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Theorem trlset 30959
Description: The set of traces of lattice translations for a fiducial co-atom  W. (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
trlset  |-  ( ( K  e.  C  /\  W  e.  H )  ->  R  =  ( f  e.  T  |->  ( iota_ x  e.  B A. p  e.  A  ( -.  p  .<_  W  ->  x  =  ( ( p 
.\/  ( f `  p ) )  ./\  W ) ) ) ) )
Distinct variable groups:    A, p    x, B    f, p, x, K    T, f    f, W, p, x
Allowed substitution hints:    A( x, f)    B( f, p)    C( x, f, p)    R( x, f, p)    T( x, p)    H( x, f, p)    .\/ ( x, f, p)    .<_ ( x, f, p)    ./\ (
x, f, p)

Proof of Theorem trlset
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 trlset.r . . 3  |-  R  =  ( ( trL `  K
) `  W )
2 trlset.b . . . . 5  |-  B  =  ( Base `  K
)
3 trlset.l . . . . 5  |-  .<_  =  ( le `  K )
4 trlset.j . . . . 5  |-  .\/  =  ( join `  K )
5 trlset.m . . . . 5  |-  ./\  =  ( meet `  K )
6 trlset.a . . . . 5  |-  A  =  ( Atoms `  K )
7 trlset.h . . . . 5  |-  H  =  ( LHyp `  K
)
82, 3, 4, 5, 6, 7trlfset 30958 . . . 4  |-  ( K  e.  C  ->  ( trL `  K )  =  ( w  e.  H  |->  ( f  e.  ( ( LTrn `  K
) `  w )  |->  ( iota_ x  e.  B A. p  e.  A  ( -.  p  .<_  w  ->  x  =  ( ( p  .\/  (
f `  p )
)  ./\  w )
) ) ) ) )
98fveq1d 5731 . . 3  |-  ( K  e.  C  ->  (
( trL `  K
) `  W )  =  ( ( w  e.  H  |->  ( f  e.  ( ( LTrn `  K ) `  w
)  |->  ( iota_ x  e.  B A. p  e.  A  ( -.  p  .<_  w  ->  x  =  ( ( p  .\/  ( f `  p
) )  ./\  w
) ) ) ) ) `  W ) )
101, 9syl5eq 2481 . 2  |-  ( K  e.  C  ->  R  =  ( ( w  e.  H  |->  ( f  e.  ( ( LTrn `  K ) `  w
)  |->  ( iota_ x  e.  B A. p  e.  A  ( -.  p  .<_  w  ->  x  =  ( ( p  .\/  ( f `  p
) )  ./\  w
) ) ) ) ) `  W ) )
11 fveq2 5729 . . . . 5  |-  ( w  =  W  ->  (
( LTrn `  K ) `  w )  =  ( ( LTrn `  K
) `  W )
)
12 breq2 4217 . . . . . . . . 9  |-  ( w  =  W  ->  (
p  .<_  w  <->  p  .<_  W ) )
1312notbid 287 . . . . . . . 8  |-  ( w  =  W  ->  ( -.  p  .<_  w  <->  -.  p  .<_  W ) )
14 oveq2 6090 . . . . . . . . 9  |-  ( w  =  W  ->  (
( p  .\/  (
f `  p )
)  ./\  w )  =  ( ( p 
.\/  ( f `  p ) )  ./\  W ) )
1514eqeq2d 2448 . . . . . . . 8  |-  ( w  =  W  ->  (
x  =  ( ( p  .\/  ( f `
 p ) ) 
./\  w )  <->  x  =  ( ( p  .\/  ( f `  p
) )  ./\  W
) ) )
1613, 15imbi12d 313 . . . . . . 7  |-  ( w  =  W  ->  (
( -.  p  .<_  w  ->  x  =  ( ( p  .\/  (
f `  p )
)  ./\  w )
)  <->  ( -.  p  .<_  W  ->  x  =  ( ( p  .\/  ( f `  p
) )  ./\  W
) ) ) )
1716ralbidv 2726 . . . . . 6  |-  ( w  =  W  ->  ( A. p  e.  A  ( -.  p  .<_  w  ->  x  =  ( ( p  .\/  (
f `  p )
)  ./\  w )
)  <->  A. p  e.  A  ( -.  p  .<_  W  ->  x  =  ( ( p  .\/  (
f `  p )
)  ./\  W )
) ) )
1817riotabidv 6552 . . . . 5  |-  ( w  =  W  ->  ( 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
) ) ) )
1911, 18mpteq12dv 4288 . . . 4  |-  ( w  =  W  ->  (
f  e.  ( (
LTrn `  K ) `  w )  |->  ( iota_ x  e.  B A. p  e.  A  ( -.  p  .<_  w  ->  x  =  ( ( p 
.\/  ( f `  p ) )  ./\  w ) ) ) )  =  ( f  e.  ( ( LTrn `  K ) `  W
)  |->  ( iota_ x  e.  B A. p  e.  A  ( -.  p  .<_  W  ->  x  =  ( ( p  .\/  ( f `  p
) )  ./\  W
) ) ) ) )
20 eqid 2437 . . . 4  |-  ( w  e.  H  |->  ( f  e.  ( ( LTrn `  K ) `  w
)  |->  ( iota_ x  e.  B A. p  e.  A  ( -.  p  .<_  w  ->  x  =  ( ( p  .\/  ( f `  p
) )  ./\  w
) ) ) ) )  =  ( w  e.  H  |->  ( f  e.  ( ( LTrn `  K ) `  w
)  |->  ( iota_ x  e.  B A. p  e.  A  ( -.  p  .<_  w  ->  x  =  ( ( p  .\/  ( f `  p
) )  ./\  w
) ) ) ) )
21 fvex 5743 . . . . 5  |-  ( (
LTrn `  K ) `  W )  e.  _V
2221mptex 5967 . . . 4  |-  ( f  e.  ( ( LTrn `  K ) `  W
)  |->  ( iota_ x  e.  B A. p  e.  A  ( -.  p  .<_  W  ->  x  =  ( ( p  .\/  ( f `  p
) )  ./\  W
) ) ) )  e.  _V
2319, 20, 22fvmpt 5807 . . 3  |-  ( W  e.  H  ->  (
( w  e.  H  |->  ( f  e.  ( ( LTrn `  K
) `  w )  |->  ( iota_ x  e.  B A. p  e.  A  ( -.  p  .<_  w  ->  x  =  ( ( p  .\/  (
f `  p )
)  ./\  w )
) ) ) ) `
 W )  =  ( f  e.  ( ( LTrn `  K
) `  W )  |->  ( iota_ x  e.  B A. p  e.  A  ( -.  p  .<_  W  ->  x  =  ( ( p  .\/  (
f `  p )
)  ./\  W )
) ) ) )
24 trlset.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
25 eqid 2437 . . . 4  |-  ( 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 )
) )
2624, 25mpteq12i 4294 . . 3  |-  ( f  e.  T  |->  ( iota_ x  e.  B A. p  e.  A  ( -.  p  .<_  W  ->  x  =  ( ( p 
.\/  ( f `  p ) )  ./\  W ) ) ) )  =  ( f  e.  ( ( LTrn `  K
) `  W )  |->  ( iota_ x  e.  B A. p  e.  A  ( -.  p  .<_  W  ->  x  =  ( ( p  .\/  (
f `  p )
)  ./\  W )
) ) )
2723, 26syl6eqr 2487 . 2  |-  ( W  e.  H  ->  (
( w  e.  H  |->  ( f  e.  ( ( LTrn `  K
) `  w )  |->  ( iota_ x  e.  B A. p  e.  A  ( -.  p  .<_  w  ->  x  =  ( ( p  .\/  (
f `  p )
)  ./\  w )
) ) ) ) `
 W )  =  ( f  e.  T  |->  ( iota_ x  e.  B A. p  e.  A  ( -.  p  .<_  W  ->  x  =  ( ( p  .\/  (
f `  p )
)  ./\  W )
) ) ) )
2810, 27sylan9eq 2489 1  |-  ( ( K  e.  C  /\  W  e.  H )  ->  R  =  ( f  e.  T  |->  ( 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 360    = wceq 1653    e. wcel 1726   A.wral 2706   class class class wbr 4213    e. cmpt 4267   ` cfv 5455  (class class class)co 6082   iota_crio 6543   Basecbs 13470   lecple 13537   joincjn 14402   meetcmee 14403   Atomscatm 30062   LHypclh 30782   LTrncltrn 30899   trLctrl 30956
This theorem is referenced by:  trlval  30960
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-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-pr 4404
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-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-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-riota 6550  df-trl 30957
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