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Theorem trlval 33161
Description: The value of the trace of a lattice translation. (Contributed by NM, 20-May-2012.)
Hypotheses
Ref Expression
trlset.b  |-  B  =  ( Base `  K
)
trlset.l  |-  .<_  =  ( le `  K )
trlset.j  |-  .\/  =  ( join `  K )
trlset.m  |-  ./\  =  ( meet `  K )
trlset.a  |-  A  =  ( Atoms `  K )
trlset.h  |-  H  =  ( LHyp `  K
)
trlset.t  |-  T  =  ( ( LTrn `  K
) `  W )
trlset.r  |-  R  =  ( ( trL `  K
) `  W )
Assertion
Ref Expression
trlval  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  F  e.  T )  ->  ( R `  F )  =  ( iota_ x  e.  B  A. p  e.  A  ( -.  p  .<_  W  ->  x  =  ( ( p  .\/  ( F `  p ) )  ./\  W )
) ) )
Distinct variable groups:    A, p    x, B    x, p, K    W, p, x    F, p, x
Allowed substitution hints:    A( x)    B( p)    R( x, p)    T( x, p)    H( x, p)    .\/ ( x, p)    .<_ ( x, p)    ./\ (
x, p)    V( x, p)

Proof of Theorem trlval
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 trlset.b . . . 4  |-  B  =  ( Base `  K
)
2 trlset.l . . . 4  |-  .<_  =  ( le `  K )
3 trlset.j . . . 4  |-  .\/  =  ( join `  K )
4 trlset.m . . . 4  |-  ./\  =  ( meet `  K )
5 trlset.a . . . 4  |-  A  =  ( Atoms `  K )
6 trlset.h . . . 4  |-  H  =  ( LHyp `  K
)
7 trlset.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
8 trlset.r . . . 4  |-  R  =  ( ( trL `  K
) `  W )
91, 2, 3, 4, 5, 6, 7, 8trlset 33160 . . 3  |-  ( ( K  e.  V  /\  W  e.  H )  ->  R  =  ( f  e.  T  |->  ( iota_ x  e.  B  A. p  e.  A  ( -.  p  .<_  W  ->  x  =  ( ( p 
.\/  ( f `  p ) )  ./\  W ) ) ) ) )
109fveq1d 5807 . 2  |-  ( ( K  e.  V  /\  W  e.  H )  ->  ( R `  F
)  =  ( ( f  e.  T  |->  (
iota_ x  e.  B  A. p  e.  A  ( -.  p  .<_  W  ->  x  =  ( ( p  .\/  (
f `  p )
)  ./\  W )
) ) ) `  F ) )
11 fveq1 5804 . . . . . . . . 9  |-  ( f  =  F  ->  (
f `  p )  =  ( F `  p ) )
1211oveq2d 6250 . . . . . . . 8  |-  ( f  =  F  ->  (
p  .\/  ( f `  p ) )  =  ( p  .\/  ( F `  p )
) )
1312oveq1d 6249 . . . . . . 7  |-  ( f  =  F  ->  (
( p  .\/  (
f `  p )
)  ./\  W )  =  ( ( p 
.\/  ( F `  p ) )  ./\  W ) )
1413eqeq2d 2416 . . . . . 6  |-  ( f  =  F  ->  (
x  =  ( ( p  .\/  ( f `
 p ) ) 
./\  W )  <->  x  =  ( ( p  .\/  ( F `  p ) )  ./\  W )
) )
1514imbi2d 314 . . . . 5  |-  ( f  =  F  ->  (
( -.  p  .<_  W  ->  x  =  ( ( p  .\/  (
f `  p )
)  ./\  W )
)  <->  ( -.  p  .<_  W  ->  x  =  ( ( p  .\/  ( F `  p ) )  ./\  W )
) ) )
1615ralbidv 2842 . . . 4  |-  ( f  =  F  ->  ( A. p  e.  A  ( -.  p  .<_  W  ->  x  =  ( ( p  .\/  (
f `  p )
)  ./\  W )
)  <->  A. p  e.  A  ( -.  p  .<_  W  ->  x  =  ( ( p  .\/  ( F `  p )
)  ./\  W )
) ) )
1716riotabidv 6198 . . 3  |-  ( f  =  F  ->  ( iota_ x  e.  B  A. p  e.  A  ( -.  p  .<_  W  ->  x  =  ( (
p  .\/  ( f `  p ) )  ./\  W ) ) )  =  ( iota_ x  e.  B  A. p  e.  A  ( -.  p  .<_  W  ->  x  =  ( ( p  .\/  ( F `  p )
)  ./\  W )
) ) )
18 eqid 2402 . . 3  |-  ( f  e.  T  |->  ( iota_ x  e.  B  A. p  e.  A  ( -.  p  .<_  W  ->  x  =  ( ( p 
.\/  ( f `  p ) )  ./\  W ) ) ) )  =  ( f  e.  T  |->  ( iota_ x  e.  B  A. p  e.  A  ( -.  p  .<_  W  ->  x  =  ( ( p  .\/  ( f `  p
) )  ./\  W
) ) ) )
19 riotaex 6200 . . 3  |-  ( iota_ x  e.  B  A. p  e.  A  ( -.  p  .<_  W  ->  x  =  ( ( p 
.\/  ( F `  p ) )  ./\  W ) ) )  e. 
_V
2017, 18, 19fvmpt 5888 . 2  |-  ( F  e.  T  ->  (
( f  e.  T  |->  ( iota_ x  e.  B  A. p  e.  A  ( -.  p  .<_  W  ->  x  =  ( ( p  .\/  (
f `  p )
)  ./\  W )
) ) ) `  F )  =  (
iota_ x  e.  B  A. p  e.  A  ( -.  p  .<_  W  ->  x  =  ( ( p  .\/  ( F `  p )
)  ./\  W )
) ) )
2110, 20sylan9eq 2463 1  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  F  e.  T )  ->  ( R `  F )  =  ( iota_ x  e.  B  A. p  e.  A  ( -.  p  .<_  W  ->  x  =  ( ( p  .\/  ( F `  p ) )  ./\  W )
) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842   A.wral 2753   class class class wbr 4394    |-> cmpt 4452   ` cfv 5525   iota_crio 6195  (class class class)co 6234   Basecbs 14733   lecple 14808   joincjn 15789   meetcmee 15790   Atomscatm 32262   LHypclh 32982   LTrncltrn 33099   trLctrl 33157
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6196  df-ov 6237  df-trl 33158
This theorem is referenced by:  trlval2  33162
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