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Theorem trlval4 30304
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 957 . 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 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
) ) )  ->  F  e.  T )
3 simp22 991 . 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 992 . 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 986 . . 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 1008 . . . . . . . 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 1078 . . . . . . . . 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 452 . . . . . . . 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 960 . . . . . . . . 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 1035 . . . . . . . . 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 30256 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  Q  e.  A
)  ->  ( F `  Q )  e.  A
)
169, 10, 8, 15syl3anc 1184 . . . . . . . 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 29491 . . . . . . . 8  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  ( F `  Q )  e.  A )  ->  Q  .<_  ( Q  .\/  ( F `  Q ) ) )
196, 8, 16, 18syl3anc 1184 . . . . . . 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 1036 . . . . . . . . 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 30282 . . . . . . . . 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 1184 . . . . . . . 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 448 . . . . . . . 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 2421 . . . . . . 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 4181 . . . . . 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 1013 . . . . . . . . 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 996 . . . . . . . . . . . . 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 452 . . . . . . . . . . . . 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 452 . . . . . . . . . . . . 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 448 . . . . . . . . . . . . 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 2389 . . . . . . . . . . . . . 14  |-  ( 0.
`  K )  =  ( 0. `  K
)
3311, 32, 12, 13, 14, 21trl0 30286 . . . . . . . . . . . . 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 1188 . . . . . . . . . . . 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 29477 . . . . . . . . . . . . . . 15  |-  ( K  e.  HL  ->  K  e.  AtLat )
366, 35syl 16 . . . . . . . . . . . . . 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 1076 . . . . . . . . . . . . . . . 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 452 . . . . . . . . . . . . . . 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 2389 . . . . . . . . . . . . . . . 16  |-  ( Base `  K )  =  (
Base `  K )
4039, 17, 12hlatjcl 29483 . . . . . . . . . . . . . . 15  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
416, 38, 8, 40syl3anc 1184 . . . . . . . . . . . . . 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 29421 . . . . . . . . . . . . . 14  |-  ( ( K  e.  AtLat  /\  ( P  .\/  Q )  e.  ( Base `  K
) )  ->  ( 0. `  K )  .<_  ( P  .\/  Q ) )
4336, 41, 42syl2anc 643 . . . . . . . . . . . . 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 452 . . . . . . . . . . . 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 4175 . . . . . . . . . . 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 424 . . . . . . . . . 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 2589 . . . . . . . . 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 30285 . . . . . . . 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 1188 . . . . . . 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 1012 . . . . . . . 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 2635 . . . . . . 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 29511 . . . . . . 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 1197 . . . . . 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 424 . . . 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 2589 . . 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 30303 . 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 1196 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 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2552   class class class wbr 4155   ` cfv 5396  (class class class)co 6022   Basecbs 13398   lecple 13465   joincjn 14330   meetcmee 14331   0.cp0 14395   Atomscatm 29380   AtLatcal 29381   HLchlt 29467   LHypclh 30100   LTrncltrn 30217   trLctrl 30274
This theorem is referenced by:  cdlemg10a  30756  cdlemg12d  30762
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-iin 4040  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-undef 6481  df-riota 6487  df-map 6958  df-poset 14332  df-plt 14344  df-lub 14360  df-glb 14361  df-join 14362  df-meet 14363  df-p0 14397  df-p1 14398  df-lat 14404  df-clat 14466  df-oposet 29293  df-ol 29295  df-oml 29296  df-covers 29383  df-ats 29384  df-atl 29415  df-cvlat 29439  df-hlat 29468  df-llines 29614  df-psubsp 29619  df-pmap 29620  df-padd 29912  df-lhyp 30104  df-laut 30105  df-ldil 30220  df-ltrn 30221  df-trl 30275
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