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Theorem climshft2 13058
Description: A shifted function converges iff the original function converges. (Contributed by Paul Chapman, 21-Nov-2007.) (Revised by Mario Carneiro, 6-Feb-2014.)
Hypotheses
Ref Expression
climshft2.1  |-  Z  =  ( ZZ>= `  M )
climshft2.2  |-  ( ph  ->  M  e.  ZZ )
climshft2.3  |-  ( ph  ->  K  e.  ZZ )
climshft2.5  |-  ( ph  ->  F  e.  W )
climshft2.6  |-  ( ph  ->  G  e.  X )
climshft2.7  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  ( k  +  K ) )  =  ( F `  k
) )
Assertion
Ref Expression
climshft2  |-  ( ph  ->  ( F  ~~>  A  <->  G  ~~>  A ) )
Distinct variable groups:    k, F    k, G    k, K    k, M    ph, k    k, Z    A, k
Allowed substitution hints:    W( k)    X( k)

Proof of Theorem climshft2
StepHypRef Expression
1 climshft2.1 . . 3  |-  Z  =  ( ZZ>= `  M )
2 ovex 6114 . . . 4  |-  ( G 
shift  -u K )  e. 
_V
32a1i 11 . . 3  |-  ( ph  ->  ( G  shift  -u K
)  e.  _V )
4 climshft2.5 . . 3  |-  ( ph  ->  F  e.  W )
5 climshft2.2 . . 3  |-  ( ph  ->  M  e.  ZZ )
6 climshft2.3 . . . . . . 7  |-  ( ph  ->  K  e.  ZZ )
76zcnd 10746 . . . . . 6  |-  ( ph  ->  K  e.  CC )
8 eluzelz 10868 . . . . . . . 8  |-  ( k  e.  ( ZZ>= `  M
)  ->  k  e.  ZZ )
98, 1eleq2s 2533 . . . . . . 7  |-  ( k  e.  Z  ->  k  e.  ZZ )
109zcnd 10746 . . . . . 6  |-  ( k  e.  Z  ->  k  e.  CC )
11 fvex 5699 . . . . . . 7  |-  (  _I 
`  G )  e. 
_V
1211shftval4 12564 . . . . . 6  |-  ( ( K  e.  CC  /\  k  e.  CC )  ->  ( ( (  _I 
`  G )  shift  -u K ) `  k
)  =  ( (  _I  `  G ) `
 ( K  +  k ) ) )
137, 10, 12syl2an 477 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  (
( (  _I  `  G )  shift  -u K
) `  k )  =  ( (  _I 
`  G ) `  ( K  +  k
) ) )
14 climshft2.6 . . . . . . . . 9  |-  ( ph  ->  G  e.  X )
15 fvi 5746 . . . . . . . . 9  |-  ( G  e.  X  ->  (  _I  `  G )  =  G )
1614, 15syl 16 . . . . . . . 8  |-  ( ph  ->  (  _I  `  G
)  =  G )
1716adantr 465 . . . . . . 7  |-  ( (
ph  /\  k  e.  Z )  ->  (  _I  `  G )  =  G )
1817oveq1d 6104 . . . . . 6  |-  ( (
ph  /\  k  e.  Z )  ->  (
(  _I  `  G
)  shift  -u K )  =  ( G  shift  -u K
) )
1918fveq1d 5691 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  (
( (  _I  `  G )  shift  -u K
) `  k )  =  ( ( G 
shift  -u K ) `  k ) )
20 addcom 9553 . . . . . . 7  |-  ( ( K  e.  CC  /\  k  e.  CC )  ->  ( K  +  k )  =  ( k  +  K ) )
217, 10, 20syl2an 477 . . . . . 6  |-  ( (
ph  /\  k  e.  Z )  ->  ( K  +  k )  =  ( k  +  K ) )
2217, 21fveq12d 5695 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  (
(  _I  `  G
) `  ( K  +  k ) )  =  ( G `  ( k  +  K
) ) )
2313, 19, 223eqtr3d 2481 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  (
( G  shift  -u K
) `  k )  =  ( G `  ( k  +  K
) ) )
24 climshft2.7 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  ( k  +  K ) )  =  ( F `  k
) )
2523, 24eqtrd 2473 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  (
( G  shift  -u K
) `  k )  =  ( F `  k ) )
261, 3, 4, 5, 25climeq 13043 . 2  |-  ( ph  ->  ( ( G  shift  -u K )  ~~>  A  <->  F  ~~>  A ) )
276znegcld 10747 . . 3  |-  ( ph  -> 
-u K  e.  ZZ )
28 climshft 13052 . . 3  |-  ( (
-u K  e.  ZZ  /\  G  e.  X )  ->  ( ( G 
shift  -u K )  ~~>  A  <->  G  ~~>  A ) )
2927, 14, 28syl2anc 661 . 2  |-  ( ph  ->  ( ( G  shift  -u K )  ~~>  A  <->  G  ~~>  A ) )
3026, 29bitr3d 255 1  |-  ( ph  ->  ( F  ~~>  A  <->  G  ~~>  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   _Vcvv 2970   class class class wbr 4290    _I cid 4629   ` cfv 5416  (class class class)co 6089   CCcc 9278    + caddc 9283   -ucneg 9594   ZZcz 10644   ZZ>=cuz 10859    shift cshi 12553    ~~> cli 12960
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-cnex 9336  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356  ax-pre-mulgt0 9357
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-om 6475  df-recs 6830  df-rdg 6864  df-er 7099  df-en 7309  df-dom 7310  df-sdom 7311  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-sub 9595  df-neg 9596  df-nn 10321  df-n0 10578  df-z 10645  df-uz 10860  df-shft 12554  df-clim 12964
This theorem is referenced by:  isercoll2  13144  trireciplem  13322  divcnvshft  27396  divcnvlin  27397
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