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Theorem isltrn 33357
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 33356 . . 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 2500 . 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 5678 . . . . . . . 8  |-  ( f  =  F  ->  (
f `  p )  =  ( F `  p ) )
1110oveq2d 6096 . . . . . . 7  |-  ( f  =  F  ->  (
p  .\/  ( f `  p ) )  =  ( p  .\/  ( F `  p )
) )
1211oveq1d 6095 . . . . . 6  |-  ( f  =  F  ->  (
( p  .\/  (
f `  p )
)  ./\  W )  =  ( ( p 
.\/  ( F `  p ) )  ./\  W ) )
13 fveq1 5678 . . . . . . . 8  |-  ( f  =  F  ->  (
f `  q )  =  ( F `  q ) )
1413oveq2d 6096 . . . . . . 7  |-  ( f  =  F  ->  (
q  .\/  ( f `  q ) )  =  ( q  .\/  ( F `  q )
) )
1514oveq1d 6095 . . . . . 6  |-  ( f  =  F  ->  (
( q  .\/  (
f `  q )
)  ./\  W )  =  ( ( q 
.\/  ( F `  q ) )  ./\  W ) )
1612, 15eqeq12d 2447 . . . . 5  |-  ( f  =  F  ->  (
( ( p  .\/  ( f `  p
) )  ./\  W
)  =  ( ( q  .\/  ( f `
 q ) ) 
./\  W )  <->  ( (
p  .\/  ( F `  p ) )  ./\  W )  =  ( ( q  .\/  ( F `
 q ) ) 
./\  W ) ) )
1716imbi2d 316 . . . 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 2747 . . 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 3106 . 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 369    = wceq 1362    e. wcel 1755   A.wral 2705   {crab 2709   class class class wbr 4280   ` cfv 5406  (class class class)co 6080   lecple 14228   joincjn 15097   meetcmee 15098   Atomscatm 32502   LHypclh 33222   LDilcldil 33338   LTrncltrn 33339
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pr 4519
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-sn 3866  df-pr 3868  df-op 3872  df-uni 4080  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-id 4623  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-ov 6083  df-ltrn 33343
This theorem is referenced by:  isltrn2N  33358  ltrnu  33359  ltrnldil  33360  ltrncnv  33384  idltrn  33388  cdleme50ltrn  33795  ltrnco  33957
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