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Theorem climshft2 12056
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 5883 . . . 4  |-  ( G 
shift  -u K )  e. 
_V
32a1i 10 . . 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 10118 . . . . . 6  |-  ( ph  ->  K  e.  CC )
8 eluzelz 10238 . . . . . . . 8  |-  ( k  e.  ( ZZ>= `  M
)  ->  k  e.  ZZ )
98, 1eleq2s 2375 . . . . . . 7  |-  ( k  e.  Z  ->  k  e.  ZZ )
109zcnd 10118 . . . . . 6  |-  ( k  e.  Z  ->  k  e.  CC )
11 fvex 5539 . . . . . . 7  |-  (  _I 
`  G )  e. 
_V
1211shftval4 11572 . . . . . 6  |-  ( ( K  e.  CC  /\  k  e.  CC )  ->  ( ( (  _I 
`  G )  shift  -u K ) `  k
)  =  ( (  _I  `  G ) `
 ( K  +  k ) ) )
137, 10, 12syl2an 463 . . . . 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 5579 . . . . . . . . 9  |-  ( G  e.  X  ->  (  _I  `  G )  =  G )
1614, 15syl 15 . . . . . . . 8  |-  ( ph  ->  (  _I  `  G
)  =  G )
1716adantr 451 . . . . . . 7  |-  ( (
ph  /\  k  e.  Z )  ->  (  _I  `  G )  =  G )
1817oveq1d 5873 . . . . . 6  |-  ( (
ph  /\  k  e.  Z )  ->  (
(  _I  `  G
)  shift  -u K )  =  ( G  shift  -u K
) )
1918fveq1d 5527 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  (
( (  _I  `  G )  shift  -u K
) `  k )  =  ( ( G 
shift  -u K ) `  k ) )
20 addcom 8998 . . . . . . 7  |-  ( ( K  e.  CC  /\  k  e.  CC )  ->  ( K  +  k )  =  ( k  +  K ) )
217, 10, 20syl2an 463 . . . . . 6  |-  ( (
ph  /\  k  e.  Z )  ->  ( K  +  k )  =  ( k  +  K ) )
2217, 21fveq12d 5531 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  (
(  _I  `  G
) `  ( K  +  k ) )  =  ( G `  ( k  +  K
) ) )
2313, 19, 223eqtr3d 2323 . . . 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 2315 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  (
( G  shift  -u K
) `  k )  =  ( F `  k ) )
261, 3, 4, 5, 25climeq 12041 . 2  |-  ( ph  ->  ( ( G  shift  -u K )  ~~>  A  <->  F  ~~>  A ) )
276znegcld 10119 . . 3  |-  ( ph  -> 
-u K  e.  ZZ )
28 climshft 12050 . . 3  |-  ( (
-u K  e.  ZZ  /\  G  e.  X )  ->  ( ( G 
shift  -u K )  ~~>  A  <->  G  ~~>  A ) )
2927, 14, 28syl2anc 642 . 2  |-  ( ph  ->  ( ( G  shift  -u K )  ~~>  A  <->  G  ~~>  A ) )
3026, 29bitr3d 246 1  |-  ( ph  ->  ( F  ~~>  A  <->  G  ~~>  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788   class class class wbr 4023    _I cid 4304   ` cfv 5255  (class class class)co 5858   CCcc 8735    + caddc 8740   -ucneg 9038   ZZcz 10024   ZZ>=cuz 10230    shift cshi 11561    ~~> cli 11958
This theorem is referenced by:  isercoll2  12142  trireciplem  12320
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-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-shft 11562  df-clim 11962
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