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Theorem ulmshft 20166
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 20165 . 2  |-  ( ph  ->  ( F ( ~~> u `  S ) G  ->  H ( ~~> u `  S ) G ) )
8 eqid 2380 . . 3  |-  ( ZZ>= `  ( ( M  +  K )  +  -u K ) )  =  ( ZZ>= `  ( ( M  +  K )  +  -u K ) )
93, 4zaddcld 10304 . . 3  |-  ( ph  ->  ( M  +  K
)  e.  ZZ )
104znegcld 10302 . . 3  |-  ( ph  -> 
-u K  e.  ZZ )
115adantr 452 . . . . . 6  |-  ( (
ph  /\  n  e.  W )  ->  F : Z --> ( CC  ^m  S ) )
123adantr 452 . . . . . . . 8  |-  ( (
ph  /\  n  e.  W )  ->  M  e.  ZZ )
134adantr 452 . . . . . . . 8  |-  ( (
ph  /\  n  e.  W )  ->  K  e.  ZZ )
14 simpr 448 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  W )  ->  n  e.  W )
1514, 2syl6eleq 2470 . . . . . . . 8  |-  ( (
ph  /\  n  e.  W )  ->  n  e.  ( ZZ>= `  ( M  +  K ) ) )
16 eluzsub 10440 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  K  e.  ZZ  /\  n  e.  ( ZZ>= `  ( M  +  K ) ) )  ->  ( n  -  K )  e.  (
ZZ>= `  M ) )
1712, 13, 15, 16syl3anc 1184 . . . . . . 7  |-  ( (
ph  /\  n  e.  W )  ->  (
n  -  K )  e.  ( ZZ>= `  M
) )
1817, 1syl6eleqr 2471 . . . . . 6  |-  ( (
ph  /\  n  e.  W )  ->  (
n  -  K )  e.  Z )
1911, 18ffvelrnd 5803 . . . . 5  |-  ( (
ph  /\  n  e.  W )  ->  ( F `  ( n  -  K ) )  e.  ( CC  ^m  S
) )
20 eqid 2380 . . . . 5  |-  ( n  e.  W  |->  ( F `
 ( n  -  K ) ) )  =  ( n  e.  W  |->  ( F `  ( n  -  K
) ) )
2119, 20fmptd 5825 . . . 4  |-  ( ph  ->  ( n  e.  W  |->  ( F `  (
n  -  K ) ) ) : W --> ( CC  ^m  S ) )
226feq1d 5513 . . . 4  |-  ( ph  ->  ( H : W --> ( CC  ^m  S )  <-> 
( n  e.  W  |->  ( F `  (
n  -  K ) ) ) : W --> ( CC  ^m  S ) ) )
2321, 22mpbird 224 . . 3  |-  ( ph  ->  H : W --> ( CC 
^m  S ) )
24 simpr 448 . . . . . . . . . . 11  |-  ( (
ph  /\  m  e.  Z )  ->  m  e.  Z )
2524, 1syl6eleq 2470 . . . . . . . . . 10  |-  ( (
ph  /\  m  e.  Z )  ->  m  e.  ( ZZ>= `  M )
)
26 eluzelz 10421 . . . . . . . . . 10  |-  ( m  e.  ( ZZ>= `  M
)  ->  m  e.  ZZ )
2725, 26syl 16 . . . . . . . . 9  |-  ( (
ph  /\  m  e.  Z )  ->  m  e.  ZZ )
2827zcnd 10301 . . . . . . . 8  |-  ( (
ph  /\  m  e.  Z )  ->  m  e.  CC )
294zcnd 10301 . . . . . . . . 9  |-  ( ph  ->  K  e.  CC )
3029adantr 452 . . . . . . . 8  |-  ( (
ph  /\  m  e.  Z )  ->  K  e.  CC )
3128, 30subnegd 9343 . . . . . . 7  |-  ( (
ph  /\  m  e.  Z )  ->  (
m  -  -u K
)  =  ( m  +  K ) )
3231fveq2d 5665 . . . . . 6  |-  ( (
ph  /\  m  e.  Z )  ->  ( H `  ( m  -  -u K ) )  =  ( H `  ( m  +  K
) ) )
336adantr 452 . . . . . . 7  |-  ( (
ph  /\  m  e.  Z )  ->  H  =  ( n  e.  W  |->  ( F `  ( n  -  K
) ) ) )
3433fveq1d 5663 . . . . . 6  |-  ( (
ph  /\  m  e.  Z )  ->  ( H `  ( m  +  K ) )  =  ( ( n  e.  W  |->  ( F `  ( n  -  K
) ) ) `  ( m  +  K
) ) )
354adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  m  e.  Z )  ->  K  e.  ZZ )
36 eluzadd 10439 . . . . . . . . . 10  |-  ( ( m  e.  ( ZZ>= `  M )  /\  K  e.  ZZ )  ->  (
m  +  K )  e.  ( ZZ>= `  ( M  +  K )
) )
3725, 35, 36syl2anc 643 . . . . . . . . 9  |-  ( (
ph  /\  m  e.  Z )  ->  (
m  +  K )  e.  ( ZZ>= `  ( M  +  K )
) )
3837, 2syl6eleqr 2471 . . . . . . . 8  |-  ( (
ph  /\  m  e.  Z )  ->  (
m  +  K )  e.  W )
39 oveq1 6020 . . . . . . . . . 10  |-  ( n  =  ( m  +  K )  ->  (
n  -  K )  =  ( ( m  +  K )  -  K ) )
4039fveq2d 5665 . . . . . . . . 9  |-  ( n  =  ( m  +  K )  ->  ( F `  ( n  -  K ) )  =  ( F `  (
( m  +  K
)  -  K ) ) )
41 fvex 5675 . . . . . . . . 9  |-  ( F `
 ( ( m  +  K )  -  K ) )  e. 
_V
4240, 20, 41fvmpt 5738 . . . . . . . 8  |-  ( ( m  +  K )  e.  W  ->  (
( n  e.  W  |->  ( F `  (
n  -  K ) ) ) `  (
m  +  K ) )  =  ( F `
 ( ( m  +  K )  -  K ) ) )
4338, 42syl 16 . . . . . . 7  |-  ( (
ph  /\  m  e.  Z )  ->  (
( n  e.  W  |->  ( F `  (
n  -  K ) ) ) `  (
m  +  K ) )  =  ( F `
 ( ( m  +  K )  -  K ) ) )
4428, 30pncand 9337 . . . . . . . 8  |-  ( (
ph  /\  m  e.  Z )  ->  (
( m  +  K
)  -  K )  =  m )
4544fveq2d 5665 . . . . . . 7  |-  ( (
ph  /\  m  e.  Z )  ->  ( F `  ( (
m  +  K )  -  K ) )  =  ( F `  m ) )
4643, 45eqtrd 2412 . . . . . 6  |-  ( (
ph  /\  m  e.  Z )  ->  (
( n  e.  W  |->  ( F `  (
n  -  K ) ) ) `  (
m  +  K ) )  =  ( F `
 m ) )
4732, 34, 463eqtrd 2416 . . . . 5  |-  ( (
ph  /\  m  e.  Z )  ->  ( H `  ( m  -  -u K ) )  =  ( F `  m ) )
4847mpteq2dva 4229 . . . 4  |-  ( ph  ->  ( m  e.  Z  |->  ( H `  (
m  -  -u K
) ) )  =  ( m  e.  Z  |->  ( F `  m
) ) )
493zcnd 10301 . . . . . . . . 9  |-  ( ph  ->  M  e.  CC )
5010zcnd 10301 . . . . . . . . 9  |-  ( ph  -> 
-u K  e.  CC )
5149, 29, 50addassd 9036 . . . . . . . 8  |-  ( ph  ->  ( ( M  +  K )  +  -u K )  =  ( M  +  ( K  +  -u K ) ) )
5229negidd 9326 . . . . . . . . 9  |-  ( ph  ->  ( K  +  -u K )  =  0 )
5352oveq2d 6029 . . . . . . . 8  |-  ( ph  ->  ( M  +  ( K  +  -u K
) )  =  ( M  +  0 ) )
5449addid1d 9191 . . . . . . . 8  |-  ( ph  ->  ( M  +  0 )  =  M )
5551, 53, 543eqtrd 2416 . . . . . . 7  |-  ( ph  ->  ( ( M  +  K )  +  -u K )  =  M )
5655fveq2d 5665 . . . . . 6  |-  ( ph  ->  ( ZZ>= `  ( ( M  +  K )  +  -u K ) )  =  ( ZZ>= `  M
) )
5756, 1syl6eqr 2430 . . . . 5  |-  ( ph  ->  ( ZZ>= `  ( ( M  +  K )  +  -u K ) )  =  Z )
5857mpteq1d 4224 . . . 4  |-  ( ph  ->  ( m  e.  (
ZZ>= `  ( ( M  +  K )  + 
-u K ) ) 
|->  ( H `  (
m  -  -u K
) ) )  =  ( m  e.  Z  |->  ( H `  (
m  -  -u K
) ) ) )
595feqmptd 5711 . . . 4  |-  ( ph  ->  F  =  ( m  e.  Z  |->  ( F `
 m ) ) )
6048, 58, 593eqtr4rd 2423 . . 3  |-  ( ph  ->  F  =  ( m  e.  ( ZZ>= `  (
( M  +  K
)  +  -u K
) )  |->  ( H `
 ( m  -  -u K ) ) ) )
612, 8, 9, 10, 23, 60ulmshftlem 20165 . 2  |-  ( ph  ->  ( H ( ~~> u `  S ) G  ->  F ( ~~> u `  S ) G ) )
627, 61impbid 184 1  |-  ( ph  ->  ( F ( ~~> u `  S ) G  <->  H ( ~~> u `  S ) G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   class class class wbr 4146    e. cmpt 4200   -->wf 5383   ` cfv 5387  (class class class)co 6013    ^m cmap 6947   CCcc 8914   0cc0 8916    + caddc 8919    - cmin 9216   -ucneg 9217   ZZcz 10207   ZZ>=cuz 10413   ~~> uculm 20152
This theorem is referenced by:  pserdvlem2  20204
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-riota 6478  df-recs 6562  df-rdg 6597  df-er 6834  df-map 6949  df-pm 6950  df-en 7039  df-dom 7040  df-sdom 7041  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-nn 9926  df-n0 10147  df-z 10208  df-uz 10414  df-ulm 20153
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