MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ulmshft Unicode version

Theorem ulmshft 19769
Description: A sequence of functions converges iff the shifted sequence converges. (Contributed by Mario Carneiro, 24-Mar-2015.)
Hypotheses
Ref Expression
ulmshft.z  |-  Z  =  ( ZZ>= `  M )
ulmshft.w  |-  W  =  ( ZZ>= `  ( M  +  K ) )
ulmshft.m  |-  ( ph  ->  M  e.  ZZ )
ulmshft.k  |-  ( ph  ->  K  e.  ZZ )
ulmshft.f  |-  ( ph  ->  F : Z --> ( CC 
^m  S ) )
ulmshft.h  |-  ( ph  ->  H  =  ( n  e.  W  |->  ( F `
 ( n  -  K ) ) ) )
Assertion
Ref Expression
ulmshft  |-  ( ph  ->  ( F ( ~~> u `  S ) G  <->  H ( ~~> u `  S ) G ) )
Distinct variable groups:    ph, n    n, W    n, F    n, K    S, n
Allowed substitution hints:    G( n)    H( n)    M( n)    Z( n)

Proof of Theorem ulmshft
Dummy variable  m is distinct from all other variables.
StepHypRef Expression
1 ulmshft.z . . 3  |-  Z  =  ( ZZ>= `  M )
2 ulmshft.w . . 3  |-  W  =  ( ZZ>= `  ( M  +  K ) )
3 ulmshft.m . . 3  |-  ( ph  ->  M  e.  ZZ )
4 ulmshft.k . . 3  |-  ( ph  ->  K  e.  ZZ )
5 ulmshft.f . . 3  |-  ( ph  ->  F : Z --> ( CC 
^m  S ) )
6 ulmshft.h . . 3  |-  ( ph  ->  H  =  ( n  e.  W  |->  ( F `
 ( n  -  K ) ) ) )
71, 2, 3, 4, 5, 6ulmshftlem 19768 . 2  |-  ( ph  ->  ( F ( ~~> u `  S ) G  ->  H ( ~~> u `  S ) G ) )
8 eqid 2283 . . 3  |-  ( ZZ>= `  ( ( M  +  K )  +  -u K ) )  =  ( ZZ>= `  ( ( M  +  K )  +  -u K ) )
93, 4zaddcld 10121 . . 3  |-  ( ph  ->  ( M  +  K
)  e.  ZZ )
104znegcld 10119 . . 3  |-  ( ph  -> 
-u K  e.  ZZ )
115adantr 451 . . . . . 6  |-  ( (
ph  /\  n  e.  W )  ->  F : Z --> ( CC  ^m  S ) )
123adantr 451 . . . . . . . 8  |-  ( (
ph  /\  n  e.  W )  ->  M  e.  ZZ )
134adantr 451 . . . . . . . 8  |-  ( (
ph  /\  n  e.  W )  ->  K  e.  ZZ )
14 simpr 447 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  W )  ->  n  e.  W )
1514, 2syl6eleq 2373 . . . . . . . 8  |-  ( (
ph  /\  n  e.  W )  ->  n  e.  ( ZZ>= `  ( M  +  K ) ) )
16 eluzsub 10257 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  K  e.  ZZ  /\  n  e.  ( ZZ>= `  ( M  +  K ) ) )  ->  ( n  -  K )  e.  (
ZZ>= `  M ) )
1712, 13, 15, 16syl3anc 1182 . . . . . . 7  |-  ( (
ph  /\  n  e.  W )  ->  (
n  -  K )  e.  ( ZZ>= `  M
) )
1817, 1syl6eleqr 2374 . . . . . 6  |-  ( (
ph  /\  n  e.  W )  ->  (
n  -  K )  e.  Z )
19 ffvelrn 5663 . . . . . 6  |-  ( ( F : Z --> ( CC 
^m  S )  /\  ( n  -  K
)  e.  Z )  ->  ( F `  ( n  -  K
) )  e.  ( CC  ^m  S ) )
2011, 18, 19syl2anc 642 . . . . 5  |-  ( (
ph  /\  n  e.  W )  ->  ( F `  ( n  -  K ) )  e.  ( CC  ^m  S
) )
21 eqid 2283 . . . . 5  |-  ( n  e.  W  |->  ( F `
 ( n  -  K ) ) )  =  ( n  e.  W  |->  ( F `  ( n  -  K
) ) )
2220, 21fmptd 5684 . . . 4  |-  ( ph  ->  ( n  e.  W  |->  ( F `  (
n  -  K ) ) ) : W --> ( CC  ^m  S ) )
236feq1d 5379 . . . 4  |-  ( ph  ->  ( H : W --> ( CC  ^m  S )  <-> 
( n  e.  W  |->  ( F `  (
n  -  K ) ) ) : W --> ( CC  ^m  S ) ) )
2422, 23mpbird 223 . . 3  |-  ( ph  ->  H : W --> ( CC 
^m  S ) )
25 simpr 447 . . . . . . . . . . 11  |-  ( (
ph  /\  m  e.  Z )  ->  m  e.  Z )
2625, 1syl6eleq 2373 . . . . . . . . . 10  |-  ( (
ph  /\  m  e.  Z )  ->  m  e.  ( ZZ>= `  M )
)
27 eluzelz 10238 . . . . . . . . . 10  |-  ( m  e.  ( ZZ>= `  M
)  ->  m  e.  ZZ )
2826, 27syl 15 . . . . . . . . 9  |-  ( (
ph  /\  m  e.  Z )  ->  m  e.  ZZ )
2928zcnd 10118 . . . . . . . 8  |-  ( (
ph  /\  m  e.  Z )  ->  m  e.  CC )
304zcnd 10118 . . . . . . . . 9  |-  ( ph  ->  K  e.  CC )
3130adantr 451 . . . . . . . 8  |-  ( (
ph  /\  m  e.  Z )  ->  K  e.  CC )
3229, 31subnegd 9164 . . . . . . 7  |-  ( (
ph  /\  m  e.  Z )  ->  (
m  -  -u K
)  =  ( m  +  K ) )
3332fveq2d 5529 . . . . . 6  |-  ( (
ph  /\  m  e.  Z )  ->  ( H `  ( m  -  -u K ) )  =  ( H `  ( m  +  K
) ) )
346adantr 451 . . . . . . 7  |-  ( (
ph  /\  m  e.  Z )  ->  H  =  ( n  e.  W  |->  ( F `  ( n  -  K
) ) ) )
3534fveq1d 5527 . . . . . 6  |-  ( (
ph  /\  m  e.  Z )  ->  ( H `  ( m  +  K ) )  =  ( ( n  e.  W  |->  ( F `  ( n  -  K
) ) ) `  ( m  +  K
) ) )
364adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  m  e.  Z )  ->  K  e.  ZZ )
37 eluzadd 10256 . . . . . . . . . 10  |-  ( ( m  e.  ( ZZ>= `  M )  /\  K  e.  ZZ )  ->  (
m  +  K )  e.  ( ZZ>= `  ( M  +  K )
) )
3826, 36, 37syl2anc 642 . . . . . . . . 9  |-  ( (
ph  /\  m  e.  Z )  ->  (
m  +  K )  e.  ( ZZ>= `  ( M  +  K )
) )
3938, 2syl6eleqr 2374 . . . . . . . 8  |-  ( (
ph  /\  m  e.  Z )  ->  (
m  +  K )  e.  W )
40 oveq1 5865 . . . . . . . . . 10  |-  ( n  =  ( m  +  K )  ->  (
n  -  K )  =  ( ( m  +  K )  -  K ) )
4140fveq2d 5529 . . . . . . . . 9  |-  ( n  =  ( m  +  K )  ->  ( F `  ( n  -  K ) )  =  ( F `  (
( m  +  K
)  -  K ) ) )
42 fvex 5539 . . . . . . . . 9  |-  ( F `
 ( ( m  +  K )  -  K ) )  e. 
_V
4341, 21, 42fvmpt 5602 . . . . . . . 8  |-  ( ( m  +  K )  e.  W  ->  (
( n  e.  W  |->  ( F `  (
n  -  K ) ) ) `  (
m  +  K ) )  =  ( F `
 ( ( m  +  K )  -  K ) ) )
4439, 43syl 15 . . . . . . 7  |-  ( (
ph  /\  m  e.  Z )  ->  (
( n  e.  W  |->  ( F `  (
n  -  K ) ) ) `  (
m  +  K ) )  =  ( F `
 ( ( m  +  K )  -  K ) ) )
4529, 31pncand 9158 . . . . . . . 8  |-  ( (
ph  /\  m  e.  Z )  ->  (
( m  +  K
)  -  K )  =  m )
4645fveq2d 5529 . . . . . . 7  |-  ( (
ph  /\  m  e.  Z )  ->  ( F `  ( (
m  +  K )  -  K ) )  =  ( F `  m ) )
4744, 46eqtrd 2315 . . . . . 6  |-  ( (
ph  /\  m  e.  Z )  ->  (
( n  e.  W  |->  ( F `  (
n  -  K ) ) ) `  (
m  +  K ) )  =  ( F `
 m ) )
4833, 35, 473eqtrd 2319 . . . . 5  |-  ( (
ph  /\  m  e.  Z )  ->  ( H `  ( m  -  -u K ) )  =  ( F `  m ) )
4948mpteq2dva 4106 . . . 4  |-  ( ph  ->  ( m  e.  Z  |->  ( H `  (
m  -  -u K
) ) )  =  ( m  e.  Z  |->  ( F `  m
) ) )
503zcnd 10118 . . . . . . . . 9  |-  ( ph  ->  M  e.  CC )
5110zcnd 10118 . . . . . . . . 9  |-  ( ph  -> 
-u K  e.  CC )
5250, 30, 51addassd 8857 . . . . . . . 8  |-  ( ph  ->  ( ( M  +  K )  +  -u K )  =  ( M  +  ( K  +  -u K ) ) )
5330negidd 9147 . . . . . . . . 9  |-  ( ph  ->  ( K  +  -u K )  =  0 )
5453oveq2d 5874 . . . . . . . 8  |-  ( ph  ->  ( M  +  ( K  +  -u K
) )  =  ( M  +  0 ) )
5550addid1d 9012 . . . . . . . 8  |-  ( ph  ->  ( M  +  0 )  =  M )
5652, 54, 553eqtrd 2319 . . . . . . 7  |-  ( ph  ->  ( ( M  +  K )  +  -u K )  =  M )
5756fveq2d 5529 . . . . . 6  |-  ( ph  ->  ( ZZ>= `  ( ( M  +  K )  +  -u K ) )  =  ( ZZ>= `  M
) )
5857, 1syl6eqr 2333 . . . . 5  |-  ( ph  ->  ( ZZ>= `  ( ( M  +  K )  +  -u K ) )  =  Z )
59 mpteq1 4100 . . . . 5  |-  ( (
ZZ>= `  ( ( M  +  K )  + 
-u K ) )  =  Z  ->  (
m  e.  ( ZZ>= `  ( ( M  +  K )  +  -u K ) )  |->  ( H `  ( m  -  -u K ) ) )  =  ( m  e.  Z  |->  ( H `
 ( m  -  -u K ) ) ) )
6058, 59syl 15 . . . 4  |-  ( ph  ->  ( m  e.  (
ZZ>= `  ( ( M  +  K )  + 
-u K ) ) 
|->  ( H `  (
m  -  -u K
) ) )  =  ( m  e.  Z  |->  ( H `  (
m  -  -u K
) ) ) )
615feqmptd 5575 . . . 4  |-  ( ph  ->  F  =  ( m  e.  Z  |->  ( F `
 m ) ) )
6249, 60, 613eqtr4rd 2326 . . 3  |-  ( ph  ->  F  =  ( m  e.  ( ZZ>= `  (
( M  +  K
)  +  -u K
) )  |->  ( H `
 ( m  -  -u K ) ) ) )
632, 8, 9, 10, 24, 62ulmshftlem 19768 . 2  |-  ( ph  ->  ( H ( ~~> u `  S ) G  ->  F ( ~~> u `  S ) G ) )
647, 63impbid 183 1  |-  ( ph  ->  ( F ( ~~> u `  S ) G  <->  H ( ~~> u `  S ) G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   class class class wbr 4023    e. cmpt 4077   -->wf 5251   ` cfv 5255  (class class class)co 5858    ^m cmap 6772   CCcc 8735   0cc0 8737    + caddc 8740    - cmin 9037   -ucneg 9038   ZZcz 10024   ZZ>=cuz 10230   ~~> uculm 19755
This theorem is referenced by:  pserdvlem2  19804
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-map 6774  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-ulm 19756
  Copyright terms: Public domain W3C validator