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Theorem isltrn 36240
Description: The predicate "is a lattice translation". Similar to definition of translation in [Crawley] p. 111. (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
isltrn  |-  ( ( K  e.  B  /\  W  e.  H )  ->  ( F  e.  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    K, p, q    W, p, q    F, p, q
Allowed substitution hints:    B( q, p)    D( q, p)    T( q, p)    H( q, p)    .\/ ( q, p)   
.<_ ( q, p)    ./\ ( q, p)

Proof of Theorem isltrn
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 ltrnset.l . . . 4  |-  .<_  =  ( le `  K )
2 ltrnset.j . . . 4  |-  .\/  =  ( join `  K )
3 ltrnset.m . . . 4  |-  ./\  =  ( meet `  K )
4 ltrnset.a . . . 4  |-  A  =  ( Atoms `  K )
5 ltrnset.h . . . 4  |-  H  =  ( LHyp `  K
)
6 ltrnset.d . . . 4  |-  D  =  ( ( LDil `  K
) `  W )
7 ltrnset.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
81, 2, 3, 4, 5, 6, 7ltrnset 36239 . . 3  |-  ( ( 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 ) ) } )
98eleq2d 2524 . 2  |-  ( ( K  e.  B  /\  W  e.  H )  ->  ( F  e.  T  <->  F  e.  { f  e.  D  |  A. p  e.  A  A. q  e.  A  ( ( -.  p  .<_  W  /\  -.  q  .<_  W )  ->  ( ( p 
.\/  ( f `  p ) )  ./\  W )  =  ( ( q  .\/  ( f `
 q ) ) 
./\  W ) ) } ) )
10 fveq1 5847 . . . . . . . 8  |-  ( f  =  F  ->  (
f `  p )  =  ( F `  p ) )
1110oveq2d 6286 . . . . . . 7  |-  ( f  =  F  ->  (
p  .\/  ( f `  p ) )  =  ( p  .\/  ( F `  p )
) )
1211oveq1d 6285 . . . . . 6  |-  ( f  =  F  ->  (
( p  .\/  (
f `  p )
)  ./\  W )  =  ( ( p 
.\/  ( F `  p ) )  ./\  W ) )
13 fveq1 5847 . . . . . . . 8  |-  ( f  =  F  ->  (
f `  q )  =  ( F `  q ) )
1413oveq2d 6286 . . . . . . 7  |-  ( f  =  F  ->  (
q  .\/  ( f `  q ) )  =  ( q  .\/  ( F `  q )
) )
1514oveq1d 6285 . . . . . 6  |-  ( f  =  F  ->  (
( q  .\/  (
f `  q )
)  ./\  W )  =  ( ( q 
.\/  ( F `  q ) )  ./\  W ) )
1612, 15eqeq12d 2476 . . . . 5  |-  ( f  =  F  ->  (
( ( p  .\/  ( f `  p
) )  ./\  W
)  =  ( ( q  .\/  ( f `
 q ) ) 
./\  W )  <->  ( (
p  .\/  ( F `  p ) )  ./\  W )  =  ( ( q  .\/  ( F `
 q ) ) 
./\  W ) ) )
1716imbi2d 314 . . . 4  |-  ( f  =  F  ->  (
( ( -.  p  .<_  W  /\  -.  q  .<_  W )  ->  (
( p  .\/  (
f `  p )
)  ./\  W )  =  ( ( q 
.\/  ( f `  q ) )  ./\  W ) )  <->  ( ( -.  p  .<_  W  /\  -.  q  .<_  W )  ->  ( ( p 
.\/  ( F `  p ) )  ./\  W )  =  ( ( q  .\/  ( F `
 q ) ) 
./\  W ) ) ) )
18172ralbidv 2898 . . 3  |-  ( f  =  F  ->  ( 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 ) ) ) )
1918elrab 3254 . 2  |-  ( F  e.  { f  e.  D  |  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 ) ) ) )
209, 19syl6bb 261 1  |-  ( ( K  e.  B  /\  W  e.  H )  ->  ( F  e.  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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823   A.wral 2804   {crab 2808   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   lecple 14791   joincjn 15772   meetcmee 15773   Atomscatm 35385   LHypclh 36105   LDilcldil 36221   LTrncltrn 36222
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-ltrn 36226
This theorem is referenced by:  isltrn2N  36241  ltrnu  36242  ltrnldil  36243  ltrncnv  36267  idltrn  36271  cdleme50ltrn  36680  ltrnco  36842
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