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Theorem isltrn 29438
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
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 29437 . . 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 2323 . 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 5422 . . . . . . . 8  |-  ( f  =  F  ->  (
f `  p )  =  ( F `  p ) )
1110oveq2d 5773 . . . . . . 7  |-  ( f  =  F  ->  (
p  .\/  ( f `  p ) )  =  ( p  .\/  ( F `  p )
) )
1211oveq1d 5772 . . . . . 6  |-  ( f  =  F  ->  (
( p  .\/  (
f `  p )
)  ./\  W )  =  ( ( p 
.\/  ( F `  p ) )  ./\  W ) )
13 fveq1 5422 . . . . . . . 8  |-  ( f  =  F  ->  (
f `  q )  =  ( F `  q ) )
1413oveq2d 5773 . . . . . . 7  |-  ( f  =  F  ->  (
q  .\/  ( f `  q ) )  =  ( q  .\/  ( F `  q )
) )
1514oveq1d 5772 . . . . . 6  |-  ( f  =  F  ->  (
( q  .\/  (
f `  q )
)  ./\  W )  =  ( ( q 
.\/  ( F `  q ) )  ./\  W ) )
1612, 15eqeq12d 2270 . . . . 5  |-  ( f  =  F  ->  (
( ( p  .\/  ( f `  p
) )  ./\  W
)  =  ( ( q  .\/  ( f `
 q ) ) 
./\  W )  <->  ( (
p  .\/  ( F `  p ) )  ./\  W )  =  ( ( q  .\/  ( F `
 q ) ) 
./\  W ) ) )
1716imbi2d 309 . . . 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 2556 . . 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 2874 . 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 254 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 set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621   A.wral 2516   {crab 2519   class class class wbr 3963   ` cfv 4638  (class class class)co 5757   lecple 13142   joincjn 14005   meetcmee 14006   Atomscatm 28583   LHypclh 29303   LDilcldil 29419   LTrncltrn 29420
This theorem is referenced by:  isltrn2N  29439  ltrnu  29440  ltrnldil  29441  ltrncnv  29465  idltrn  29469  cdleme50ltrn  29876  ltrnco  30038
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4071  ax-sep 4081  ax-nul 4089  ax-pr 4152  ax-un 4449
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3769  df-iun 3848  df-br 3964  df-opab 4018  df-mpt 4019  df-id 4246  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-ov 5760  df-ltrn 29424
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