3.a . . . . . . . . . 10  |-  A  =  ( Atoms `  K )
13 trlval3.h . . . . . . . . . 10  |-  H  =  ( LHyp `  K
)
14 trlval3.t . . . . . . . . . 10  |-  T  =  ( ( LTrn `  K
) `  W )
1511, 12, 13, 14ltrnat 33121 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  Q  e.  A
)  ->  ( F `  Q )  e.  A
)
169, 10, 8, 15syl3anc 1228 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  ( R `  F )  .<_  ( P 
.\/  Q ) ) )  /\  ( P 
.\/  ( F `  P ) )  =  ( Q  .\/  ( F `  Q )
) )  ->  ( F `  Q )  e.  A )
17 trlval3.j . . . . . . . . 9  |-  .\/  =  ( join `  K )
1811, 17, 12hlatlej1 32356 . . . . . . . 8  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  ( F `  Q )  e.  A )  ->  Q  .<_  ( Q  .\/  ( F `  Q ) ) )
196, 8, 16, 18syl3anc 1228 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  ( R `  F )  .<_  ( P 
.\/  Q ) ) )  /\  ( P 
.\/  ( F `  P ) )  =  ( Q  .\/  ( F `  Q )
) )  ->  Q  .<_  ( Q  .\/  ( F `  Q )
) )
20 simpl22 1074 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  ( R `  F )  .<_  ( P 
.\/  Q ) ) )  /\  ( P 
.\/  ( F `  P ) )  =  ( Q  .\/  ( F `  Q )
) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
21 trlval3.r . . . . . . . . . 10  |-  R  =  ( ( trL `  K
) `  W )
2211, 17, 12, 13, 14, 21trljat1 33148 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( P  .\/  ( R `  F
) )  =  ( P  .\/  ( F `
 P ) ) )
239, 10, 20, 22syl3anc 1228 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  ( R `  F )  .<_  ( P 
.\/  Q ) ) )  /\  ( P 
.\/  ( F `  P ) )  =  ( Q  .\/  ( F `  Q )
) )  ->  ( P  .\/  ( R `  F ) )  =  ( P  .\/  ( F `  P )
) )
24 simpr 459 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  ( R `  F )  .<_  ( P 
.\/  Q ) ) )  /\  ( P 
.\/  ( F `  P ) )  =  ( Q  .\/  ( F `  Q )
) )  ->  ( P  .\/  ( F `  P ) )  =  ( Q  .\/  ( F `  Q )
) )
2523, 24eqtrd 2441 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  ( R `  F )  .<_  ( P 
.\/  Q ) ) )  /\  ( P 
.\/  ( F `  P ) )  =  ( Q  .\/  ( F `  Q )
) )  ->  ( P  .\/  ( R `  F ) )  =  ( Q  .\/  ( F `  Q )
) )
2619, 25breqtrrd 4418 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  ( R `  F )  .<_  ( P 
.\/  Q ) ) )  /\  ( P 
.\/  ( F `  P ) )  =  ( Q  .\/  ( F `  Q )
) )  ->  Q  .<_  ( P  .\/  ( R `  F )
) )
27 simpl3r 1051 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  ( R `  F )  .<_  ( P 
.\/  Q ) ) )  /\  ( P 
.\/  ( F `  P ) )  =  ( Q  .\/  ( F `  Q )
) )  ->  -.  ( R `  F ) 
.<_  ( P  .\/  Q
) )
28 simpll1 1034 . . . . . . . . . . . . 13  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  ( R `  F )  .<_  ( P 
.\/  Q ) ) )  /\  ( P 
.\/  ( F `  P ) )  =  ( Q  .\/  ( F `  Q )
) )  /\  ( F `  P )  =  P )  ->  ( K  e.  HL  /\  W  e.  H ) )
2920adantr 463 . . . . . . . . . . . . 13  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  ( R `  F )  .<_  ( P 
.\/  Q ) ) )  /\  ( P 
.\/  ( F `  P ) )  =  ( Q  .\/  ( F `  Q )
) )  /\  ( F `  P )  =  P )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
3010adantr 463 . . . . . . . . . . . . 13  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  ( R `  F )  .<_  ( P 
.\/  Q ) ) )  /\  ( P 
.\/  ( F `  P ) )  =  ( Q  .\/  ( F `  Q )
) )  /\  ( F `  P )  =  P )  ->  F  e.  T )
31 simpr 459 . . . . . . . . . . . . 13  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  ( R `  F )  .<_  ( P 
.\/  Q ) ) )  /\  ( P 
.\/  ( F `  P ) )  =  ( Q  .\/  ( F `  Q )
) )  /\  ( F `  P )  =  P )  ->  ( F `  P )  =  P )
32 eqid 2400 . . . . . . . . . . . . . 14  |-  ( 0.
`  K )  =  ( 0. `  K
)
3311, 32, 12, 13, 14, 21trl0 33152 . . . . . . . . . . . . 13  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F  e.  T  /\  ( F `  P )  =  P ) )  ->  ( R `  F )  =  ( 0. `  K ) )
3428, 29, 30, 31, 33syl112anc 1232 . . . . . . . . . . . 12  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  ( R `  F )  .<_  ( P 
.\/  Q ) ) )  /\  ( P 
.\/  ( F `  P ) )  =  ( Q  .\/  ( F `  Q )
) )  /\  ( F `  P )  =  P )  ->  ( R `  F )  =  ( 0. `  K ) )
35 hlatl 32342 . . . . . . . . . . . . . . 15  |-  ( K  e.  HL  ->  K  e.  AtLat )
366, 35syl 17 . . . . . . . . . . . . . 14  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  ( R `  F )  .<_  ( P 
.\/  Q ) ) )  /\  ( P 
.\/  ( F `  P ) )  =  ( Q  .\/  ( F `  Q )
) )  ->  K  e.  AtLat )
37 simp22l 1114 . . . . . . . . . . . . . . . 16  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  ( R `  F ) 
.<_  ( P  .\/  Q
) ) )  ->  P  e.  A )
3837adantr 463 . . . . . . . . . . . . . . 15  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  ( R `  F )  .<_  ( P 
.\/  Q ) ) )  /\  ( P 
.\/  ( F `  P ) )  =  ( Q  .\/  ( F `  Q )
) )  ->  P  e.  A )
39 eqid 2400 . . . . . . . . . . . . . . . 16  |-  ( Base `  K )  =  (
Base `  K )
4039, 17, 12hlatjcl 32348 . . . . . . . . . . . . . . 15  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
416, 38, 8, 40syl3anc 1228 . . . . . . . . . . . . . 14  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  ( R `  F )  .<_  ( P 
.\/  Q ) ) )  /\  ( P 
.\/  ( F `  P ) )  =  ( Q  .\/  ( F `  Q )
) )  ->  ( P  .\/  Q )  e.  ( Base `  K
) )
4239, 11, 32atl0le 32286 . . . . . . . . . . . . . 14  |-  ( ( K  e.  AtLat  /\  ( P  .\/  Q )  e.  ( Base `  K
) )  ->  ( 0. `  K )  .<_  ( P  .\/  Q ) )
4336, 41, 42syl2anc 659 . . . . . . . . . . . . 13  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  ( R `  F )  .<_  ( P 
.\/  Q ) ) )  /\  ( P 
.\/  ( F `  P ) )  =  ( Q  .\/  ( F `  Q )
) )  ->  ( 0. `  K )  .<_  ( P  .\/  Q ) )
4443adantr 463 . . . . . . . . . . . 12  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  ( R `  F )  .<_  ( P 
.\/  Q ) ) )  /\  ( P 
.\/  ( F `  P ) )  =  ( Q  .\/  ( F `  Q )
) )  /\  ( F `  P )  =  P )  ->  ( 0. `  K )  .<_  ( P  .\/  Q ) )
4534, 44eqbrtrd 4412 . . . . . . . . . . 11  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  ( R `  F )  .<_  ( P 
.\/  Q ) ) )  /\  ( P 
.\/  ( F `  P ) )  =  ( Q  .\/  ( F `  Q )
) )  /\  ( F `  P )  =  P )  ->  ( R `  F )  .<_  ( P  .\/  Q
) )
4645ex 432 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  ( R `  F )  .<_  ( P 
.\/  Q ) ) )  /\  ( P 
.\/  ( F `  P ) )  =  ( Q  .\/  ( F `  Q )
) )  ->  (
( F `  P
)  =  P  -> 
( R `  F
)  .<_  ( P  .\/  Q ) ) )
4746necon3bd 2613 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  ( R `  F )  .<_  ( P 
.\/  Q ) ) )  /\  ( P 
.\/  ( F `  P ) )  =  ( Q  .\/  ( F `  Q )
) )  ->  ( -.  ( R `  F
)  .<_  ( P  .\/  Q )  ->  ( F `  P )  =/=  P
) )
4827, 47mpd 15 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  ( R `  F )  .<_  ( P 
.\/  Q ) ) )  /\  ( P 
.\/  ( F `  P ) )  =  ( Q  .\/  ( F `  Q )
) )  ->  ( F `  P )  =/=  P )
4911, 12, 13, 14, 21trlat 33151 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F  e.  T  /\  ( F `  P )  =/=  P ) )  ->  ( R `  F )  e.  A
)
509, 20, 10, 48, 49syl112anc 1232 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  ( R `  F )  .<_  ( P 
.\/  Q ) ) )  /\  ( P 
.\/  ( F `  P ) )  =  ( Q  .\/  ( F `  Q )
) )  ->  ( R `  F )  e.  A )
51 simpl3l 1050 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  ( R `  F )  .<_  ( P 
.\/  Q ) ) )  /\  ( P 
.\/  ( F `  P ) )  =  ( Q  .\/  ( F `  Q )
) )  ->  P  =/=  Q )
5251necomd 2672 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  ( R `  F )  .<_  ( P 
.\/  Q ) ) )  /\  ( P 
.\/  ( F `  P ) )  =  ( Q  .\/  ( F `  Q )
) )  ->  Q  =/=  P )
5311, 17, 12hlatexch1 32376 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  ( R `  F
)  e.  A  /\  P  e.  A )  /\  Q  =/=  P
)  ->  ( Q  .<_  ( P  .\/  ( R `  F )
)  ->  ( R `  F )  .<_  ( P 
.\/  Q ) ) )
546, 8, 50, 38, 52, 53syl131anc 1241 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  ( R `  F )  .<_  ( P 
.\/  Q ) ) )  /\  ( P 
.\/  ( F `  P ) )  =  ( Q  .\/  ( F `  Q )
) )  ->  ( Q  .<_  ( P  .\/  ( R `  F ) )  ->  ( R `  F )  .<_  ( P 
.\/  Q ) ) )
5526, 54mpd 15 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  ( R `  F )  .<_  ( P 
.\/  Q ) ) )  /\  ( P 
.\/  ( F `  P ) )  =  ( Q  .\/  ( F `  Q )
) )  ->  ( R `  F )  .<_  ( P  .\/  Q
) )
5655ex 432 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  ( R `  F ) 
.<_  ( P  .\/  Q
) ) )  -> 
( ( P  .\/  ( F `  P ) )  =  ( Q 
.\/  ( F `  Q ) )  -> 
( R `  F
)  .<_  ( P  .\/  Q ) ) )
5756necon3bd 2613 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  ( R `  F ) 
.<_  ( P  .\/  Q
) ) )  -> 
( -.  ( R `
 F )  .<_  ( P  .\/  Q )  ->  ( P  .\/  ( F `  P ) )  =/=  ( Q 
.\/  ( F `  Q ) ) ) )
585, 57mpd 15 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  ( R `  F ) 
.<_  ( P  .\/  Q
) ) )  -> 
( P  .\/  ( F `  P )
)  =/=  ( Q 
.\/  ( F `  Q ) ) )
59 trlval3.m . . 3  |-  ./\  =  ( meet `  K )
6011, 17, 59, 12, 13, 14, 21trlval3 33169 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  .\/  ( F `  P ) )  =/=  ( Q 
.\/  ( F `  Q ) ) ) )  ->  ( R `  F )  =  ( ( P  .\/  ( F `  P )
)  ./\  ( Q  .\/  ( F `  Q
) ) ) )
611, 2, 3, 4, 58, 60syl113anc 1240 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  ( R `  F ) 
.<_  ( P  .\/  Q
) ) )  -> 
( R `  F
)  =  ( ( P  .\/  ( F `
 P ) ) 
./\  ( Q  .\/  ( F `  Q ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367    /\ w3a 972    = wceq 1403    e. wcel 1840    =/= wne 2596   class class class wbr 4392   ` cfv 5523  (class class class)co 6232   Basecbs 14731   lecple 14806   joincjn 15787   meetcmee 15788   0.cp0 15881   Atomscatm 32245   AtLatcal 32246   HLchlt 32332   LHypclh 32965   LTrncltrn 33082   trLctrl 33140
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-reu 2758  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-iun 4270  df-iin 4271  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4735  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-riota 6194  df-ov 6235  df-oprab 6236  df-mpt2 6237  df-1st 6736  df-2nd 6737  df-map 7377  df-preset 15771  df-poset 15789  df-plt 15802  df-lub 15818  df-glb 15819  df-join 15820  df-meet 15821  df-p0 15883  df-p1 15884  df-lat 15890  df-clat 15952  df-oposet 32158  df-ol 32160  df-oml 32161  df-covers 32248  df-ats 32249  df-atl 32280  df-cvlat 32304  df-hlat 32333  df-llines 32479  df-psubsp 32484  df-pmap 32485  df-padd 32777  df-lhyp 32969  df-laut 32970  df-ldil 33085  df-ltrn 33086  df-trl 33141
This theorem is referenced by:  cdlemg10a  33623  cdlemg12d  33629
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