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Theorem ltrnset 30307
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 30306 . . . 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 5527 . . 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 2327 . 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 5525 . . . . 5  |-  ( w  =  W  ->  (
( LDil `  K ) `  w )  =  ( ( LDil `  K
) `  W )
)
11 ltrnset.d . . . . 5  |-  D  =  ( ( LDil `  K
) `  W )
1210, 11syl6eqr 2333 . . . 4  |-  ( w  =  W  ->  (
( LDil `  K ) `  w )  =  D )
13 breq2 4027 . . . . . . . 8  |-  ( w  =  W  ->  (
p  .<_  w  <->  p  .<_  W ) )
1413notbid 285 . . . . . . 7  |-  ( w  =  W  ->  ( -.  p  .<_  w  <->  -.  p  .<_  W ) )
15 breq2 4027 . . . . . . . 8  |-  ( w  =  W  ->  (
q  .<_  w  <->  q  .<_  W ) )
1615notbid 285 . . . . . . 7  |-  ( w  =  W  ->  ( -.  q  .<_  w  <->  -.  q  .<_  W ) )
1714, 16anbi12d 691 . . . . . 6  |-  ( w  =  W  ->  (
( -.  p  .<_  w  /\  -.  q  .<_  w )  <->  ( -.  p  .<_  W  /\  -.  q  .<_  W ) ) )
18 oveq2 5866 . . . . . . 7  |-  ( w  =  W  ->  (
( p  .\/  (
f `  p )
)  ./\  w )  =  ( ( p 
.\/  ( f `  p ) )  ./\  W ) )
19 oveq2 5866 . . . . . . 7  |-  ( w  =  W  ->  (
( q  .\/  (
f `  q )
)  ./\  w )  =  ( ( q 
.\/  ( f `  q ) )  ./\  W ) )
2018, 19eqeq12d 2297 . . . . . 6  |-  ( w  =  W  ->  (
( ( p  .\/  ( f `  p
) )  ./\  w
)  =  ( ( q  .\/  ( f `
 q ) ) 
./\  w )  <->  ( (
p  .\/  ( f `  p ) )  ./\  W )  =  ( ( q  .\/  ( f `
 q ) ) 
./\  W ) ) )
2117, 20imbi12d 311 . . . . 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 2585 . . . 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 2783 . . 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 2283 . . 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 5539 . . . . 5  |-  ( (
LDil `  K ) `  W )  e.  _V
2611, 25eqeltri 2353 . . . 4  |-  D  e. 
_V
2726rabex 4165 . . 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 5602 . 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 2335 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 358    = wceq 1623    e. wcel 1684   A.wral 2543   {crab 2547   _Vcvv 2788   class class class wbr 4023    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   lecple 13215   joincjn 14078   meetcmee 14079   Atomscatm 29453   LHypclh 30173   LDilcldil 30289   LTrncltrn 30290
This theorem is referenced by:  isltrn  30308
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-ltrn 30294
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