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Theorem trlval4 34190
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 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
) ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
2 simp21 1021 . 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 1022 . 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 1023 . 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 1017 . . 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 1039 . . . . . . . 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 1109 . . . . . . . . 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 465 . . . . . . . 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 991 . . . . . . . . 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 1066 . . . . . . . . 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 34142 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  Q  e.  A
)  ->  ( F `  Q )  e.  A
)
169, 10, 8, 15syl3anc 1219 . . . . . . . 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 33377 . . . . . . . 8  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  ( F `  Q )  e.  A )  ->  Q  .<_  ( Q  .\/  ( F `  Q ) ) )
196, 8, 16, 18syl3anc 1219 . . . . . . 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 1067 . . . . . . . . 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 34168 . . . . . . . . 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 1219 . . . . . . . 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 461 . . . . . . . 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 2495 . . . . . . 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 4429 . . . . . 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 1044 . . . . . . . . 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 1027 . . . . . . . . . . . . 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 465 . . . . . . . . . . . . 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 465 . . . . . . . . . . . . 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 461 . . . . . . . . . . . . 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 2454 . . . . . . . . . . . . . 14  |-  ( 0.
`  K )  =  ( 0. `  K
)
3311, 32, 12, 13, 14, 21trl0 34172 . . . . . . . . . . . . 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 1223 . . . . . . . . . . . 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 33363 . . . . . . . . . . . . . . 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 1107 . . . . . . . . . . . . . . . 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 465 . . . . . . . . . . . . . . 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 2454 . . . . . . . . . . . . . . . 16  |-  ( Base `  K )  =  (
Base `  K )
4039, 17, 12hlatjcl 33369 . . . . . . . . . . . . . . 15  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
416, 38, 8, 40syl3anc 1219 . . . . . . . . . . . . . 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 33307 . . . . . . . . . . . . . 14  |-  ( ( K  e.  AtLat  /\  ( P  .\/  Q )  e.  ( Base `  K
) )  ->  ( 0. `  K )  .<_  ( P  .\/  Q ) )
4336, 41, 42syl2anc 661 . . . . . . . . . . . . 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 465 . . . . . . . . . . . 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 4423 . . . . . . . . . . 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 434 . . . . . . . . . 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 2664 . . . . . . . . 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 34171 . . . . . . . 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 1223 . . . . . . 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 1043 . . . . . . . 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 2723 . . . . . . 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 33397 . . . . . . 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 1232 . . . . . 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 434 . . . 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 2664 . . 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 34189 . 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 1231 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 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2648   class class class wbr 4403   ` cfv 5529  (class class class)co 6203   Basecbs 14295   lecple 14367   joincjn 15236   meetcmee 15237   0.cp0 15329   Atomscatm 33266   AtLatcal 33267   HLchlt 33353   LHypclh 33986   LTrncltrn 34103   trLctrl 34160
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-iin 4285  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-1st 6690  df-2nd 6691  df-map 7329  df-poset 15238  df-plt 15250  df-lub 15266  df-glb 15267  df-join 15268  df-meet 15269  df-p0 15331  df-p1 15332  df-lat 15338  df-clat 15400  df-oposet 33179  df-ol 33181  df-oml 33182  df-covers 33269  df-ats 33270  df-atl 33301  df-cvlat 33325  df-hlat 33354  df-llines 33500  df-psubsp 33505  df-pmap 33506  df-padd 33798  df-lhyp 33990  df-laut 33991  df-ldil 34106  df-ltrn 34107  df-trl 34161
This theorem is referenced by:  cdlemg10a  34642  cdlemg12d  34648
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