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Theorem trlval3 33797
Description: The value of the trace of a lattice translation in terms of 2 atoms. TODO: Try to shorten proof. (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
trlval3  |-  ( ( ( 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
) ) ) )

Proof of Theorem trlval3
StepHypRef Expression
1 simpl1 1017 . . . 4  |-  ( ( ( ( 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 )
) ) )  /\  ( F `  P )  =  P )  -> 
( K  e.  HL  /\  W  e.  H ) )
2 simpl31 1095 . . . 4  |-  ( ( ( ( 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 )
) ) )  /\  ( F `  P )  =  P )  -> 
( P  e.  A  /\  -.  P  .<_  W ) )
3 simpl2 1018 . . . 4  |-  ( ( ( ( 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 )
) ) )  /\  ( F `  P )  =  P )  ->  F  e.  T )
4 simpr 467 . . . 4  |-  ( ( ( ( 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 )
) ) )  /\  ( F `  P )  =  P )  -> 
( F `  P
)  =  P )
5 trlval3.l . . . . 5  |-  .<_  =  ( le `  K )
6 eqid 2461 . . . . 5  |-  ( 0.
`  K )  =  ( 0. `  K
)
7 trlval3.a . . . . 5  |-  A  =  ( Atoms `  K )
8 trlval3.h . . . . 5  |-  H  =  ( LHyp `  K
)
9 trlval3.t . . . . 5  |-  T  =  ( ( LTrn `  K
) `  W )
10 trlval3.r . . . . 5  |-  R  =  ( ( trL `  K
) `  W )
115, 6, 7, 8, 9, 10trl0 33780 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F  e.  T  /\  ( F `  P )  =  P ) )  ->  ( R `  F )  =  ( 0. `  K ) )
121, 2, 3, 4, 11syl112anc 1280 . . 3  |-  ( ( ( ( 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 )
) ) )  /\  ( F `  P )  =  P )  -> 
( R `  F
)  =  ( 0.
`  K ) )
13 simpl33 1097 . . . 4  |-  ( ( ( ( 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 )
) ) )  /\  ( F `  P )  =  P )  -> 
( P  .\/  ( F `  P )
)  =/=  ( Q 
.\/  ( F `  Q ) ) )
14 simpl1l 1065 . . . . . 6  |-  ( ( ( ( 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 )
) ) )  /\  ( F `  P )  =  P )  ->  K  e.  HL )
15 hlatl 32970 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  AtLat )
1614, 15syl 17 . . . . 5  |-  ( ( ( ( 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 )
) ) )  /\  ( F `  P )  =  P )  ->  K  e.  AtLat )
174oveq2d 6330 . . . . . . 7  |-  ( ( ( ( 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 )
) ) )  /\  ( F `  P )  =  P )  -> 
( P  .\/  ( F `  P )
)  =  ( P 
.\/  P ) )
18 simp31l 1137 . . . . . . . . 9  |-  ( ( ( 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 ) ) ) )  ->  P  e.  A )
1918adantr 471 . . . . . . . 8  |-  ( ( ( ( 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 )
) ) )  /\  ( F `  P )  =  P )  ->  P  e.  A )
20 trlval3.j . . . . . . . . 9  |-  .\/  =  ( join `  K )
2120, 7hlatjidm 32978 . . . . . . . 8  |-  ( ( K  e.  HL  /\  P  e.  A )  ->  ( P  .\/  P
)  =  P )
2214, 19, 21syl2anc 671 . . . . . . 7  |-  ( ( ( ( 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 )
) ) )  /\  ( F `  P )  =  P )  -> 
( P  .\/  P
)  =  P )
2317, 22eqtrd 2495 . . . . . 6  |-  ( ( ( ( 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 )
) ) )  /\  ( F `  P )  =  P )  -> 
( P  .\/  ( F `  P )
)  =  P )
2423, 19eqeltrd 2539 . . . . 5  |-  ( ( ( ( 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 )
) ) )  /\  ( F `  P )  =  P )  -> 
( P  .\/  ( F `  P )
)  e.  A )
25 simp1 1014 . . . . . . . . . 10  |-  ( ( ( 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 ) ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
26 simp2 1015 . . . . . . . . . 10  |-  ( ( ( 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 ) ) ) )  ->  F  e.  T )
27 simp31 1050 . . . . . . . . . 10  |-  ( ( ( 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 ) ) ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
28 simp32 1051 . . . . . . . . . 10  |-  ( ( ( 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 ) ) ) )  ->  ( Q  e.  A  /\  -.  Q  .<_  W ) )
295, 7, 8, 9ltrn2ateq 33790 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  (
( F `  P
)  =  P  <->  ( F `  Q )  =  Q ) )
3025, 26, 27, 28, 29syl13anc 1278 . . . . . . . . 9  |-  ( ( ( 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 ) ) ) )  ->  ( ( F `  P )  =  P  <->  ( F `  Q )  =  Q ) )
3130biimpa 491 . . . . . . . 8  |-  ( ( ( ( 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 )
) ) )  /\  ( F `  P )  =  P )  -> 
( F `  Q
)  =  Q )
3231oveq2d 6330 . . . . . . 7  |-  ( ( ( ( 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 )
) ) )  /\  ( F `  P )  =  P )  -> 
( Q  .\/  ( F `  Q )
)  =  ( Q 
.\/  Q ) )
33 simp32l 1139 . . . . . . . . 9  |-  ( ( ( 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 ) ) ) )  ->  Q  e.  A )
3433adantr 471 . . . . . . . 8  |-  ( ( ( ( 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 )
) ) )  /\  ( F `  P )  =  P )  ->  Q  e.  A )
3520, 7hlatjidm 32978 . . . . . . . 8  |-  ( ( K  e.  HL  /\  Q  e.  A )  ->  ( Q  .\/  Q
)  =  Q )
3614, 34, 35syl2anc 671 . . . . . . 7  |-  ( ( ( ( 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 )
) ) )  /\  ( F `  P )  =  P )  -> 
( Q  .\/  Q
)  =  Q )
3732, 36eqtrd 2495 . . . . . 6  |-  ( ( ( ( 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 )
) ) )  /\  ( F `  P )  =  P )  -> 
( Q  .\/  ( F `  Q )
)  =  Q )
3837, 34eqeltrd 2539 . . . . 5  |-  ( ( ( ( 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 )
) ) )  /\  ( F `  P )  =  P )  -> 
( Q  .\/  ( F `  Q )
)  e.  A )
39 trlval3.m . . . . . 6  |-  ./\  =  ( meet `  K )
4039, 6, 7atnem0 32928 . . . . 5  |-  ( ( K  e.  AtLat  /\  ( P  .\/  ( F `  P ) )  e.  A  /\  ( Q 
.\/  ( F `  Q ) )  e.  A )  ->  (
( P  .\/  ( F `  P )
)  =/=  ( Q 
.\/  ( F `  Q ) )  <->  ( ( P  .\/  ( F `  P ) )  ./\  ( Q  .\/  ( F `
 Q ) ) )  =  ( 0.
`  K ) ) )
4116, 24, 38, 40syl3anc 1276 . . . 4  |-  ( ( ( ( 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 )
) ) )  /\  ( F `  P )  =  P )  -> 
( ( P  .\/  ( F `  P ) )  =/=  ( Q 
.\/  ( F `  Q ) )  <->  ( ( P  .\/  ( F `  P ) )  ./\  ( Q  .\/  ( F `
 Q ) ) )  =  ( 0.
`  K ) ) )
4213, 41mpbid 215 . . 3  |-  ( ( ( ( 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 )
) ) )  /\  ( F `  P )  =  P )  -> 
( ( P  .\/  ( F `  P ) )  ./\  ( Q  .\/  ( F `  Q
) ) )  =  ( 0. `  K
) )
4312, 42eqtr4d 2498 . 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 )
) ) )  /\  ( F `  P )  =  P )  -> 
( R `  F
)  =  ( ( P  .\/  ( F `
 P ) ) 
./\  ( Q  .\/  ( F `  Q ) ) ) )
44 simpl1 1017 . . . . . 6  |-  ( ( ( ( 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 )
) ) )  /\  ( F `  P )  =/=  P )  -> 
( K  e.  HL  /\  W  e.  H ) )
45 simpl2 1018 . . . . . 6  |-  ( ( ( ( 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 )
) ) )  /\  ( F `  P )  =/=  P )  ->  F  e.  T )
46 simpl31 1095 . . . . . 6  |-  ( ( ( ( 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 )
) ) )  /\  ( F `  P )  =/=  P )  -> 
( P  e.  A  /\  -.  P  .<_  W ) )
475, 20, 39, 7, 8, 9, 10trlval2 33773 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( R `  F )  =  ( ( P  .\/  ( F `  P )
)  ./\  W )
)
4844, 45, 46, 47syl3anc 1276 . . . . 5  |-  ( ( ( ( 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 )
) ) )  /\  ( F `  P )  =/=  P )  -> 
( R `  F
)  =  ( ( P  .\/  ( F `
 P ) ) 
./\  W ) )
49 simpl1l 1065 . . . . . . 7  |-  ( ( ( ( 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 )
) ) )  /\  ( F `  P )  =/=  P )  ->  K  e.  HL )
50 hllat 32973 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  Lat )
5149, 50syl 17 . . . . . 6  |-  ( ( ( ( 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 )
) ) )  /\  ( F `  P )  =/=  P )  ->  K  e.  Lat )
5218adantr 471 . . . . . . 7  |-  ( ( ( ( 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 )
) ) )  /\  ( F `  P )  =/=  P )  ->  P  e.  A )
535, 7, 8, 9ltrnat 33749 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  A
)  ->  ( F `  P )  e.  A
)
5444, 45, 52, 53syl3anc 1276 . . . . . . 7  |-  ( ( ( ( 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 )
) ) )  /\  ( F `  P )  =/=  P )  -> 
( F `  P
)  e.  A )
55 eqid 2461 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  K )
5655, 20, 7hlatjcl 32976 . . . . . . 7  |-  ( ( K  e.  HL  /\  P  e.  A  /\  ( F `  P )  e.  A )  -> 
( P  .\/  ( F `  P )
)  e.  ( Base `  K ) )
5749, 52, 54, 56syl3anc 1276 . . . . . 6  |-  ( ( ( ( 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 )
) ) )  /\  ( F `  P )  =/=  P )  -> 
( P  .\/  ( F `  P )
)  e.  ( Base `  K ) )
58 simpl1r 1066 . . . . . . 7  |-  ( ( ( ( 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 )
) ) )  /\  ( F `  P )  =/=  P )  ->  W  e.  H )
5955, 8lhpbase 33607 . . . . . . 7  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
6058, 59syl 17 . . . . . 6  |-  ( ( ( ( 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 )
) ) )  /\  ( F `  P )  =/=  P )  ->  W  e.  ( Base `  K ) )
6155, 5, 39latmle1 16370 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( P  .\/  ( F `
 P ) )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  ( F `  P ) )  ./\  W )  .<_  ( P  .\/  ( F `  P
) ) )
6251, 57, 60, 61syl3anc 1276 . . . . 5  |-  ( ( ( ( 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 )
) ) )  /\  ( F `  P )  =/=  P )  -> 
( ( P  .\/  ( F `  P ) )  ./\  W )  .<_  ( P  .\/  ( F `  P )
) )
6348, 62eqbrtrd 4436 . . . 4  |-  ( ( ( ( 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 )
) ) )  /\  ( F `  P )  =/=  P )  -> 
( R `  F
)  .<_  ( P  .\/  ( F `  P ) ) )
64 simpl32 1096 . . . . . 6  |-  ( ( ( ( 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 )
) ) )  /\  ( F `  P )  =/=  P )  -> 
( Q  e.  A  /\  -.  Q  .<_  W ) )
655, 20, 39, 7, 8, 9, 10trlval2 33773 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  ( R `  F )  =  ( ( Q  .\/  ( F `  Q )
)  ./\  W )
)
6644, 45, 64, 65syl3anc 1276 . . . . 5  |-  ( ( ( ( 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 )
) ) )  /\  ( F `  P )  =/=  P )  -> 
( R `  F
)  =  ( ( Q  .\/  ( F `
 Q ) ) 
./\  W ) )
6733adantr 471 . . . . . . 7  |-  ( ( ( ( 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 )
) ) )  /\  ( F `  P )  =/=  P )  ->  Q  e.  A )
685, 7, 8, 9ltrnat 33749 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  Q  e.  A
)  ->  ( F `  Q )  e.  A
)
6944, 45, 67, 68syl3anc 1276 . . . . . . 7  |-  ( ( ( ( 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 )
) ) )  /\  ( F `  P )  =/=  P )  -> 
( F `  Q
)  e.  A )
7055, 20, 7hlatjcl 32976 . . . . . . 7  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  ( F `  Q )  e.  A )  -> 
( Q  .\/  ( F `  Q )
)  e.  ( Base `  K ) )
7149, 67, 69, 70syl3anc 1276 . . . . . 6  |-  ( ( ( ( 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 )
) ) )  /\  ( F `  P )  =/=  P )  -> 
( Q  .\/  ( F `  Q )
)  e.  ( Base `  K ) )
7255, 5, 39latmle1 16370 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( Q  .\/  ( F `
 Q ) )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( Q  .\/  ( F `  Q ) )  ./\  W )  .<_  ( Q  .\/  ( F `  Q
) ) )
7351, 71, 60, 72syl3anc 1276 . . . . 5  |-  ( ( ( ( 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 )
) ) )  /\  ( F `  P )  =/=  P )  -> 
( ( Q  .\/  ( F `  Q ) )  ./\  W )  .<_  ( Q  .\/  ( F `  Q )
) )
7466, 73eqbrtrd 4436 . . . 4  |-  ( ( ( ( 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 )
) ) )  /\  ( F `  P )  =/=  P )  -> 
( R `  F
)  .<_  ( Q  .\/  ( F `  Q ) ) )
7555, 8, 9, 10trlcl 33774 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( R `  F )  e.  (
Base `  K )
)
7644, 45, 75syl2anc 671 . . . . 5  |-  ( ( ( ( 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 )
) ) )  /\  ( F `  P )  =/=  P )  -> 
( R `  F
)  e.  ( Base `  K ) )
7755, 5, 39latlem12 16372 . . . . 5  |-  ( ( K  e.  Lat  /\  ( ( R `  F )  e.  (
Base `  K )  /\  ( P  .\/  ( F `  P )
)  e.  ( Base `  K )  /\  ( Q  .\/  ( F `  Q ) )  e.  ( Base `  K
) ) )  -> 
( ( ( R `
 F )  .<_  ( P  .\/  ( F `
 P ) )  /\  ( R `  F )  .<_  ( Q 
.\/  ( F `  Q ) ) )  <-> 
( R `  F
)  .<_  ( ( P 
.\/  ( F `  P ) )  ./\  ( Q  .\/  ( F `
 Q ) ) ) ) )
7851, 76, 57, 71, 77syl13anc 1278 . . . 4  |-  ( ( ( ( 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 )
) ) )  /\  ( F `  P )  =/=  P )  -> 
( ( ( R `
 F )  .<_  ( P  .\/  ( F `
 P ) )  /\  ( R `  F )  .<_  ( Q 
.\/  ( F `  Q ) ) )  <-> 
( R `  F
)  .<_  ( ( P 
.\/  ( F `  P ) )  ./\  ( Q  .\/  ( F `
 Q ) ) ) ) )
7963, 74, 78mpbi2and 937 . . 3  |-  ( ( ( ( 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 )
) ) )  /\  ( F `  P )  =/=  P )  -> 
( R `  F
)  .<_  ( ( P 
.\/  ( F `  P ) )  ./\  ( Q  .\/  ( F `
 Q ) ) ) )
8049, 15syl 17 . . . 4  |-  ( ( ( ( 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 )
) ) )  /\  ( F `  P )  =/=  P )  ->  K  e.  AtLat )
81 simpr 467 . . . . 5  |-  ( ( ( ( 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 )
) ) )  /\  ( F `  P )  =/=  P )  -> 
( F `  P
)  =/=  P )
825, 7, 8, 9, 10trlat 33779 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F  e.  T  /\  ( F `  P )  =/=  P ) )  ->  ( R `  F )  e.  A
)
8344, 46, 45, 81, 82syl112anc 1280 . . . 4  |-  ( ( ( ( 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 )
) ) )  /\  ( F `  P )  =/=  P )  -> 
( R `  F
)  e.  A )
8455, 39latmcl 16346 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( P  .\/  ( F `
 P ) )  e.  ( Base `  K
)  /\  ( Q  .\/  ( F `  Q
) )  e.  (
Base `  K )
)  ->  ( ( P  .\/  ( F `  P ) )  ./\  ( Q  .\/  ( F `
 Q ) ) )  e.  ( Base `  K ) )
8551, 57, 71, 84syl3anc 1276 . . . . . . 7  |-  ( ( ( ( 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 )
) ) )  /\  ( F `  P )  =/=  P )  -> 
( ( P  .\/  ( F `  P ) )  ./\  ( Q  .\/  ( F `  Q
) ) )  e.  ( Base `  K
) )
8655, 5, 6, 7atlen0 32920 . . . . . . 7  |-  ( ( ( K  e.  AtLat  /\  ( ( P  .\/  ( F `  P ) )  ./\  ( Q  .\/  ( F `  Q
) ) )  e.  ( Base `  K
)  /\  ( R `  F )  e.  A
)  /\  ( R `  F )  .<_  ( ( P  .\/  ( F `
 P ) ) 
./\  ( Q  .\/  ( F `  Q ) ) ) )  -> 
( ( P  .\/  ( F `  P ) )  ./\  ( Q  .\/  ( F `  Q
) ) )  =/=  ( 0. `  K
) )
8780, 85, 83, 79, 86syl31anc 1279 . . . . . 6  |-  ( ( ( ( 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 )
) ) )  /\  ( F `  P )  =/=  P )  -> 
( ( P  .\/  ( F `  P ) )  ./\  ( Q  .\/  ( F `  Q
) ) )  =/=  ( 0. `  K
) )
8887neneqd 2639 . . . . 5  |-  ( ( ( ( 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 )
) ) )  /\  ( F `  P )  =/=  P )  ->  -.  ( ( P  .\/  ( F `  P ) )  ./\  ( Q  .\/  ( F `  Q
) ) )  =  ( 0. `  K
) )
89 simpl33 1097 . . . . . . 7  |-  ( ( ( ( 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 )
) ) )  /\  ( F `  P )  =/=  P )  -> 
( P  .\/  ( F `  P )
)  =/=  ( Q 
.\/  ( F `  Q ) ) )
9020, 39, 6, 72atmat0 33135 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  ( F `  P )  e.  A )  /\  ( Q  e.  A  /\  ( F `  Q
)  e.  A  /\  ( P  .\/  ( F `
 P ) )  =/=  ( Q  .\/  ( F `  Q ) ) ) )  -> 
( ( ( P 
.\/  ( F `  P ) )  ./\  ( Q  .\/  ( F `
 Q ) ) )  e.  A  \/  ( ( P  .\/  ( F `  P ) )  ./\  ( Q  .\/  ( F `  Q
) ) )  =  ( 0. `  K
) ) )
9149, 52, 54, 67, 69, 89, 90syl33anc 1291 . . . . . 6  |-  ( ( ( ( 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 )
) ) )  /\  ( F `  P )  =/=  P )  -> 
( ( ( P 
.\/  ( F `  P ) )  ./\  ( Q  .\/  ( F `
 Q ) ) )  e.  A  \/  ( ( P  .\/  ( F `  P ) )  ./\  ( Q  .\/  ( F `  Q
) ) )  =  ( 0. `  K
) ) )
9291ord 383 . . . . 5  |-  ( ( ( ( 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 )
) ) )  /\  ( F `  P )  =/=  P )  -> 
( -.  ( ( P  .\/  ( F `
 P ) ) 
./\  ( Q  .\/  ( F `  Q ) ) )  e.  A  ->  ( ( P  .\/  ( F `  P ) )  ./\  ( Q  .\/  ( F `  Q
) ) )  =  ( 0. `  K
) ) )
9388, 92mt3d 130 . . . 4  |-  ( ( ( ( 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 )
) ) )  /\  ( F `  P )  =/=  P )  -> 
( ( P  .\/  ( F `  P ) )  ./\  ( Q  .\/  ( F `  Q
) ) )  e.  A )
945, 7atcmp 32921 . . . 4  |-  ( ( K  e.  AtLat  /\  ( R `  F )  e.  A  /\  (
( P  .\/  ( F `  P )
)  ./\  ( Q  .\/  ( F `  Q
) ) )  e.  A )  ->  (
( R `  F
)  .<_  ( ( P 
.\/  ( F `  P ) )  ./\  ( Q  .\/  ( F `
 Q ) ) )  <->  ( R `  F )  =  ( ( P  .\/  ( F `  P )
)  ./\  ( Q  .\/  ( F `  Q
) ) ) ) )
9580, 83, 93, 94syl3anc 1276 . . 3  |-  ( ( ( ( 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 )
) ) )  /\  ( F `  P )  =/=  P )  -> 
( ( R `  F )  .<_  ( ( P  .\/  ( F `
 P ) ) 
./\  ( Q  .\/  ( F `  Q ) ) )  <->  ( R `  F )  =  ( ( P  .\/  ( F `  P )
)  ./\  ( Q  .\/  ( F `  Q
) ) ) ) )
9679, 95mpbid 215 . 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 )
) ) )  /\  ( F `  P )  =/=  P )  -> 
( R `  F
)  =  ( ( P  .\/  ( F `
 P ) ) 
./\  ( Q  .\/  ( F `  Q ) ) ) )
9743, 96pm2.61dane 2722 1  |-  ( ( ( 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
) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    \/ wo 374    /\ wa 375    /\ w3a 991    = wceq 1454    e. wcel 1897    =/= wne 2632   class class class wbr 4415   ` cfv 5600  (class class class)co 6314   Basecbs 15169   lecple 15245   joincjn 16237   meetcmee 16238   0.cp0 16331   Latclat 16339   Atomscatm 32873   AtLatcal 32874   HLchlt 32960   LHypclh 33593   LTrncltrn 33710   trLctrl 33768
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-rep 4528  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-ral 2753  df-rex 2754  df-reu 2755  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-nul 3743  df-if 3893  df-pw 3964  df-sn 3980  df-pr 3982  df-op 3986  df-uni 4212  df-iun 4293  df-br 4416  df-opab 4475  df-mpt 4476  df-id 4767  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-riota 6276  df-ov 6317  df-oprab 6318  df-mpt2 6319  df-map 7499  df-preset 16221  df-poset 16239  df-plt 16252  df-lub 16268  df-glb 16269  df-join 16270  df-meet 16271  df-p0 16333  df-p1 16334  df-lat 16340  df-clat 16402  df-oposet 32786  df-ol 32788  df-oml 32789  df-covers 32876  df-ats 32877  df-atl 32908  df-cvlat 32932  df-hlat 32961  df-llines 33107  df-lhyp 33597  df-laut 33598  df-ldil 33713  df-ltrn 33714  df-trl 33769
This theorem is referenced by:  trlval4  33798
  Copyright terms: Public domain W3C validator