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Theorem trlval 30973
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 30972 . . 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 5543 . 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 5540 . . . . . . . . 9  |-  ( f  =  F  ->  (
f `  p )  =  ( F `  p ) )
1211oveq2d 5890 . . . . . . . 8  |-  ( f  =  F  ->  (
p  .\/  ( f `  p ) )  =  ( p  .\/  ( F `  p )
) )
1312oveq1d 5889 . . . . . . 7  |-  ( f  =  F  ->  (
( p  .\/  (
f `  p )
)  ./\  W )  =  ( ( p 
.\/  ( F `  p ) )  ./\  W ) )
1413eqeq2d 2307 . . . . . 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 2576 . . . 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 6322 . . 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 2296 . . 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 6324 . . 3  |-  ( iota_ x  e.  B A. p  e.  A  ( -.  p  .<_  W  ->  x  =  ( ( p 
.\/  ( F `  p ) )  ./\  W ) ) )  e. 
_V
2017, 18, 19fvmpt 5618 . 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 2348 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 1632    e. wcel 1696   A.wral 2556   class class class wbr 4039    e. cmpt 4093   ` cfv 5271  (class class class)co 5874   iota_crio 6313   Basecbs 13164   lecple 13231   joincjn 14094   meetcmee 14095   Atomscatm 30075   LHypclh 30795   LTrncltrn 30912   trLctrl 30969
This theorem is referenced by:  trlval2  30974
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-riota 6320  df-trl 30970
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