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Theorem trlval 30351
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 30350 . . 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 5527 . 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 5524 . . . . . . . . 9  |-  ( f  =  F  ->  (
f `  p )  =  ( F `  p ) )
1211oveq2d 5874 . . . . . . . 8  |-  ( f  =  F  ->  (
p  .\/  ( f `  p ) )  =  ( p  .\/  ( F `  p )
) )
1312oveq1d 5873 . . . . . . 7  |-  ( f  =  F  ->  (
( p  .\/  (
f `  p )
)  ./\  W )  =  ( ( p 
.\/  ( F `  p ) )  ./\  W ) )
1413eqeq2d 2294 . . . . . 6  |-  ( f  =  F  ->  (
x  =  ( ( p  .\/  ( f `
 p ) ) 
./\  W )  <->  x  =  ( ( p  .\/  ( F `  p ) )  ./\  W )
) )
1514imbi2d 307 . . . . 5  |-  ( f  =  F  ->  (
( -.  p  .<_  W  ->  x  =  ( ( p  .\/  (
f `  p )
)  ./\  W )
)  <->  ( -.  p  .<_  W  ->  x  =  ( ( p  .\/  ( F `  p ) )  ./\  W )
) ) )
1615ralbidv 2563 . . . 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 6306 . . 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 2283 . . 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 6308 . . 3  |-  ( iota_ x  e.  B A. p  e.  A  ( -.  p  .<_  W  ->  x  =  ( ( p 
.\/  ( F `  p ) )  ./\  W ) ) )  e. 
_V
2017, 18, 19fvmpt 5602 . 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 2335 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 set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   class class class wbr 4023    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   iota_crio 6297   Basecbs 13148   lecple 13215   joincjn 14078   meetcmee 14079   Atomscatm 29453   LHypclh 30173   LTrncltrn 30290   trLctrl 30347
This theorem is referenced by:  trlval2  30352
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-riota 6304  df-trl 30348
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