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Theorem trlfset 33801
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 2979 . 2  |-  ( K  e.  C  ->  K  e.  _V )
2 fveq2 5689 . . . . 5  |-  ( k  =  K  ->  ( LHyp `  k )  =  ( LHyp `  K
) )
3 trlset.h . . . . 5  |-  H  =  ( LHyp `  K
)
42, 3syl6eqr 2491 . . . 4  |-  ( k  =  K  ->  ( LHyp `  k )  =  H )
5 fveq2 5689 . . . . . 6  |-  ( k  =  K  ->  ( LTrn `  k )  =  ( LTrn `  K
) )
65fveq1d 5691 . . . . 5  |-  ( k  =  K  ->  (
( LTrn `  k ) `  w )  =  ( ( LTrn `  K
) `  w )
)
7 fveq2 5689 . . . . . . 7  |-  ( k  =  K  ->  ( Base `  k )  =  ( Base `  K
) )
8 trlset.b . . . . . . 7  |-  B  =  ( Base `  K
)
97, 8syl6eqr 2491 . . . . . 6  |-  ( k  =  K  ->  ( Base `  k )  =  B )
10 fveq2 5689 . . . . . . . 8  |-  ( k  =  K  ->  ( Atoms `  k )  =  ( Atoms `  K )
)
11 trlset.a . . . . . . . 8  |-  A  =  ( Atoms `  K )
1210, 11syl6eqr 2491 . . . . . . 7  |-  ( k  =  K  ->  ( Atoms `  k )  =  A )
13 fveq2 5689 . . . . . . . . . . 11  |-  ( k  =  K  ->  ( le `  k )  =  ( le `  K
) )
14 trlset.l . . . . . . . . . . 11  |-  .<_  =  ( le `  K )
1513, 14syl6eqr 2491 . . . . . . . . . 10  |-  ( k  =  K  ->  ( le `  k )  = 
.<_  )
1615breqd 4301 . . . . . . . . 9  |-  ( k  =  K  ->  (
p ( le `  k ) w  <->  p  .<_  w ) )
1716notbid 294 . . . . . . . 8  |-  ( k  =  K  ->  ( -.  p ( le `  k ) w  <->  -.  p  .<_  w ) )
18 fveq2 5689 . . . . . . . . . . 11  |-  ( k  =  K  ->  ( meet `  k )  =  ( meet `  K
) )
19 trlset.m . . . . . . . . . . 11  |-  ./\  =  ( meet `  K )
2018, 19syl6eqr 2491 . . . . . . . . . 10  |-  ( k  =  K  ->  ( meet `  k )  = 
./\  )
21 fveq2 5689 . . . . . . . . . . . 12  |-  ( k  =  K  ->  ( join `  k )  =  ( join `  K
) )
22 trlset.j . . . . . . . . . . . 12  |-  .\/  =  ( join `  K )
2321, 22syl6eqr 2491 . . . . . . . . . . 11  |-  ( k  =  K  ->  ( join `  k )  = 
.\/  )
2423oveqd 6106 . . . . . . . . . 10  |-  ( k  =  K  ->  (
p ( join `  k
) ( f `  p ) )  =  ( p  .\/  (
f `  p )
) )
25 eqidd 2442 . . . . . . . . . 10  |-  ( k  =  K  ->  w  =  w )
2620, 24, 25oveq123d 6110 . . . . . . . . 9  |-  ( k  =  K  ->  (
( p ( join `  k ) ( f `
 p ) ) ( meet `  k
) w )  =  ( ( p  .\/  ( f `  p
) )  ./\  w
) )
2726eqeq2d 2452 . . . . . . . 8  |-  ( k  =  K  ->  (
x  =  ( ( p ( join `  k
) ( f `  p ) ) (
meet `  k )
w )  <->  x  =  ( ( p  .\/  ( f `  p
) )  ./\  w
) ) )
2817, 27imbi12d 320 . . . . . . 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 2929 . . . . . 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 6053 . . . . 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 4368 . . . 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 4368 . . 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 33800 . . 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 5699 . . . . 5  |-  ( LHyp `  K )  e.  _V
353, 34eqeltri 2511 . . . 4  |-  H  e. 
_V
3635mptex 5946 . . 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 5772 . 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1369    e. wcel 1756   A.wral 2713   _Vcvv 2970   class class class wbr 4290    e. cmpt 4348   ` cfv 5416   iota_crio 6049  (class class class)co 6089   Basecbs 14172   lecple 14243   joincjn 15112   meetcmee 15113   Atomscatm 32905   LHypclh 33625   LTrncltrn 33742   trLctrl 33799
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pr 4529
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-riota 6050  df-ov 6092  df-trl 33800
This theorem is referenced by:  trlset  33802
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