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Theorem ltrnset 30233
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 30232 . . . 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 5671 . . 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 2432 . 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 5669 . . . . 5  |-  ( w  =  W  ->  (
( LDil `  K ) `  w )  =  ( ( LDil `  K
) `  W )
)
11 ltrnset.d . . . . 5  |-  D  =  ( ( LDil `  K
) `  W )
1210, 11syl6eqr 2438 . . . 4  |-  ( w  =  W  ->  (
( LDil `  K ) `  w )  =  D )
13 breq2 4158 . . . . . . . 8  |-  ( w  =  W  ->  (
p  .<_  w  <->  p  .<_  W ) )
1413notbid 286 . . . . . . 7  |-  ( w  =  W  ->  ( -.  p  .<_  w  <->  -.  p  .<_  W ) )
15 breq2 4158 . . . . . . . 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 6029 . . . . . . 7  |-  ( w  =  W  ->  (
( p  .\/  (
f `  p )
)  ./\  w )  =  ( ( p 
.\/  ( f `  p ) )  ./\  W ) )
19 oveq2 6029 . . . . . . 7  |-  ( w  =  W  ->  (
( q  .\/  (
f `  q )
)  ./\  w )  =  ( ( q 
.\/  ( f `  q ) )  ./\  W ) )
2018, 19eqeq12d 2402 . . . . . 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 2692 . . . 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 2895 . . 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 2388 . . 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 5683 . . . . 5  |-  ( (
LDil `  K ) `  W )  e.  _V
2611, 25eqeltri 2458 . . . 4  |-  D  e. 
_V
2726rabex 4296 . . 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 5746 . 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 2440 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 1717   A.wral 2650   {crab 2654   _Vcvv 2900   class class class wbr 4154    e. cmpt 4208   ` cfv 5395  (class class class)co 6021   lecple 13464   joincjn 14329   meetcmee 14330   Atomscatm 29379   LHypclh 30099   LDilcldil 30215   LTrncltrn 30216
This theorem is referenced by:  isltrn  30234
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 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pr 4345
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-ltrn 30220
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