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Theorem trlval3 35384
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 999 . . . 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 1077 . . . 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 1000 . . . 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 461 . . . 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 2467 . . . . 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 35367 . . . 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 1232 . . 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 1079 . . . 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 1047 . . . . . 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 34558 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  AtLat )
1614, 15syl 16 . . . . 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 6311 . . . . . . 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 1119 . . . . . . . . 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 465 . . . . . . . 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 34566 . . . . . . . 8  |-  ( ( K  e.  HL  /\  P  e.  A )  ->  ( P  .\/  P
)  =  P )
2214, 19, 21syl2anc 661 . . . . . . 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 2508 . . . . . 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 2555 . . . . 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 996 . . . . . . . . . 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 997 . . . . . . . . . 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 1032 . . . . . . . . . 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 1033 . . . . . . . . . 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 35377 . . . . . . . . . 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 1230 . . . . . . . . 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 484 . . . . . . . 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 6311 . . . . . . 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 1121 . . . . . . . . 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 465 . . . . . . . 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 34566 . . . . . . . 8  |-  ( ( K  e.  HL  /\  Q  e.  A )  ->  ( Q  .\/  Q
)  =  Q )
3614, 34, 35syl2anc 661 . . . . . . 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 2508 . . . . . 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 2555 . . . . 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 34516 . . . . 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 1228 . . . 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 210 . . 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 2511 . 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 999 . . . . . 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 1000 . . . . . 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 1077 . . . . . 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 35360 . . . . . 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 1228 . . . . 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 1047 . . . . . . 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 34561 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  Lat )
5149, 50syl 16 . . . . . 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 465 . . . . . . 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 35337 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  A
)  ->  ( F `  P )  e.  A
)
5444, 45, 52, 53syl3anc 1228 . . . . . . 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 2467 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  K )
5655, 20, 7hlatjcl 34564 . . . . . . 7  |-  ( ( K  e.  HL  /\  P  e.  A  /\  ( F `  P )  e.  A )  -> 
( P  .\/  ( F `  P )
)  e.  ( Base `  K ) )
5749, 52, 54, 56syl3anc 1228 . . . . . 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 1048 . . . . . . 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 35195 . . . . . . 7  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
6058, 59syl 16 . . . . . 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 15580 . . . . . 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 1228 . . . . 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 4473 . . . 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 1078 . . . . . 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 35360 . . . . . 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 1228 . . . . 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 465 . . . . . . 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 35337 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  Q  e.  A
)  ->  ( F `  Q )  e.  A
)
6944, 45, 67, 68syl3anc 1228 . . . . . . 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 34564 . . . . . . 7  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  ( F `  Q )  e.  A )  -> 
( Q  .\/  ( F `  Q )
)  e.  ( Base `  K ) )
7149, 67, 69, 70syl3anc 1228 . . . . . 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 15580 . . . . . 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 1228 . . . . 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 4473 . . . 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 35361 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( R `  F )  e.  (
Base `  K )
)
7644, 45, 75syl2anc 661 . . . . 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 15582 . . . . 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 1230 . . . 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 919 . . 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 16 . . . 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 461 . . . . 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 35366 . . . . 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 1232 . . . 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 15556 . . . . . . . 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 1228 . . . . . . 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 34508 . . . . . . 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 1231 . . . . . 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 2669 . . . . 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 1079 . . . . . . 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 34723 . . . . . . 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 1243 . . . . . 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 377 . . . . 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 125 . . . 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 34509 . . . 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 1228 . . 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 210 . 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 2785 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 184    \/ wo 368    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   class class class wbr 4453   ` cfv 5594  (class class class)co 6295   Basecbs 14507   lecple 14579   joincjn 15448   meetcmee 15449   0.cp0 15541   Latclat 15549   Atomscatm 34461   AtLatcal 34462   HLchlt 34548   LHypclh 35181   LTrncltrn 35298   trLctrl 35355
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-map 7434  df-poset 15450  df-plt 15462  df-lub 15478  df-glb 15479  df-join 15480  df-meet 15481  df-p0 15543  df-p1 15544  df-lat 15550  df-clat 15612  df-oposet 34374  df-ol 34376  df-oml 34377  df-covers 34464  df-ats 34465  df-atl 34496  df-cvlat 34520  df-hlat 34549  df-llines 34695  df-lhyp 35185  df-laut 35186  df-ldil 35301  df-ltrn 35302  df-trl 35356
This theorem is referenced by:  trlval4  35385
  Copyright terms: Public domain W3C validator