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Theorem ltrnfset 30928
Description: The set of all lattice translations for a lattice  K. (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
)
Assertion
Ref Expression
ltrnfset  |-  ( K  e.  C  ->  ( 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 ) ) } ) )
Distinct variable groups:    q, p, A    w, H    f, p, q, w, K
Allowed substitution hints:    A( w, f)    C( w, f, q, p)    H( f, q, p)    .\/ ( w, f, q, p)    .<_ ( w, f, q, p)    ./\ ( w, f, q, p)

Proof of Theorem ltrnfset
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 elex 2809 . 2  |-  ( K  e.  C  ->  K  e.  _V )
2 fveq2 5541 . . . . 5  |-  ( k  =  K  ->  ( LHyp `  k )  =  ( LHyp `  K
) )
3 ltrnset.h . . . . 5  |-  H  =  ( LHyp `  K
)
42, 3syl6eqr 2346 . . . 4  |-  ( k  =  K  ->  ( LHyp `  k )  =  H )
5 fveq2 5541 . . . . . 6  |-  ( k  =  K  ->  ( LDil `  k )  =  ( LDil `  K
) )
65fveq1d 5543 . . . . 5  |-  ( k  =  K  ->  (
( LDil `  k ) `  w )  =  ( ( LDil `  K
) `  w )
)
7 fveq2 5541 . . . . . . 7  |-  ( k  =  K  ->  ( Atoms `  k )  =  ( Atoms `  K )
)
8 ltrnset.a . . . . . . 7  |-  A  =  ( Atoms `  K )
97, 8syl6eqr 2346 . . . . . 6  |-  ( k  =  K  ->  ( Atoms `  k )  =  A )
10 fveq2 5541 . . . . . . . . . . . 12  |-  ( k  =  K  ->  ( le `  k )  =  ( le `  K
) )
11 ltrnset.l . . . . . . . . . . . 12  |-  .<_  =  ( le `  K )
1210, 11syl6eqr 2346 . . . . . . . . . . 11  |-  ( k  =  K  ->  ( le `  k )  = 
.<_  )
1312breqd 4050 . . . . . . . . . 10  |-  ( k  =  K  ->  (
p ( le `  k ) w  <->  p  .<_  w ) )
1413notbid 285 . . . . . . . . 9  |-  ( k  =  K  ->  ( -.  p ( le `  k ) w  <->  -.  p  .<_  w ) )
1512breqd 4050 . . . . . . . . . 10  |-  ( k  =  K  ->  (
q ( le `  k ) w  <->  q  .<_  w ) )
1615notbid 285 . . . . . . . . 9  |-  ( k  =  K  ->  ( -.  q ( le `  k ) w  <->  -.  q  .<_  w ) )
1714, 16anbi12d 691 . . . . . . . 8  |-  ( k  =  K  ->  (
( -.  p ( le `  k ) w  /\  -.  q
( le `  k
) w )  <->  ( -.  p  .<_  w  /\  -.  q  .<_  w ) ) )
18 fveq2 5541 . . . . . . . . . . 11  |-  ( k  =  K  ->  ( meet `  k )  =  ( meet `  K
) )
19 ltrnset.m . . . . . . . . . . 11  |-  ./\  =  ( meet `  K )
2018, 19syl6eqr 2346 . . . . . . . . . 10  |-  ( k  =  K  ->  ( meet `  k )  = 
./\  )
21 fveq2 5541 . . . . . . . . . . . 12  |-  ( k  =  K  ->  ( join `  k )  =  ( join `  K
) )
22 ltrnset.j . . . . . . . . . . . 12  |-  .\/  =  ( join `  K )
2321, 22syl6eqr 2346 . . . . . . . . . . 11  |-  ( k  =  K  ->  ( join `  k )  = 
.\/  )
2423oveqd 5891 . . . . . . . . . 10  |-  ( k  =  K  ->  (
p ( join `  k
) ( f `  p ) )  =  ( p  .\/  (
f `  p )
) )
25 eqidd 2297 . . . . . . . . . 10  |-  ( k  =  K  ->  w  =  w )
2620, 24, 25oveq123d 5895 . . . . . . . . 9  |-  ( k  =  K  ->  (
( p ( join `  k ) ( f `
 p ) ) ( meet `  k
) w )  =  ( ( p  .\/  ( f `  p
) )  ./\  w
) )
2723oveqd 5891 . . . . . . . . . 10  |-  ( k  =  K  ->  (
q ( join `  k
) ( f `  q ) )  =  ( q  .\/  (
f `  q )
) )
2820, 27, 25oveq123d 5895 . . . . . . . . 9  |-  ( k  =  K  ->  (
( q ( join `  k ) ( f `
 q ) ) ( meet `  k
) w )  =  ( ( q  .\/  ( f `  q
) )  ./\  w
) )
2926, 28eqeq12d 2310 . . . . . . . 8  |-  ( k  =  K  ->  (
( ( p (
join `  k )
( f `  p
) ) ( meet `  k ) w )  =  ( ( q ( join `  k
) ( f `  q ) ) (
meet `  k )
w )  <->  ( (
p  .\/  ( f `  p ) )  ./\  w )  =  ( ( q  .\/  (
f `  q )
)  ./\  w )
) )
3017, 29imbi12d 311 . . . . . . 7  |-  ( k  =  K  ->  (
( ( -.  p
( le `  k
) w  /\  -.  q ( le `  k ) w )  ->  ( ( p ( join `  k
) ( f `  p ) ) (
meet `  k )
w )  =  ( ( q ( join `  k ) ( f `
 q ) ) ( meet `  k
) w ) )  <-> 
( ( -.  p  .<_  w  /\  -.  q  .<_  w )  ->  (
( p  .\/  (
f `  p )
)  ./\  w )  =  ( ( q 
.\/  ( f `  q ) )  ./\  w ) ) ) )
319, 30raleqbidv 2761 . . . . . 6  |-  ( k  =  K  ->  ( A. q  e.  ( Atoms `  k ) ( ( -.  p ( le `  k ) w  /\  -.  q
( le `  k
) w )  -> 
( ( p (
join `  k )
( f `  p
) ) ( meet `  k ) w )  =  ( ( q ( join `  k
) ( f `  q ) ) (
meet `  k )
w ) )  <->  A. q  e.  A  ( ( -.  p  .<_  w  /\  -.  q  .<_  w )  ->  ( ( p 
.\/  ( f `  p ) )  ./\  w )  =  ( ( q  .\/  (
f `  q )
)  ./\  w )
) ) )
329, 31raleqbidv 2761 . . . . 5  |-  ( k  =  K  ->  ( A. p  e.  ( Atoms `  k ) A. q  e.  ( Atoms `  k ) ( ( -.  p ( le
`  k ) w  /\  -.  q ( le `  k ) w )  ->  (
( p ( join `  k ) ( f `
 p ) ) ( meet `  k
) w )  =  ( ( q (
join `  k )
( f `  q
) ) ( meet `  k ) w ) )  <->  A. p  e.  A  A. q  e.  A  ( ( -.  p  .<_  w  /\  -.  q  .<_  w )  ->  (
( p  .\/  (
f `  p )
)  ./\  w )  =  ( ( q 
.\/  ( f `  q ) )  ./\  w ) ) ) )
336, 32rabeqbidv 2796 . . . 4  |-  ( k  =  K  ->  { f  e.  ( ( LDil `  k ) `  w
)  |  A. p  e.  ( Atoms `  k ) A. q  e.  ( Atoms `  k ) ( ( -.  p ( le `  k ) w  /\  -.  q
( le `  k
) w )  -> 
( ( p (
join `  k )
( f `  p
) ) ( meet `  k ) w )  =  ( ( q ( join `  k
) ( f `  q ) ) (
meet `  k )
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 ) ) } )
344, 33mpteq12dv 4114 . . 3  |-  ( k  =  K  ->  (
w  e.  ( LHyp `  k )  |->  { f  e.  ( ( LDil `  k ) `  w
)  |  A. p  e.  ( Atoms `  k ) A. q  e.  ( Atoms `  k ) ( ( -.  p ( le `  k ) w  /\  -.  q
( le `  k
) w )  -> 
( ( p (
join `  k )
( f `  p
) ) ( meet `  k ) w )  =  ( ( q ( join `  k
) ( f `  q ) ) (
meet `  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 )
) } ) )
35 df-ltrn 30916 . . 3  |-  LTrn  =  ( k  e.  _V  |->  ( w  e.  ( LHyp `  k )  |->  { f  e.  ( (
LDil `  k ) `  w )  |  A. p  e.  ( Atoms `  k ) A. q  e.  ( Atoms `  k )
( ( -.  p
( le `  k
) w  /\  -.  q ( le `  k ) w )  ->  ( ( p ( join `  k
) ( f `  p ) ) (
meet `  k )
w )  =  ( ( q ( join `  k ) ( f `
 q ) ) ( meet `  k
) w ) ) } ) )
36 fvex 5555 . . . . 5  |-  ( LHyp `  K )  e.  _V
373, 36eqeltri 2366 . . . 4  |-  H  e. 
_V
3837mptex 5762 . . 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 )
) } )  e. 
_V
3934, 35, 38fvmpt 5618 . 2  |-  ( K  e.  _V  ->  ( 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 ) ) } ) )
401, 39syl 15 1  |-  ( K  e.  C  ->  ( 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 ) ) } ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   {crab 2560   _Vcvv 2801   class class class wbr 4039    e. cmpt 4093   ` cfv 5271  (class class class)co 5874   lecple 13231   joincjn 14094   meetcmee 14095   Atomscatm 30075   LHypclh 30795   LDilcldil 30911   LTrncltrn 30912
This theorem is referenced by:  ltrnset  30929
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-ltrn 30916
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