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Theorem ulmshft 20259
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 20258 . 2  |-  ( ph  ->  ( F ( ~~> u `  S ) G  ->  H ( ~~> u `  S ) G ) )
8 eqid 2404 . . 3  |-  ( ZZ>= `  ( ( M  +  K )  +  -u K ) )  =  ( ZZ>= `  ( ( M  +  K )  +  -u K ) )
93, 4zaddcld 10335 . . 3  |-  ( ph  ->  ( M  +  K
)  e.  ZZ )
104znegcld 10333 . . 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 2494 . . . . . . . 8  |-  ( (
ph  /\  n  e.  W )  ->  n  e.  ( ZZ>= `  ( M  +  K ) ) )
16 eluzsub 10471 . . . . . . . 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 2495 . . . . . 6  |-  ( (
ph  /\  n  e.  W )  ->  (
n  -  K )  e.  Z )
1911, 18ffvelrnd 5830 . . . . 5  |-  ( (
ph  /\  n  e.  W )  ->  ( F `  ( n  -  K ) )  e.  ( CC  ^m  S
) )
20 eqid 2404 . . . . 5  |-  ( n  e.  W  |->  ( F `
 ( n  -  K ) ) )  =  ( n  e.  W  |->  ( F `  ( n  -  K
) ) )
2119, 20fmptd 5852 . . . 4  |-  ( ph  ->  ( n  e.  W  |->  ( F `  (
n  -  K ) ) ) : W --> ( CC  ^m  S ) )
226feq1d 5539 . . . 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 2494 . . . . . . . . . 10  |-  ( (
ph  /\  m  e.  Z )  ->  m  e.  ( ZZ>= `  M )
)
26 eluzelz 10452 . . . . . . . . . 10  |-  ( m  e.  ( ZZ>= `  M
)  ->  m  e.  ZZ )
2725, 26syl 16 . . . . . . . . 9  |-  ( (
ph  /\  m  e.  Z )  ->  m  e.  ZZ )
2827zcnd 10332 . . . . . . . 8  |-  ( (
ph  /\  m  e.  Z )  ->  m  e.  CC )
294zcnd 10332 . . . . . . . . 9  |-  ( ph  ->  K  e.  CC )
3029adantr 452 . . . . . . . 8  |-  ( (
ph  /\  m  e.  Z )  ->  K  e.  CC )
3128, 30subnegd 9374 . . . . . . 7  |-  ( (
ph  /\  m  e.  Z )  ->  (
m  -  -u K
)  =  ( m  +  K ) )
3231fveq2d 5691 . . . . . 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 5689 . . . . . 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 10470 . . . . . . . . . 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 2495 . . . . . . . 8  |-  ( (
ph  /\  m  e.  Z )  ->  (
m  +  K )  e.  W )
39 oveq1 6047 . . . . . . . . . 10  |-  ( n  =  ( m  +  K )  ->  (
n  -  K )  =  ( ( m  +  K )  -  K ) )
4039fveq2d 5691 . . . . . . . . 9  |-  ( n  =  ( m  +  K )  ->  ( F `  ( n  -  K ) )  =  ( F `  (
( m  +  K
)  -  K ) ) )
41 fvex 5701 . . . . . . . . 9  |-  ( F `
 ( ( m  +  K )  -  K ) )  e. 
_V
4240, 20, 41fvmpt 5765 . . . . . . . 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 9368 . . . . . . . 8  |-  ( (
ph  /\  m  e.  Z )  ->  (
( m  +  K
)  -  K )  =  m )
4544fveq2d 5691 . . . . . . 7  |-  ( (
ph  /\  m  e.  Z )  ->  ( F `  ( (
m  +  K )  -  K ) )  =  ( F `  m ) )
4643, 45eqtrd 2436 . . . . . 6  |-  ( (
ph  /\  m  e.  Z )  ->  (
( n  e.  W  |->  ( F `  (
n  -  K ) ) ) `  (
m  +  K ) )  =  ( F `
 m ) )
4732, 34, 463eqtrd 2440 . . . . 5  |-  ( (
ph  /\  m  e.  Z )  ->  ( H `  ( m  -  -u K ) )  =  ( F `  m ) )
4847mpteq2dva 4255 . . . 4  |-  ( ph  ->  ( m  e.  Z  |->  ( H `  (
m  -  -u K
) ) )  =  ( m  e.  Z  |->  ( F `  m
) ) )
493zcnd 10332 . . . . . . . . 9  |-  ( ph  ->  M  e.  CC )
5010zcnd 10332 . . . . . . . . 9  |-  ( ph  -> 
-u K  e.  CC )
5149, 29, 50addassd 9066 . . . . . . . 8  |-  ( ph  ->  ( ( M  +  K )  +  -u K )  =  ( M  +  ( K  +  -u K ) ) )
5229negidd 9357 . . . . . . . . 9  |-  ( ph  ->  ( K  +  -u K )  =  0 )
5352oveq2d 6056 . . . . . . . 8  |-  ( ph  ->  ( M  +  ( K  +  -u K
) )  =  ( M  +  0 ) )
5449addid1d 9222 . . . . . . . 8  |-  ( ph  ->  ( M  +  0 )  =  M )
5551, 53, 543eqtrd 2440 . . . . . . 7  |-  ( ph  ->  ( ( M  +  K )  +  -u K )  =  M )
5655fveq2d 5691 . . . . . 6  |-  ( ph  ->  ( ZZ>= `  ( ( M  +  K )  +  -u K ) )  =  ( ZZ>= `  M
) )
5756, 1syl6eqr 2454 . . . . 5  |-  ( ph  ->  ( ZZ>= `  ( ( M  +  K )  +  -u K ) )  =  Z )
5857mpteq1d 4250 . . . 4  |-  ( ph  ->  ( m  e.  (
ZZ>= `  ( ( M  +  K )  + 
-u K ) ) 
|->  ( H `  (
m  -  -u K
) ) )  =  ( m  e.  Z  |->  ( H `  (
m  -  -u K
) ) ) )
595feqmptd 5738 . . . 4  |-  ( ph  ->  F  =  ( m  e.  Z  |->  ( F `
 m ) ) )
6048, 58, 593eqtr4rd 2447 . . 3  |-  ( ph  ->  F  =  ( m  e.  ( ZZ>= `  (
( M  +  K
)  +  -u K
) )  |->  ( H `
 ( m  -  -u K ) ) ) )
612, 8, 9, 10, 23, 60ulmshftlem 20258 . 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 1721   class class class wbr 4172    e. cmpt 4226   -->wf 5409   ` cfv 5413  (class class class)co 6040    ^m cmap 6977   CCcc 8944   0cc0 8946    + caddc 8949    - cmin 9247   -ucneg 9248   ZZcz 10238   ZZ>=cuz 10444   ~~> uculm 20245
This theorem is referenced by:  pserdvlem2  20297
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-map 6979  df-pm 6980  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-n0 10178  df-z 10239  df-uz 10445  df-ulm 20246
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