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Theorem ltrnfset 30841
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 2956 . 2  |-  ( K  e.  C  ->  K  e.  _V )
2 fveq2 5720 . . . . 5  |-  ( k  =  K  ->  ( LHyp `  k )  =  ( LHyp `  K
) )
3 ltrnset.h . . . . 5  |-  H  =  ( LHyp `  K
)
42, 3syl6eqr 2485 . . . 4  |-  ( k  =  K  ->  ( LHyp `  k )  =  H )
5 fveq2 5720 . . . . . 6  |-  ( k  =  K  ->  ( LDil `  k )  =  ( LDil `  K
) )
65fveq1d 5722 . . . . 5  |-  ( k  =  K  ->  (
( LDil `  k ) `  w )  =  ( ( LDil `  K
) `  w )
)
7 fveq2 5720 . . . . . . 7  |-  ( k  =  K  ->  ( Atoms `  k )  =  ( Atoms `  K )
)
8 ltrnset.a . . . . . . 7  |-  A  =  ( Atoms `  K )
97, 8syl6eqr 2485 . . . . . 6  |-  ( k  =  K  ->  ( Atoms `  k )  =  A )
10 fveq2 5720 . . . . . . . . . . . 12  |-  ( k  =  K  ->  ( le `  k )  =  ( le `  K
) )
11 ltrnset.l . . . . . . . . . . . 12  |-  .<_  =  ( le `  K )
1210, 11syl6eqr 2485 . . . . . . . . . . 11  |-  ( k  =  K  ->  ( le `  k )  = 
.<_  )
1312breqd 4215 . . . . . . . . . 10  |-  ( k  =  K  ->  (
p ( le `  k ) w  <->  p  .<_  w ) )
1413notbid 286 . . . . . . . . 9  |-  ( k  =  K  ->  ( -.  p ( le `  k ) w  <->  -.  p  .<_  w ) )
1512breqd 4215 . . . . . . . . . 10  |-  ( k  =  K  ->  (
q ( le `  k ) w  <->  q  .<_  w ) )
1615notbid 286 . . . . . . . . 9  |-  ( k  =  K  ->  ( -.  q ( le `  k ) w  <->  -.  q  .<_  w ) )
1714, 16anbi12d 692 . . . . . . . 8  |-  ( k  =  K  ->  (
( -.  p ( le `  k ) w  /\  -.  q
( le `  k
) w )  <->  ( -.  p  .<_  w  /\  -.  q  .<_  w ) ) )
18 fveq2 5720 . . . . . . . . . . 11  |-  ( k  =  K  ->  ( meet `  k )  =  ( meet `  K
) )
19 ltrnset.m . . . . . . . . . . 11  |-  ./\  =  ( meet `  K )
2018, 19syl6eqr 2485 . . . . . . . . . 10  |-  ( k  =  K  ->  ( meet `  k )  = 
./\  )
21 fveq2 5720 . . . . . . . . . . . 12  |-  ( k  =  K  ->  ( join `  k )  =  ( join `  K
) )
22 ltrnset.j . . . . . . . . . . . 12  |-  .\/  =  ( join `  K )
2321, 22syl6eqr 2485 . . . . . . . . . . 11  |-  ( k  =  K  ->  ( join `  k )  = 
.\/  )
2423oveqd 6090 . . . . . . . . . 10  |-  ( k  =  K  ->  (
p ( join `  k
) ( f `  p ) )  =  ( p  .\/  (
f `  p )
) )
25 eqidd 2436 . . . . . . . . . 10  |-  ( k  =  K  ->  w  =  w )
2620, 24, 25oveq123d 6094 . . . . . . . . 9  |-  ( k  =  K  ->  (
( p ( join `  k ) ( f `
 p ) ) ( meet `  k
) w )  =  ( ( p  .\/  ( f `  p
) )  ./\  w
) )
2723oveqd 6090 . . . . . . . . . 10  |-  ( k  =  K  ->  (
q ( join `  k
) ( f `  q ) )  =  ( q  .\/  (
f `  q )
) )
2820, 27, 25oveq123d 6094 . . . . . . . . 9  |-  ( k  =  K  ->  (
( q ( join `  k ) ( f `
 q ) ) ( meet `  k
) w )  =  ( ( q  .\/  ( f `  q
) )  ./\  w
) )
2926, 28eqeq12d 2449 . . . . . . . 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 312 . . . . . . 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 2908 . . . . . 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 2908 . . . . 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 2943 . . . 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 4279 . . 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 30829 . . 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 5734 . . . . 5  |-  ( LHyp `  K )  e.  _V
373, 36eqeltri 2505 . . . 4  |-  H  e. 
_V
3837mptex 5958 . . 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 5798 . 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 16 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 359    = wceq 1652    e. wcel 1725   A.wral 2697   {crab 2701   _Vcvv 2948   class class class wbr 4204    e. cmpt 4258   ` cfv 5446  (class class class)co 6073   lecple 13528   joincjn 14393   meetcmee 14394   Atomscatm 29988   LHypclh 30708   LDilcldil 30824   LTrncltrn 30825
This theorem is referenced by:  ltrnset  30842
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-ltrn 30829
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