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Theorem trlval 34833
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 34832 . . 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 5859 . 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 5856 . . . . . . . . 9  |-  ( f  =  F  ->  (
f `  p )  =  ( F `  p ) )
1211oveq2d 6291 . . . . . . . 8  |-  ( f  =  F  ->  (
p  .\/  ( f `  p ) )  =  ( p  .\/  ( F `  p )
) )
1312oveq1d 6290 . . . . . . 7  |-  ( f  =  F  ->  (
( p  .\/  (
f `  p )
)  ./\  W )  =  ( ( p 
.\/  ( F `  p ) )  ./\  W ) )
1413eqeq2d 2474 . . . . . 6  |-  ( f  =  F  ->  (
x  =  ( ( p  .\/  ( f `
 p ) ) 
./\  W )  <->  x  =  ( ( p  .\/  ( F `  p ) )  ./\  W )
) )
1514imbi2d 316 . . . . 5  |-  ( f  =  F  ->  (
( -.  p  .<_  W  ->  x  =  ( ( p  .\/  (
f `  p )
)  ./\  W )
)  <->  ( -.  p  .<_  W  ->  x  =  ( ( p  .\/  ( F `  p ) )  ./\  W )
) ) )
1615ralbidv 2896 . . . 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 6238 . . 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 2460 . . 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 6240 . . 3  |-  ( iota_ x  e.  B  A. p  e.  A  ( -.  p  .<_  W  ->  x  =  ( ( p 
.\/  ( F `  p ) )  ./\  W ) ) )  e. 
_V
2017, 18, 19fvmpt 5941 . 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 2521 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 369    = wceq 1374    e. wcel 1762   A.wral 2807   class class class wbr 4440    |-> cmpt 4498   ` cfv 5579   iota_crio 6235  (class class class)co 6275   Basecbs 14479   lecple 14551   joincjn 15420   meetcmee 15421   Atomscatm 33935   LHypclh 34655   LTrncltrn 34772   trLctrl 34829
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-trl 34830
This theorem is referenced by:  trlval2  34834
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