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Theorem ltrnset 30600
Description: The set of lattice translations for a fiducial co-atom 
W. (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
)
ltrnset.d  |-  D  =  ( ( LDil `  K
) `  W )
ltrnset.t  |-  T  =  ( ( LTrn `  K
) `  W )
Assertion
Ref Expression
ltrnset  |-  ( ( K  e.  B  /\  W  e.  H )  ->  T  =  { f  e.  D  |  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    D, f    f, p, q, K    f, W, p, q
Allowed substitution hints:    A( f)    B( f, q, p)    D( q, p)    T( f, q, p)    H( f, q, p)    .\/ ( f,
q, p)    .<_ ( f, q, p)    ./\ ( f, q, p)

Proof of Theorem ltrnset
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 ltrnset.t . . 3  |-  T  =  ( ( LTrn `  K
) `  W )
2 ltrnset.l . . . . 5  |-  .<_  =  ( le `  K )
3 ltrnset.j . . . . 5  |-  .\/  =  ( join `  K )
4 ltrnset.m . . . . 5  |-  ./\  =  ( meet `  K )
5 ltrnset.a . . . . 5  |-  A  =  ( Atoms `  K )
6 ltrnset.h . . . . 5  |-  H  =  ( LHyp `  K
)
72, 3, 4, 5, 6ltrnfset 30599 . . . 4  |-  ( K  e.  B  ->  ( 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 ) ) } ) )
87fveq1d 5689 . . 3  |-  ( K  e.  B  ->  (
( LTrn `  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 ) ) } ) `  W ) )
91, 8syl5eq 2448 . 2  |-  ( K  e.  B  ->  T  =  ( ( 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 )
) } ) `  W ) )
10 fveq2 5687 . . . . 5  |-  ( w  =  W  ->  (
( LDil `  K ) `  w )  =  ( ( LDil `  K
) `  W )
)
11 ltrnset.d . . . . 5  |-  D  =  ( ( LDil `  K
) `  W )
1210, 11syl6eqr 2454 . . . 4  |-  ( w  =  W  ->  (
( LDil `  K ) `  w )  =  D )
13 breq2 4176 . . . . . . . 8  |-  ( w  =  W  ->  (
p  .<_  w  <->  p  .<_  W ) )
1413notbid 286 . . . . . . 7  |-  ( w  =  W  ->  ( -.  p  .<_  w  <->  -.  p  .<_  W ) )
15 breq2 4176 . . . . . . . 8  |-  ( w  =  W  ->  (
q  .<_  w  <->  q  .<_  W ) )
1615notbid 286 . . . . . . 7  |-  ( w  =  W  ->  ( -.  q  .<_  w  <->  -.  q  .<_  W ) )
1714, 16anbi12d 692 . . . . . 6  |-  ( w  =  W  ->  (
( -.  p  .<_  w  /\  -.  q  .<_  w )  <->  ( -.  p  .<_  W  /\  -.  q  .<_  W ) ) )
18 oveq2 6048 . . . . . . 7  |-  ( w  =  W  ->  (
( p  .\/  (
f `  p )
)  ./\  w )  =  ( ( p 
.\/  ( f `  p ) )  ./\  W ) )
19 oveq2 6048 . . . . . . 7  |-  ( w  =  W  ->  (
( q  .\/  (
f `  q )
)  ./\  w )  =  ( ( q 
.\/  ( f `  q ) )  ./\  W ) )
2018, 19eqeq12d 2418 . . . . . 6  |-  ( w  =  W  ->  (
( ( p  .\/  ( f `  p
) )  ./\  w
)  =  ( ( q  .\/  ( f `
 q ) ) 
./\  w )  <->  ( (
p  .\/  ( f `  p ) )  ./\  W )  =  ( ( q  .\/  ( f `
 q ) ) 
./\  W ) ) )
2117, 20imbi12d 312 . . . . 5  |-  ( w  =  W  ->  (
( ( -.  p  .<_  w  /\  -.  q  .<_  w )  ->  (
( p  .\/  (
f `  p )
)  ./\  w )  =  ( ( q 
.\/  ( f `  q ) )  ./\  w ) )  <->  ( ( -.  p  .<_  W  /\  -.  q  .<_  W )  ->  ( ( p 
.\/  ( f `  p ) )  ./\  W )  =  ( ( q  .\/  ( f `
 q ) ) 
./\  W ) ) ) )
22212ralbidv 2708 . . . 4  |-  ( w  =  W  ->  ( A. p  e.  A  A. q  e.  A  ( ( -.  p  .<_  w  /\  -.  q  .<_  w )  ->  (
( p  .\/  (
f `  p )
)  ./\  w )  =  ( ( q 
.\/  ( f `  q ) )  ./\  w ) )  <->  A. p  e.  A  A. q  e.  A  ( ( -.  p  .<_  W  /\  -.  q  .<_  W )  ->  ( ( p 
.\/  ( f `  p ) )  ./\  W )  =  ( ( q  .\/  ( f `
 q ) ) 
./\  W ) ) ) )
2312, 22rabeqbidv 2911 . . 3  |-  ( w  =  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 )
) }  =  {
f  e.  D  |  A. p  e.  A  A. q  e.  A  ( ( -.  p  .<_  W  /\  -.  q  .<_  W )  ->  (
( p  .\/  (
f `  p )
)  ./\  W )  =  ( ( q 
.\/  ( f `  q ) )  ./\  W ) ) } )
24 eqid 2404 . . 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 )
) } )  =  ( 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 ) ) } )
25 fvex 5701 . . . . 5  |-  ( (
LDil `  K ) `  W )  e.  _V
2611, 25eqeltri 2474 . . . 4  |-  D  e. 
_V
2726rabex 4314 . . 3  |-  { f  e.  D  |  A. p  e.  A  A. q  e.  A  (
( -.  p  .<_  W  /\  -.  q  .<_  W )  ->  (
( p  .\/  (
f `  p )
)  ./\  W )  =  ( ( q 
.\/  ( f `  q ) )  ./\  W ) ) }  e.  _V
2823, 24, 27fvmpt 5765 . 2  |-  ( W  e.  H  ->  (
( 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 ) ) } ) `  W )  =  { f  e.  D  |  A. p  e.  A  A. q  e.  A  ( ( -.  p  .<_  W  /\  -.  q  .<_  W )  ->  ( ( p 
.\/  ( f `  p ) )  ./\  W )  =  ( ( q  .\/  ( f `
 q ) ) 
./\  W ) ) } )
299, 28sylan9eq 2456 1  |-  ( ( K  e.  B  /\  W  e.  H )  ->  T  =  { f  e.  D  |  A. p  e.  A  A. q  e.  A  (
( -.  p  .<_  W  /\  -.  q  .<_  W )  ->  (
( p  .\/  (
f `  p )
)  ./\  W )  =  ( ( q 
.\/  ( f `  q ) )  ./\  W ) ) } )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2666   {crab 2670   _Vcvv 2916   class class class wbr 4172    e. cmpt 4226   ` cfv 5413  (class class class)co 6040   lecple 13491   joincjn 14356   meetcmee 14357   Atomscatm 29746   LHypclh 30466   LDilcldil 30582   LTrncltrn 30583
This theorem is referenced by:  isltrn  30601
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-ltrn 30587
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