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Theorem climshft2 13416
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 6324 . . . 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 10991 . . . . . 6  |-  ( ph  ->  K  e.  CC )
8 eluzelz 11115 . . . . . . . 8  |-  ( k  e.  ( ZZ>= `  M
)  ->  k  e.  ZZ )
98, 1eleq2s 2565 . . . . . . 7  |-  ( k  e.  Z  ->  k  e.  ZZ )
109zcnd 10991 . . . . . 6  |-  ( k  e.  Z  ->  k  e.  CC )
11 fvex 5882 . . . . . . 7  |-  (  _I 
`  G )  e. 
_V
1211shftval4 12921 . . . . . 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 5930 . . . . . . . . 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 6311 . . . . . 6  |-  ( (
ph  /\  k  e.  Z )  ->  (
(  _I  `  G
)  shift  -u K )  =  ( G  shift  -u K
) )
1918fveq1d 5874 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  (
( (  _I  `  G )  shift  -u K
) `  k )  =  ( ( G 
shift  -u K ) `  k ) )
20 addcom 9783 . . . . . . 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 5878 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  (
(  _I  `  G
) `  ( K  +  k ) )  =  ( G `  ( k  +  K
) ) )
2313, 19, 223eqtr3d 2506 . . . 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 2498 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  (
( G  shift  -u K
) `  k )  =  ( F `  k ) )
261, 3, 4, 5, 25climeq 13401 . 2  |-  ( ph  ->  ( ( G  shift  -u K )  ~~>  A  <->  F  ~~>  A ) )
276znegcld 10992 . . 3  |-  ( ph  -> 
-u K  e.  ZZ )
28 climshft 13410 . . 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 1395    e. wcel 1819   _Vcvv 3109   class class class wbr 4456    _I cid 4799   ` cfv 5594  (class class class)co 6296   CCcc 9507    + caddc 9512   -ucneg 9825   ZZcz 10885   ZZ>=cuz 11106    shift cshi 12910    ~~> cli 13318
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-recs 7060  df-rdg 7094  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-n0 10817  df-z 10886  df-uz 11107  df-shft 12911  df-clim 13322
This theorem is referenced by:  isercoll2  13502  trireciplem  13684  divcnvshft  29292  divcnvlin  29293
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