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Theorem trlval4 30886
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 30838 . . . . . . . . 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 30073 . . . . . . . 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 30864 . . . . . . . . 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 2467 . . . . . . 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 4230 . . . . . 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 2435 . . . . . . . . . . . . . 14  |-  ( 0.
`  K )  =  ( 0. `  K
)
3311, 32, 12, 13, 14, 21trl0 30868 . . . . . . . . . . . . 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 30059 . . . . . . . . . . . . . . 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 2435 . . . . . . . . . . . . . . . 16  |-  ( Base `  K )  =  (
Base `  K )
4039, 17, 12hlatjcl 30065 . . . . . . . . . . . . . . 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 30003 . . . . . . . . . . . . . 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 4224 . . . . . . . . . . 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 2635 . . . . . . . . 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 30867 . . . . . . . 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 2681 . . . . . . 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 30093 . . . . . . 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 2635 . . 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 30885 . 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 1652    e. wcel 1725    =/= wne 2598   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   Basecbs 13459   lecple 13526   joincjn 14391   meetcmee 14392   0.cp0 14456   Atomscatm 29962   AtLatcal 29963   HLchlt 30049   LHypclh 30682   LTrncltrn 30799   trLctrl 30856
This theorem is referenced by:  cdlemg10a  31338  cdlemg12d  31344
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 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-undef 6535  df-riota 6541  df-map 7012  df-poset 14393  df-plt 14405  df-lub 14421  df-glb 14422  df-join 14423  df-meet 14424  df-p0 14458  df-p1 14459  df-lat 14465  df-clat 14527  df-oposet 29875  df-ol 29877  df-oml 29878  df-covers 29965  df-ats 29966  df-atl 29997  df-cvlat 30021  df-hlat 30050  df-llines 30196  df-psubsp 30201  df-pmap 30202  df-padd 30494  df-lhyp 30686  df-laut 30687  df-ldil 30802  df-ltrn 30803  df-trl 30857
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