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Theorem trlfset 30957
Description: The set of all traces of lattice translations for a lattice 
K. (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
)
Assertion
Ref Expression
trlfset  |-  ( 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 )
) ) ) ) )
Distinct variable groups:    A, p    x, B    w, H    f, p, w, x, K
Allowed substitution hints:    A( x, w, f)    B( w, f, p)    C( x, w, f, p)    H( x, f, p)    .\/ ( x, w, f, p)    .<_ ( x, w, f, p)    ./\ ( x, w, f, p)

Proof of Theorem trlfset
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 elex 2964 . 2  |-  ( K  e.  C  ->  K  e.  _V )
2 fveq2 5728 . . . . 5  |-  ( k  =  K  ->  ( LHyp `  k )  =  ( LHyp `  K
) )
3 trlset.h . . . . 5  |-  H  =  ( LHyp `  K
)
42, 3syl6eqr 2486 . . . 4  |-  ( k  =  K  ->  ( LHyp `  k )  =  H )
5 fveq2 5728 . . . . . 6  |-  ( k  =  K  ->  ( LTrn `  k )  =  ( LTrn `  K
) )
65fveq1d 5730 . . . . 5  |-  ( k  =  K  ->  (
( LTrn `  k ) `  w )  =  ( ( LTrn `  K
) `  w )
)
7 fveq2 5728 . . . . . . 7  |-  ( k  =  K  ->  ( Base `  k )  =  ( Base `  K
) )
8 trlset.b . . . . . . 7  |-  B  =  ( Base `  K
)
97, 8syl6eqr 2486 . . . . . 6  |-  ( k  =  K  ->  ( Base `  k )  =  B )
10 fveq2 5728 . . . . . . . 8  |-  ( k  =  K  ->  ( Atoms `  k )  =  ( Atoms `  K )
)
11 trlset.a . . . . . . . 8  |-  A  =  ( Atoms `  K )
1210, 11syl6eqr 2486 . . . . . . 7  |-  ( k  =  K  ->  ( Atoms `  k )  =  A )
13 fveq2 5728 . . . . . . . . . . 11  |-  ( k  =  K  ->  ( le `  k )  =  ( le `  K
) )
14 trlset.l . . . . . . . . . . 11  |-  .<_  =  ( le `  K )
1513, 14syl6eqr 2486 . . . . . . . . . 10  |-  ( k  =  K  ->  ( le `  k )  = 
.<_  )
1615breqd 4223 . . . . . . . . 9  |-  ( k  =  K  ->  (
p ( le `  k ) w  <->  p  .<_  w ) )
1716notbid 286 . . . . . . . 8  |-  ( k  =  K  ->  ( -.  p ( le `  k ) w  <->  -.  p  .<_  w ) )
18 fveq2 5728 . . . . . . . . . . 11  |-  ( k  =  K  ->  ( meet `  k )  =  ( meet `  K
) )
19 trlset.m . . . . . . . . . . 11  |-  ./\  =  ( meet `  K )
2018, 19syl6eqr 2486 . . . . . . . . . 10  |-  ( k  =  K  ->  ( meet `  k )  = 
./\  )
21 fveq2 5728 . . . . . . . . . . . 12  |-  ( k  =  K  ->  ( join `  k )  =  ( join `  K
) )
22 trlset.j . . . . . . . . . . . 12  |-  .\/  =  ( join `  K )
2321, 22syl6eqr 2486 . . . . . . . . . . 11  |-  ( k  =  K  ->  ( join `  k )  = 
.\/  )
2423oveqd 6098 . . . . . . . . . 10  |-  ( k  =  K  ->  (
p ( join `  k
) ( f `  p ) )  =  ( p  .\/  (
f `  p )
) )
25 eqidd 2437 . . . . . . . . . 10  |-  ( k  =  K  ->  w  =  w )
2620, 24, 25oveq123d 6102 . . . . . . . . 9  |-  ( k  =  K  ->  (
( p ( join `  k ) ( f `
 p ) ) ( meet `  k
) w )  =  ( ( p  .\/  ( f `  p
) )  ./\  w
) )
2726eqeq2d 2447 . . . . . . . 8  |-  ( k  =  K  ->  (
x  =  ( ( p ( join `  k
) ( f `  p ) ) (
meet `  k )
w )  <->  x  =  ( ( p  .\/  ( f `  p
) )  ./\  w
) ) )
2817, 27imbi12d 312 . . . . . . 7  |-  ( k  =  K  ->  (
( -.  p ( le `  k ) w  ->  x  =  ( ( p (
join `  k )
( f `  p
) ) ( meet `  k ) w ) )  <->  ( -.  p  .<_  w  ->  x  =  ( ( p  .\/  ( f `  p
) )  ./\  w
) ) ) )
2912, 28raleqbidv 2916 . . . . . 6  |-  ( k  =  K  ->  ( A. p  e.  ( Atoms `  k ) ( -.  p ( le
`  k ) w  ->  x  =  ( ( p ( join `  k ) ( f `
 p ) ) ( meet `  k
) w ) )  <->  A. p  e.  A  ( -.  p  .<_  w  ->  x  =  ( ( p  .\/  (
f `  p )
)  ./\  w )
) ) )
309, 29riotaeqbidv 6552 . . . . 5  |-  ( k  =  K  ->  ( iota_ x  e.  ( Base `  k ) A. p  e.  ( Atoms `  k )
( -.  p ( le `  k ) w  ->  x  =  ( ( p (
join `  k )
( f `  p
) ) ( meet `  k ) w ) ) )  =  (
iota_ x  e.  B A. p  e.  A  ( -.  p  .<_  w  ->  x  =  ( ( p  .\/  (
f `  p )
)  ./\  w )
) ) )
316, 30mpteq12dv 4287 . . . 4  |-  ( k  =  K  ->  (
f  e.  ( (
LTrn `  k ) `  w )  |->  ( iota_ x  e.  ( Base `  k
) A. p  e.  ( Atoms `  k )
( -.  p ( le `  k ) w  ->  x  =  ( ( p (
join `  k )
( f `  p
) ) ( meet `  k ) w ) ) ) )  =  ( f  e.  ( ( LTrn `  K
) `  w )  |->  ( iota_ x  e.  B A. p  e.  A  ( -.  p  .<_  w  ->  x  =  ( ( p  .\/  (
f `  p )
)  ./\  w )
) ) ) )
324, 31mpteq12dv 4287 . . 3  |-  ( k  =  K  ->  (
w  e.  ( LHyp `  k )  |->  ( f  e.  ( ( LTrn `  k ) `  w
)  |->  ( iota_ x  e.  ( Base `  k
) A. p  e.  ( Atoms `  k )
( -.  p ( le `  k ) w  ->  x  =  ( ( p (
join `  k )
( f `  p
) ) ( meet `  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 )
) ) ) ) )
33 df-trl 30956 . . 3  |-  trL  =  ( k  e.  _V  |->  ( w  e.  ( LHyp `  k )  |->  ( f  e.  ( (
LTrn `  k ) `  w )  |->  ( iota_ x  e.  ( Base `  k
) A. p  e.  ( Atoms `  k )
( -.  p ( le `  k ) w  ->  x  =  ( ( p (
join `  k )
( f `  p
) ) ( meet `  k ) w ) ) ) ) ) )
34 fvex 5742 . . . . 5  |-  ( LHyp `  K )  e.  _V
353, 34eqeltri 2506 . . . 4  |-  H  e. 
_V
3635mptex 5966 . . 3  |-  ( w  e.  H  |->  ( f  e.  ( ( LTrn `  K ) `  w
)  |->  ( iota_ x  e.  B A. p  e.  A  ( -.  p  .<_  w  ->  x  =  ( ( p  .\/  ( f `  p
) )  ./\  w
) ) ) ) )  e.  _V
3732, 33, 36fvmpt 5806 . 2  |-  ( K  e.  _V  ->  ( 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 )
) ) ) ) )
381, 37syl 16 1  |-  ( 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 )
) ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1652    e. wcel 1725   A.wral 2705   _Vcvv 2956   class class class wbr 4212    e. cmpt 4266   ` cfv 5454  (class class class)co 6081   iota_crio 6542   Basecbs 13469   lecple 13536   joincjn 14401   meetcmee 14402   Atomscatm 30061   LHypclh 30781   LTrncltrn 30898   trLctrl 30955
This theorem is referenced by:  trlset  30958
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-riota 6549  df-trl 30956
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