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Theorem ltrnfset 33766
Description: The set of all lattice translations for a lattice  K. (Contributed by NM, 11-May-2012.)
Hypotheses
Ref Expression
ltrnset.l  |-  .<_  =  ( le `  K )
ltrnset.j  |-  .\/  =  ( join `  K )
ltrnset.m  |-  ./\  =  ( meet `  K )
ltrnset.a  |-  A  =  ( Atoms `  K )
ltrnset.h  |-  H  =  ( LHyp `  K
)
Assertion
Ref Expression
ltrnfset  |-  ( K  e.  C  ->  ( LTrn `  K )  =  ( w  e.  H  |->  { f  e.  ( ( LDil `  K
) `  w )  |  A. p  e.  A  A. q  e.  A  ( ( -.  p  .<_  w  /\  -.  q  .<_  w )  ->  (
( p  .\/  (
f `  p )
)  ./\  w )  =  ( ( q 
.\/  ( f `  q ) )  ./\  w ) ) } ) )
Distinct variable groups:    q, p, A    w, H    f, p, q, w, K
Allowed substitution hints:    A( w, f)    C( w, f, q, p)    H( f, q, p)    .\/ ( w, f, q, p)    .<_ ( w, f, q, p)    ./\ ( w, f, q, p)

Proof of Theorem ltrnfset
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 elex 2986 . 2  |-  ( K  e.  C  ->  K  e.  _V )
2 fveq2 5696 . . . . 5  |-  ( k  =  K  ->  ( LHyp `  k )  =  ( LHyp `  K
) )
3 ltrnset.h . . . . 5  |-  H  =  ( LHyp `  K
)
42, 3syl6eqr 2493 . . . 4  |-  ( k  =  K  ->  ( LHyp `  k )  =  H )
5 fveq2 5696 . . . . . 6  |-  ( k  =  K  ->  ( LDil `  k )  =  ( LDil `  K
) )
65fveq1d 5698 . . . . 5  |-  ( k  =  K  ->  (
( LDil `  k ) `  w )  =  ( ( LDil `  K
) `  w )
)
7 fveq2 5696 . . . . . . 7  |-  ( k  =  K  ->  ( Atoms `  k )  =  ( Atoms `  K )
)
8 ltrnset.a . . . . . . 7  |-  A  =  ( Atoms `  K )
97, 8syl6eqr 2493 . . . . . 6  |-  ( k  =  K  ->  ( Atoms `  k )  =  A )
10 fveq2 5696 . . . . . . . . . . . 12  |-  ( k  =  K  ->  ( le `  k )  =  ( le `  K
) )
11 ltrnset.l . . . . . . . . . . . 12  |-  .<_  =  ( le `  K )
1210, 11syl6eqr 2493 . . . . . . . . . . 11  |-  ( k  =  K  ->  ( le `  k )  = 
.<_  )
1312breqd 4308 . . . . . . . . . 10  |-  ( k  =  K  ->  (
p ( le `  k ) w  <->  p  .<_  w ) )
1413notbid 294 . . . . . . . . 9  |-  ( k  =  K  ->  ( -.  p ( le `  k ) w  <->  -.  p  .<_  w ) )
1512breqd 4308 . . . . . . . . . 10  |-  ( k  =  K  ->  (
q ( le `  k ) w  <->  q  .<_  w ) )
1615notbid 294 . . . . . . . . 9  |-  ( k  =  K  ->  ( -.  q ( le `  k ) w  <->  -.  q  .<_  w ) )
1714, 16anbi12d 710 . . . . . . . 8  |-  ( k  =  K  ->  (
( -.  p ( le `  k ) w  /\  -.  q
( le `  k
) w )  <->  ( -.  p  .<_  w  /\  -.  q  .<_  w ) ) )
18 fveq2 5696 . . . . . . . . . . 11  |-  ( k  =  K  ->  ( meet `  k )  =  ( meet `  K
) )
19 ltrnset.m . . . . . . . . . . 11  |-  ./\  =  ( meet `  K )
2018, 19syl6eqr 2493 . . . . . . . . . 10  |-  ( k  =  K  ->  ( meet `  k )  = 
./\  )
21 fveq2 5696 . . . . . . . . . . . 12  |-  ( k  =  K  ->  ( join `  k )  =  ( join `  K
) )
22 ltrnset.j . . . . . . . . . . . 12  |-  .\/  =  ( join `  K )
2321, 22syl6eqr 2493 . . . . . . . . . . 11  |-  ( k  =  K  ->  ( join `  k )  = 
.\/  )
2423oveqd 6113 . . . . . . . . . 10  |-  ( k  =  K  ->  (
p ( join `  k
) ( f `  p ) )  =  ( p  .\/  (
f `  p )
) )
25 eqidd 2444 . . . . . . . . . 10  |-  ( k  =  K  ->  w  =  w )
2620, 24, 25oveq123d 6117 . . . . . . . . 9  |-  ( k  =  K  ->  (
( p ( join `  k ) ( f `
 p ) ) ( meet `  k
) w )  =  ( ( p  .\/  ( f `  p
) )  ./\  w
) )
2723oveqd 6113 . . . . . . . . . 10  |-  ( k  =  K  ->  (
q ( join `  k
) ( f `  q ) )  =  ( q  .\/  (
f `  q )
) )
2820, 27, 25oveq123d 6117 . . . . . . . . 9  |-  ( k  =  K  ->  (
( q ( join `  k ) ( f `
 q ) ) ( meet `  k
) w )  =  ( ( q  .\/  ( f `  q
) )  ./\  w
) )
2926, 28eqeq12d 2457 . . . . . . . 8  |-  ( k  =  K  ->  (
( ( p (
join `  k )
( f `  p
) ) ( meet `  k ) w )  =  ( ( q ( join `  k
) ( f `  q ) ) (
meet `  k )
w )  <->  ( (
p  .\/  ( f `  p ) )  ./\  w )  =  ( ( q  .\/  (
f `  q )
)  ./\  w )
) )
3017, 29imbi12d 320 . . . . . . 7  |-  ( k  =  K  ->  (
( ( -.  p
( le `  k
) w  /\  -.  q ( le `  k ) w )  ->  ( ( p ( join `  k
) ( f `  p ) ) (
meet `  k )
w )  =  ( ( q ( join `  k ) ( f `
 q ) ) ( meet `  k
) w ) )  <-> 
( ( -.  p  .<_  w  /\  -.  q  .<_  w )  ->  (
( p  .\/  (
f `  p )
)  ./\  w )  =  ( ( q 
.\/  ( f `  q ) )  ./\  w ) ) ) )
319, 30raleqbidv 2936 . . . . . 6  |-  ( k  =  K  ->  ( A. q  e.  ( Atoms `  k ) ( ( -.  p ( le `  k ) w  /\  -.  q
( le `  k
) w )  -> 
( ( p (
join `  k )
( f `  p
) ) ( meet `  k ) w )  =  ( ( q ( join `  k
) ( f `  q ) ) (
meet `  k )
w ) )  <->  A. q  e.  A  ( ( -.  p  .<_  w  /\  -.  q  .<_  w )  ->  ( ( p 
.\/  ( f `  p ) )  ./\  w )  =  ( ( q  .\/  (
f `  q )
)  ./\  w )
) ) )
329, 31raleqbidv 2936 . . . . 5  |-  ( k  =  K  ->  ( A. p  e.  ( Atoms `  k ) A. q  e.  ( Atoms `  k ) ( ( -.  p ( le
`  k ) w  /\  -.  q ( le `  k ) w )  ->  (
( p ( join `  k ) ( f `
 p ) ) ( meet `  k
) w )  =  ( ( q (
join `  k )
( f `  q
) ) ( meet `  k ) w ) )  <->  A. p  e.  A  A. q  e.  A  ( ( -.  p  .<_  w  /\  -.  q  .<_  w )  ->  (
( p  .\/  (
f `  p )
)  ./\  w )  =  ( ( q 
.\/  ( f `  q ) )  ./\  w ) ) ) )
336, 32rabeqbidv 2972 . . . 4  |-  ( k  =  K  ->  { f  e.  ( ( LDil `  k ) `  w
)  |  A. p  e.  ( Atoms `  k ) A. q  e.  ( Atoms `  k ) ( ( -.  p ( le `  k ) w  /\  -.  q
( le `  k
) w )  -> 
( ( p (
join `  k )
( f `  p
) ) ( meet `  k ) w )  =  ( ( q ( join `  k
) ( f `  q ) ) (
meet `  k )
w ) ) }  =  { f  e.  ( ( LDil `  K
) `  w )  |  A. p  e.  A  A. q  e.  A  ( ( -.  p  .<_  w  /\  -.  q  .<_  w )  ->  (
( p  .\/  (
f `  p )
)  ./\  w )  =  ( ( q 
.\/  ( f `  q ) )  ./\  w ) ) } )
344, 33mpteq12dv 4375 . . 3  |-  ( k  =  K  ->  (
w  e.  ( LHyp `  k )  |->  { f  e.  ( ( LDil `  k ) `  w
)  |  A. p  e.  ( Atoms `  k ) A. q  e.  ( Atoms `  k ) ( ( -.  p ( le `  k ) w  /\  -.  q
( le `  k
) w )  -> 
( ( p (
join `  k )
( f `  p
) ) ( meet `  k ) w )  =  ( ( q ( join `  k
) ( f `  q ) ) (
meet `  k )
w ) ) } )  =  ( w  e.  H  |->  { f  e.  ( ( LDil `  K ) `  w
)  |  A. p  e.  A  A. q  e.  A  ( ( -.  p  .<_  w  /\  -.  q  .<_  w )  ->  ( ( p 
.\/  ( f `  p ) )  ./\  w )  =  ( ( q  .\/  (
f `  q )
)  ./\  w )
) } ) )
35 df-ltrn 33754 . . 3  |-  LTrn  =  ( k  e.  _V  |->  ( w  e.  ( LHyp `  k )  |->  { f  e.  ( (
LDil `  k ) `  w )  |  A. p  e.  ( Atoms `  k ) A. q  e.  ( Atoms `  k )
( ( -.  p
( le `  k
) w  /\  -.  q ( le `  k ) w )  ->  ( ( p ( join `  k
) ( f `  p ) ) (
meet `  k )
w )  =  ( ( q ( join `  k ) ( f `
 q ) ) ( meet `  k
) w ) ) } ) )
36 fvex 5706 . . . . 5  |-  ( LHyp `  K )  e.  _V
373, 36eqeltri 2513 . . . 4  |-  H  e. 
_V
3837mptex 5953 . . 3  |-  ( w  e.  H  |->  { f  e.  ( ( LDil `  K ) `  w
)  |  A. p  e.  A  A. q  e.  A  ( ( -.  p  .<_  w  /\  -.  q  .<_  w )  ->  ( ( p 
.\/  ( f `  p ) )  ./\  w )  =  ( ( q  .\/  (
f `  q )
)  ./\  w )
) } )  e. 
_V
3934, 35, 38fvmpt 5779 . 2  |-  ( K  e.  _V  ->  ( LTrn `  K )  =  ( w  e.  H  |->  { f  e.  ( ( LDil `  K
) `  w )  |  A. p  e.  A  A. q  e.  A  ( ( -.  p  .<_  w  /\  -.  q  .<_  w )  ->  (
( p  .\/  (
f `  p )
)  ./\  w )  =  ( ( q 
.\/  ( f `  q ) )  ./\  w ) ) } ) )
401, 39syl 16 1  |-  ( K  e.  C  ->  ( LTrn `  K )  =  ( w  e.  H  |->  { f  e.  ( ( LDil `  K
) `  w )  |  A. p  e.  A  A. q  e.  A  ( ( -.  p  .<_  w  /\  -.  q  .<_  w )  ->  (
( p  .\/  (
f `  p )
)  ./\  w )  =  ( ( q 
.\/  ( f `  q ) )  ./\  w ) ) } ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2720   {crab 2724   _Vcvv 2977   class class class wbr 4297    e. cmpt 4355   ` cfv 5423  (class class class)co 6096   lecple 14250   joincjn 15119   meetcmee 15120   Atomscatm 32913   LHypclh 33633   LDilcldil 33749   LTrncltrn 33750
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 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pr 4536
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 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-ov 6099  df-ltrn 33754
This theorem is referenced by:  ltrnset  33767
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