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Theorem trlval4 30377
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 955 . 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 988 . 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 989 . 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 990 . 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 984 . . 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 1006 . . . . . . . 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 1076 . . . . . . . . 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 451 . . . . . . . 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 958 . . . . . . . . 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 1033 . . . . . . . . 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 30329 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  Q  e.  A
)  ->  ( F `  Q )  e.  A
)
169, 10, 8, 15syl3anc 1182 . . . . . . . 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 29564 . . . . . . . 8  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  ( F `  Q )  e.  A )  ->  Q  .<_  ( Q  .\/  ( F `  Q ) ) )
196, 8, 16, 18syl3anc 1182 . . . . . . 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 1034 . . . . . . . . 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 30355 . . . . . . . . 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 1182 . . . . . . . 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 447 . . . . . . . 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 2315 . . . . . . 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 4049 . . . . . 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 1011 . . . . . . . . 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 994 . . . . . . . . . . . . 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 451 . . . . . . . . . . . . 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 451 . . . . . . . . . . . . 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 447 . . . . . . . . . . . . 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 2283 . . . . . . . . . . . . . 14  |-  ( 0.
`  K )  =  ( 0. `  K
)
3311, 32, 12, 13, 14, 21trl0 30359 . . . . . . . . . . . . 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 1186 . . . . . . . . . . . 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 29550 . . . . . . . . . . . . . . 15  |-  ( K  e.  HL  ->  K  e.  AtLat )
366, 35syl 15 . . . . . . . . . . . . . 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 1074 . . . . . . . . . . . . . . . 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 451 . . . . . . . . . . . . . . 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 2283 . . . . . . . . . . . . . . . 16  |-  ( Base `  K )  =  (
Base `  K )
4039, 17, 12hlatjcl 29556 . . . . . . . . . . . . . . 15  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
416, 38, 8, 40syl3anc 1182 . . . . . . . . . . . . . 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 29494 . . . . . . . . . . . . . 14  |-  ( ( K  e.  AtLat  /\  ( P  .\/  Q )  e.  ( Base `  K
) )  ->  ( 0. `  K )  .<_  ( P  .\/  Q ) )
4336, 41, 42syl2anc 642 . . . . . . . . . . . . 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 451 . . . . . . . . . . . 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 4043 . . . . . . . . . . 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 423 . . . . . . . . . 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 2483 . . . . . . . . 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 14 . . . . . . . 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 30358 . . . . . . . 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 1186 . . . . . . 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 1010 . . . . . . . 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 2529 . . . . . . 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 29584 . . . . . . 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 1195 . . . . . 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 14 . . . . 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 423 . . . 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 2483 . . 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 14 . 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 30376 . 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 1194 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 set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215   joincjn 14078   meetcmee 14079   0.cp0 14143   Atomscatm 29453   AtLatcal 29454   HLchlt 29540   LHypclh 30173   LTrncltrn 30290   trLctrl 30347
This theorem is referenced by:  cdlemg10a  30829  cdlemg12d  30835
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-map 6774  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-p1 14146  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-llines 29687  df-psubsp 29692  df-pmap 29693  df-padd 29985  df-lhyp 30177  df-laut 30178  df-ldil 30293  df-ltrn 30294  df-trl 30348
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