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Theorem trlset 35987
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 35986 . . . 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 5874 . . 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 2510 . 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 5872 . . . . 5  |-  ( w  =  W  ->  (
( LTrn `  K ) `  w )  =  ( ( LTrn `  K
) `  W )
)
12 breq2 4460 . . . . . . . . 9  |-  ( w  =  W  ->  (
p  .<_  w  <->  p  .<_  W ) )
1312notbid 294 . . . . . . . 8  |-  ( w  =  W  ->  ( -.  p  .<_  w  <->  -.  p  .<_  W ) )
14 oveq2 6304 . . . . . . . . 9  |-  ( w  =  W  ->  (
( p  .\/  (
f `  p )
)  ./\  w )  =  ( ( p 
.\/  ( f `  p ) )  ./\  W ) )
1514eqeq2d 2471 . . . . . . . 8  |-  ( w  =  W  ->  (
x  =  ( ( p  .\/  ( f `
 p ) ) 
./\  w )  <->  x  =  ( ( p  .\/  ( f `  p
) )  ./\  W
) ) )
1613, 15imbi12d 320 . . . . . . 7  |-  ( w  =  W  ->  (
( -.  p  .<_  w  ->  x  =  ( ( p  .\/  (
f `  p )
)  ./\  w )
)  <->  ( -.  p  .<_  W  ->  x  =  ( ( p  .\/  ( f `  p
) )  ./\  W
) ) ) )
1716ralbidv 2896 . . . . . 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 6260 . . . . 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 4535 . . . 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 2457 . . . 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 5882 . . . . 5  |-  ( (
LTrn `  K ) `  W )  e.  _V
2221mptex 6144 . . . 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 5956 . . 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 2457 . . . 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 4541 . . 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 2516 . 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 2518 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819   A.wral 2807   class class class wbr 4456    |-> cmpt 4515   ` cfv 5594   iota_crio 6257  (class class class)co 6296   Basecbs 14643   lecple 14718   joincjn 15699   meetcmee 15700   Atomscatm 35089   LHypclh 35809   LTrncltrn 35926   trLctrl 35984
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-trl 35985
This theorem is referenced by:  trlval  35988
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