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Theorem trlval3 30985
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 961 . . . 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 1039 . . . 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 962 . . . 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 449 . . . 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 2437 . . . . 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 30968 . . . 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 1189 . . 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 1041 . . . 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 1009 . . . . . 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 30159 . . . . . 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 6098 . . . . . . 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 1081 . . . . . . . . 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 453 . . . . . . . 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 30167 . . . . . . . 8  |-  ( ( K  e.  HL  /\  P  e.  A )  ->  ( P  .\/  P
)  =  P )
2214, 19, 21syl2anc 644 . . . . . . 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 2469 . . . . . 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 2511 . . . . 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 958 . . . . . . . . . 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 959 . . . . . . . . . 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 994 . . . . . . . . . 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 995 . . . . . . . . . 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 30978 . . . . . . . . . 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 1187 . . . . . . . . 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 472 . . . . . . . 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 6098 . . . . . . 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 1083 . . . . . . . . 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 453 . . . . . . . 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 30167 . . . . . . . 8  |-  ( ( K  e.  HL  /\  Q  e.  A )  ->  ( Q  .\/  Q
)  =  Q )
3614, 34, 35syl2anc 644 . . . . . . 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 2469 . . . . . 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 2511 . . . . 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 30117 . . . . 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 1185 . . . 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 203 . . 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 2472 . 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 961 . . . . . 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 962 . . . . . 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 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 )  -> 
( P  e.  A  /\  -.  P  .<_  W ) )
475, 20, 39, 7, 8, 9, 10trlval2 30961 . . . . . 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 1185 . . . . 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 1009 . . . . . . 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 30162 . . . . . . 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 453 . . . . . . 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 30938 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  A
)  ->  ( F `  P )  e.  A
)
5444, 45, 52, 53syl3anc 1185 . . . . . . 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 2437 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  K )
5655, 20, 7hlatjcl 30165 . . . . . . 7  |-  ( ( K  e.  HL  /\  P  e.  A  /\  ( F `  P )  e.  A )  -> 
( P  .\/  ( F `  P )
)  e.  ( Base `  K ) )
5749, 52, 54, 56syl3anc 1185 . . . . . 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 1010 . . . . . . 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 30796 . . . . . . 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 14506 . . . . . 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 1185 . . . . 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 4233 . . . 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 1040 . . . . . 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 30961 . . . . . 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 1185 . . . . 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 453 . . . . . . 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 30938 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  Q  e.  A
)  ->  ( F `  Q )  e.  A
)
6944, 45, 67, 68syl3anc 1185 . . . . . . 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 30165 . . . . . . 7  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  ( F `  Q )  e.  A )  -> 
( Q  .\/  ( F `  Q )
)  e.  ( Base `  K ) )
7149, 67, 69, 70syl3anc 1185 . . . . . 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 14506 . . . . . 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 1185 . . . . 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 4233 . . . 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 30962 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( R `  F )  e.  (
Base `  K )
)
7644, 45, 75syl2anc 644 . . . . 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 14508 . . . . 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 1187 . . . 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 889 . . 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 449 . . . . 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 30967 . . . . 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 1189 . . . 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 14481 . . . . . . . 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 1185 . . . . . . 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 30109 . . . . . . 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 1188 . . . . . 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 2618 . . . . 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 1041 . . . . . . 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 30324 . . . . . . 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 1200 . . . . . 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 368 . . . . 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 120 . . . 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 30110 . . . 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 1185 . . 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 203 . 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 2683 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 set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    \/ wo 359    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2600   class class class wbr 4213   ` cfv 5455  (class class class)co 6082   Basecbs 13470   lecple 13537   joincjn 14402   meetcmee 14403   0.cp0 14467   Latclat 14475   Atomscatm 30062   AtLatcal 30063   HLchlt 30149   LHypclh 30782   LTrncltrn 30899   trLctrl 30956
This theorem is referenced by:  trlval4  30986
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351  df-undef 6544  df-riota 6550  df-map 7021  df-poset 14404  df-plt 14416  df-lub 14432  df-glb 14433  df-join 14434  df-meet 14435  df-p0 14469  df-p1 14470  df-lat 14476  df-clat 14538  df-oposet 29975  df-ol 29977  df-oml 29978  df-covers 30065  df-ats 30066  df-atl 30097  df-cvlat 30121  df-hlat 30150  df-llines 30296  df-lhyp 30786  df-laut 30787  df-ldil 30902  df-ltrn 30903  df-trl 30957
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