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Theorem ulmshft 20298
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 20297 . 2  |-  ( ph  ->  ( F ( ~~> u `  S ) G  ->  H ( ~~> u `  S ) G ) )
8 eqid 2435 . . 3  |-  ( ZZ>= `  ( ( M  +  K )  +  -u K ) )  =  ( ZZ>= `  ( ( M  +  K )  +  -u K ) )
93, 4zaddcld 10371 . . 3  |-  ( ph  ->  ( M  +  K
)  e.  ZZ )
104znegcld 10369 . . 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 2525 . . . . . . . 8  |-  ( (
ph  /\  n  e.  W )  ->  n  e.  ( ZZ>= `  ( M  +  K ) ) )
16 eluzsub 10507 . . . . . . . 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 2526 . . . . . 6  |-  ( (
ph  /\  n  e.  W )  ->  (
n  -  K )  e.  Z )
1911, 18ffvelrnd 5863 . . . . 5  |-  ( (
ph  /\  n  e.  W )  ->  ( F `  ( n  -  K ) )  e.  ( CC  ^m  S
) )
20 eqid 2435 . . . . 5  |-  ( n  e.  W  |->  ( F `
 ( n  -  K ) ) )  =  ( n  e.  W  |->  ( F `  ( n  -  K
) ) )
2119, 20fmptd 5885 . . . 4  |-  ( ph  ->  ( n  e.  W  |->  ( F `  (
n  -  K ) ) ) : W --> ( CC  ^m  S ) )
226feq1d 5572 . . . 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 2525 . . . . . . . . . 10  |-  ( (
ph  /\  m  e.  Z )  ->  m  e.  ( ZZ>= `  M )
)
26 eluzelz 10488 . . . . . . . . . 10  |-  ( m  e.  ( ZZ>= `  M
)  ->  m  e.  ZZ )
2725, 26syl 16 . . . . . . . . 9  |-  ( (
ph  /\  m  e.  Z )  ->  m  e.  ZZ )
2827zcnd 10368 . . . . . . . 8  |-  ( (
ph  /\  m  e.  Z )  ->  m  e.  CC )
294zcnd 10368 . . . . . . . . 9  |-  ( ph  ->  K  e.  CC )
3029adantr 452 . . . . . . . 8  |-  ( (
ph  /\  m  e.  Z )  ->  K  e.  CC )
3128, 30subnegd 9410 . . . . . . 7  |-  ( (
ph  /\  m  e.  Z )  ->  (
m  -  -u K
)  =  ( m  +  K ) )
3231fveq2d 5724 . . . . . 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 5722 . . . . . 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 10506 . . . . . . . . . 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 2526 . . . . . . . 8  |-  ( (
ph  /\  m  e.  Z )  ->  (
m  +  K )  e.  W )
39 oveq1 6080 . . . . . . . . . 10  |-  ( n  =  ( m  +  K )  ->  (
n  -  K )  =  ( ( m  +  K )  -  K ) )
4039fveq2d 5724 . . . . . . . . 9  |-  ( n  =  ( m  +  K )  ->  ( F `  ( n  -  K ) )  =  ( F `  (
( m  +  K
)  -  K ) ) )
41 fvex 5734 . . . . . . . . 9  |-  ( F `
 ( ( m  +  K )  -  K ) )  e. 
_V
4240, 20, 41fvmpt 5798 . . . . . . . 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 9404 . . . . . . . 8  |-  ( (
ph  /\  m  e.  Z )  ->  (
( m  +  K
)  -  K )  =  m )
4544fveq2d 5724 . . . . . . 7  |-  ( (
ph  /\  m  e.  Z )  ->  ( F `  ( (
m  +  K )  -  K ) )  =  ( F `  m ) )
4643, 45eqtrd 2467 . . . . . 6  |-  ( (
ph  /\  m  e.  Z )  ->  (
( n  e.  W  |->  ( F `  (
n  -  K ) ) ) `  (
m  +  K ) )  =  ( F `
 m ) )
4732, 34, 463eqtrd 2471 . . . . 5  |-  ( (
ph  /\  m  e.  Z )  ->  ( H `  ( m  -  -u K ) )  =  ( F `  m ) )
4847mpteq2dva 4287 . . . 4  |-  ( ph  ->  ( m  e.  Z  |->  ( H `  (
m  -  -u K
) ) )  =  ( m  e.  Z  |->  ( F `  m
) ) )
493zcnd 10368 . . . . . . . . 9  |-  ( ph  ->  M  e.  CC )
5010zcnd 10368 . . . . . . . . 9  |-  ( ph  -> 
-u K  e.  CC )
5149, 29, 50addassd 9102 . . . . . . . 8  |-  ( ph  ->  ( ( M  +  K )  +  -u K )  =  ( M  +  ( K  +  -u K ) ) )
5229negidd 9393 . . . . . . . . 9  |-  ( ph  ->  ( K  +  -u K )  =  0 )
5352oveq2d 6089 . . . . . . . 8  |-  ( ph  ->  ( M  +  ( K  +  -u K
) )  =  ( M  +  0 ) )
5449addid1d 9258 . . . . . . . 8  |-  ( ph  ->  ( M  +  0 )  =  M )
5551, 53, 543eqtrd 2471 . . . . . . 7  |-  ( ph  ->  ( ( M  +  K )  +  -u K )  =  M )
5655fveq2d 5724 . . . . . 6  |-  ( ph  ->  ( ZZ>= `  ( ( M  +  K )  +  -u K ) )  =  ( ZZ>= `  M
) )
5756, 1syl6eqr 2485 . . . . 5  |-  ( ph  ->  ( ZZ>= `  ( ( M  +  K )  +  -u K ) )  =  Z )
5857mpteq1d 4282 . . . 4  |-  ( ph  ->  ( m  e.  (
ZZ>= `  ( ( M  +  K )  + 
-u K ) ) 
|->  ( H `  (
m  -  -u K
) ) )  =  ( m  e.  Z  |->  ( H `  (
m  -  -u K
) ) ) )
595feqmptd 5771 . . . 4  |-  ( ph  ->  F  =  ( m  e.  Z  |->  ( F `
 m ) ) )
6048, 58, 593eqtr4rd 2478 . . 3  |-  ( ph  ->  F  =  ( m  e.  ( ZZ>= `  (
( M  +  K
)  +  -u K
) )  |->  ( H `
 ( m  -  -u K ) ) ) )
612, 8, 9, 10, 23, 60ulmshftlem 20297 . 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 1652    e. wcel 1725   class class class wbr 4204    e. cmpt 4258   -->wf 5442   ` cfv 5446  (class class class)co 6073    ^m cmap 7010   CCcc 8980   0cc0 8982    + caddc 8985    - cmin 9283   -ucneg 9284   ZZcz 10274   ZZ>=cuz 10480   ~~> uculm 20284
This theorem is referenced by:  pserdvlem2  20336
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-13 1727  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-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  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-nel 2601  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-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  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-oprab 6077  df-mpt2 6078  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-map 7012  df-pm 7013  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-n0 10214  df-z 10275  df-uz 10481  df-ulm 20285
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