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Theorem climshft2 10691
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 climshft2.6 . . . 4  |-  ( ph  ->  G  e.  X )
3 climshft2.3 . . . . . 6  |-  ( ph  ->  K  e.  ZZ )
43zcnd 8867 . . . . 5  |-  ( ph  ->  K  e.  CC )
54negcld 7778 . . . 4  |-  ( ph  -> 
-u K  e.  CC )
6 ovshftex 10249 . . . 4  |-  ( ( G  e.  X  /\  -u K  e.  CC )  ->  ( G  shift  -u K )  e.  _V )
72, 5, 6syl2anc 403 . . 3  |-  ( ph  ->  ( G  shift  -u K
)  e.  _V )
8 climshft2.5 . . 3  |-  ( ph  ->  F  e.  W )
9 climshft2.2 . . 3  |-  ( ph  ->  M  e.  ZZ )
10 funi 5046 . . . . . . . 8  |-  Fun  _I
11 elex 2630 . . . . . . . . . 10  |-  ( G  e.  X  ->  G  e.  _V )
122, 11syl 14 . . . . . . . . 9  |-  ( ph  ->  G  e.  _V )
13 dmi 4651 . . . . . . . . 9  |-  dom  _I  =  _V
1412, 13syl6eleqr 2181 . . . . . . . 8  |-  ( ph  ->  G  e.  dom  _I  )
15 funfvex 5322 . . . . . . . 8  |-  ( ( Fun  _I  /\  G  e.  dom  _I  )  -> 
(  _I  `  G
)  e.  _V )
1610, 14, 15sylancr 405 . . . . . . 7  |-  ( ph  ->  (  _I  `  G
)  e.  _V )
1716adantr 270 . . . . . 6  |-  ( (
ph  /\  k  e.  Z )  ->  (  _I  `  G )  e. 
_V )
184adantr 270 . . . . . 6  |-  ( (
ph  /\  k  e.  Z )  ->  K  e.  CC )
19 eluzelz 9026 . . . . . . . . 9  |-  ( k  e.  ( ZZ>= `  M
)  ->  k  e.  ZZ )
2019, 1eleq2s 2182 . . . . . . . 8  |-  ( k  e.  Z  ->  k  e.  ZZ )
2120zcnd 8867 . . . . . . 7  |-  ( k  e.  Z  ->  k  e.  CC )
2221adantl 271 . . . . . 6  |-  ( (
ph  /\  k  e.  Z )  ->  k  e.  CC )
23 shftval4g 10267 . . . . . 6  |-  ( ( (  _I  `  G
)  e.  _V  /\  K  e.  CC  /\  k  e.  CC )  ->  (
( (  _I  `  G )  shift  -u K
) `  k )  =  ( (  _I 
`  G ) `  ( K  +  k
) ) )
2417, 18, 22, 23syl3anc 1174 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  (
( (  _I  `  G )  shift  -u K
) `  k )  =  ( (  _I 
`  G ) `  ( K  +  k
) ) )
25 fvi 5361 . . . . . . . . 9  |-  ( G  e.  X  ->  (  _I  `  G )  =  G )
262, 25syl 14 . . . . . . . 8  |-  ( ph  ->  (  _I  `  G
)  =  G )
2726adantr 270 . . . . . . 7  |-  ( (
ph  /\  k  e.  Z )  ->  (  _I  `  G )  =  G )
2827oveq1d 5667 . . . . . 6  |-  ( (
ph  /\  k  e.  Z )  ->  (
(  _I  `  G
)  shift  -u K )  =  ( G  shift  -u K
) )
2928fveq1d 5307 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  (
( (  _I  `  G )  shift  -u K
) `  k )  =  ( ( G 
shift  -u K ) `  k ) )
30 addcom 7617 . . . . . . 7  |-  ( ( K  e.  CC  /\  k  e.  CC )  ->  ( K  +  k )  =  ( k  +  K ) )
314, 21, 30syl2an 283 . . . . . 6  |-  ( (
ph  /\  k  e.  Z )  ->  ( K  +  k )  =  ( k  +  K ) )
3227, 31fveq12d 5312 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  (
(  _I  `  G
) `  ( K  +  k ) )  =  ( G `  ( k  +  K
) ) )
3324, 29, 323eqtr3d 2128 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  (
( G  shift  -u K
) `  k )  =  ( G `  ( k  +  K
) ) )
34 climshft2.7 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  ( k  +  K ) )  =  ( F `  k
) )
3533, 34eqtrd 2120 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  (
( G  shift  -u K
) `  k )  =  ( F `  k ) )
361, 7, 8, 9, 35climeq 10683 . 2  |-  ( ph  ->  ( ( G  shift  -u K )  ~~>  A  <->  F  ~~>  A ) )
373znegcld 8868 . . 3  |-  ( ph  -> 
-u K  e.  ZZ )
38 climshft 10688 . . 3  |-  ( (
-u K  e.  ZZ  /\  G  e.  X )  ->  ( ( G 
shift  -u K )  ~~>  A  <->  G  ~~>  A ) )
3937, 2, 38syl2anc 403 . 2  |-  ( ph  ->  ( ( G  shift  -u K )  ~~>  A  <->  G  ~~>  A ) )
4036, 39bitr3d 188 1  |-  ( ph  ->  ( F  ~~>  A  <->  G  ~~>  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1289    e. wcel 1438   _Vcvv 2619   class class class wbr 3845    _I cid 4115   dom cdm 4438   Fun wfun 5009   ` cfv 5015  (class class class)co 5652   CCcc 7346    + caddc 7351   -ucneg 7652   ZZcz 8748   ZZ>=cuz 9017    shift cshi 10244    ~~> cli 10662
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-cnex 7434  ax-resscn 7435  ax-1cn 7436  ax-1re 7437  ax-icn 7438  ax-addcl 7439  ax-addrcl 7440  ax-mulcl 7441  ax-addcom 7443  ax-addass 7445  ax-distr 7447  ax-i2m1 7448  ax-0lt1 7449  ax-0id 7451  ax-rnegex 7452  ax-cnre 7454  ax-pre-ltirr 7455  ax-pre-ltwlin 7456  ax-pre-lttrn 7457  ax-pre-apti 7458  ax-pre-ltadd 7459
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-if 3394  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-riota 5608  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-pnf 7522  df-mnf 7523  df-xr 7524  df-ltxr 7525  df-le 7526  df-sub 7653  df-neg 7654  df-inn 8421  df-n0 8672  df-z 8749  df-uz 9018  df-shft 10245  df-clim 10663
This theorem is referenced by:  trireciplem  10890
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