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Theorem trlval3 33831
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 991 . . . 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 1069 . . . 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 992 . . . 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 2443 . . . . 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 33814 . . . 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 1222 . . 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 1071 . . . 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 1039 . . . . . 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 33005 . . . . . 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 6107 . . . . . . 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 1111 . . . . . . . . 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 33013 . . . . . . . 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 2475 . . . . . 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 2517 . . . . 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 988 . . . . . . . . . 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 989 . . . . . . . . . 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 1024 . . . . . . . . . 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 1025 . . . . . . . . . 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 33824 . . . . . . . . . 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 1220 . . . . . . . . 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 6107 . . . . . . 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 1113 . . . . . . . . 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 33013 . . . . . . . 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 2475 . . . . . 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 2517 . . . . 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 32963 . . . . 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 1218 . . . 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 2478 . 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 991 . . . . . 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 992 . . . . . 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 1069 . . . . . 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 33807 . . . . . 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 1218 . . . . 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 1039 . . . . . . 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 33008 . . . . . . 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 33784 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  A
)  ->  ( F `  P )  e.  A
)
5444, 45, 52, 53syl3anc 1218 . . . . . . 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 2443 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  K )
5655, 20, 7hlatjcl 33011 . . . . . . 7  |-  ( ( K  e.  HL  /\  P  e.  A  /\  ( F `  P )  e.  A )  -> 
( P  .\/  ( F `  P )
)  e.  ( Base `  K ) )
5749, 52, 54, 56syl3anc 1218 . . . . . 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 1040 . . . . . . 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 33642 . . . . . . 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 15246 . . . . . 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 1218 . . . . 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 4312 . . . 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 1070 . . . . . 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 33807 . . . . . 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 1218 . . . . 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 33784 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  Q  e.  A
)  ->  ( F `  Q )  e.  A
)
6944, 45, 67, 68syl3anc 1218 . . . . . . 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 33011 . . . . . . 7  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  ( F `  Q )  e.  A )  -> 
( Q  .\/  ( F `  Q )
)  e.  ( Base `  K ) )
7149, 67, 69, 70syl3anc 1218 . . . . . 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 15246 . . . . . 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 1218 . . . . 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 4312 . . . 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 33808 . . . . . 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 15248 . . . . 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 1220 . . . 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 912 . . 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 33813 . . . . 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 1222 . . . 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 15222 . . . . . . . 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 1218 . . . . . . 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 32955 . . . . . . 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 1221 . . . . . 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 2624 . . . . 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 1071 . . . . . . 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 33170 . . . . . . 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 1233 . . . . . 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 32956 . . . 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 1218 . . 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 2689 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 965    = wceq 1369    e. wcel 1756    =/= wne 2606   class class class wbr 4292   ` cfv 5418  (class class class)co 6091   Basecbs 14174   lecple 14245   joincjn 15114   meetcmee 15115   0.cp0 15207   Latclat 15215   Atomscatm 32908   AtLatcal 32909   HLchlt 32995   LHypclh 33628   LTrncltrn 33745   trLctrl 33802
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-map 7216  df-poset 15116  df-plt 15128  df-lub 15144  df-glb 15145  df-join 15146  df-meet 15147  df-p0 15209  df-p1 15210  df-lat 15216  df-clat 15278  df-oposet 32821  df-ol 32823  df-oml 32824  df-covers 32911  df-ats 32912  df-atl 32943  df-cvlat 32967  df-hlat 32996  df-llines 33142  df-lhyp 33632  df-laut 33633  df-ldil 33748  df-ltrn 33749  df-trl 33803
This theorem is referenced by:  trlval4  33832
  Copyright terms: Public domain W3C validator