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Theorem climshft 11260
Description: A shifted function converges iff the original function converges. (Contributed by NM, 16-Aug-2005.) (Revised by Mario Carneiro, 31-Jan-2014.)
Assertion
Ref Expression
climshft  |-  ( ( M  e.  ZZ  /\  F  e.  V )  ->  ( ( F  shift  M )  ~~>  A  <->  F  ~~>  A ) )

Proof of Theorem climshft
Dummy variables  f  k are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq1 5858 . . . . . 6  |-  ( f  =  F  ->  (
f  shift  M )  =  ( F  shift  M ) )
21breq1d 3997 . . . . 5  |-  ( f  =  F  ->  (
( f  shift  M )  ~~>  A  <->  ( F  shift  M )  ~~>  A ) )
3 breq1 3990 . . . . 5  |-  ( f  =  F  ->  (
f  ~~>  A  <->  F  ~~>  A ) )
42, 3bibi12d 234 . . . 4  |-  ( f  =  F  ->  (
( ( f  shift  M )  ~~>  A  <->  f  ~~>  A )  <-> 
( ( F  shift  M )  ~~>  A  <->  F  ~~>  A ) ) )
54imbi2d 229 . . 3  |-  ( f  =  F  ->  (
( M  e.  ZZ  ->  ( ( f  shift  M )  ~~>  A  <->  f  ~~>  A ) )  <->  ( M  e.  ZZ  ->  ( ( F  shift  M )  ~~>  A  <->  F  ~~>  A ) ) ) )
6 znegcl 9236 . . . . . 6  |-  ( M  e.  ZZ  ->  -u M  e.  ZZ )
7 vex 2733 . . . . . . 7  |-  f  e. 
_V
8 zcn 9210 . . . . . . 7  |-  ( M  e.  ZZ  ->  M  e.  CC )
9 ovshftex 10776 . . . . . . 7  |-  ( ( f  e.  _V  /\  M  e.  CC )  ->  ( f  shift  M )  e.  _V )
107, 8, 9sylancr 412 . . . . . 6  |-  ( M  e.  ZZ  ->  (
f  shift  M )  e. 
_V )
11 climshftlemg 11258 . . . . . 6  |-  ( (
-u M  e.  ZZ  /\  ( f  shift  M )  e.  _V )  -> 
( ( f  shift  M )  ~~>  A  ->  (
( f  shift  M ) 
shift  -u M )  ~~>  A ) )
126, 10, 11syl2anc 409 . . . . 5  |-  ( M  e.  ZZ  ->  (
( f  shift  M )  ~~>  A  ->  ( (
f  shift  M )  shift  -u M )  ~~>  A ) )
13 eqid 2170 . . . . . 6  |-  ( ZZ>= `  M )  =  (
ZZ>= `  M )
148negcld 8210 . . . . . . 7  |-  ( M  e.  ZZ  ->  -u M  e.  CC )
15 ovshftex 10776 . . . . . . 7  |-  ( ( ( f  shift  M )  e.  _V  /\  -u M  e.  CC )  ->  (
( f  shift  M ) 
shift  -u M )  e. 
_V )
1610, 14, 15syl2anc 409 . . . . . 6  |-  ( M  e.  ZZ  ->  (
( f  shift  M ) 
shift  -u M )  e. 
_V )
177a1i 9 . . . . . 6  |-  ( M  e.  ZZ  ->  f  e.  _V )
18 id 19 . . . . . 6  |-  ( M  e.  ZZ  ->  M  e.  ZZ )
19 eluzelcn 9491 . . . . . . 7  |-  ( k  e.  ( ZZ>= `  M
)  ->  k  e.  CC )
207shftcan1 10791 . . . . . . 7  |-  ( ( M  e.  CC  /\  k  e.  CC )  ->  ( ( ( f 
shift  M )  shift  -u M
) `  k )  =  ( f `  k ) )
218, 19, 20syl2an 287 . . . . . 6  |-  ( ( M  e.  ZZ  /\  k  e.  ( ZZ>= `  M ) )  -> 
( ( ( f 
shift  M )  shift  -u M
) `  k )  =  ( f `  k ) )
2213, 16, 17, 18, 21climeq 11255 . . . . 5  |-  ( M  e.  ZZ  ->  (
( ( f  shift  M )  shift  -u M )  ~~>  A  <->  f  ~~>  A ) )
2312, 22sylibd 148 . . . 4  |-  ( M  e.  ZZ  ->  (
( f  shift  M )  ~~>  A  ->  f  ~~>  A ) )
24 climshftlemg 11258 . . . . 5  |-  ( ( M  e.  ZZ  /\  f  e.  _V )  ->  ( f  ~~>  A  -> 
( f  shift  M )  ~~>  A ) )
257, 24mpan2 423 . . . 4  |-  ( M  e.  ZZ  ->  (
f  ~~>  A  ->  (
f  shift  M )  ~~>  A ) )
2623, 25impbid 128 . . 3  |-  ( M  e.  ZZ  ->  (
( f  shift  M )  ~~>  A  <->  f  ~~>  A ) )
275, 26vtoclg 2790 . 2  |-  ( F  e.  V  ->  ( M  e.  ZZ  ->  ( ( F  shift  M )  ~~>  A  <->  F  ~~>  A )
) )
2827impcom 124 1  |-  ( ( M  e.  ZZ  /\  F  e.  V )  ->  ( ( F  shift  M )  ~~>  A  <->  F  ~~>  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1348    e. wcel 2141   _Vcvv 2730   class class class wbr 3987   ` cfv 5196  (class class class)co 5851   CCcc 7765   -ucneg 8084   ZZcz 9205   ZZ>=cuz 9480    shift cshi 10771    ~~> cli 11234
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-cnex 7858  ax-resscn 7859  ax-1cn 7860  ax-1re 7861  ax-icn 7862  ax-addcl 7863  ax-addrcl 7864  ax-mulcl 7865  ax-addcom 7867  ax-addass 7869  ax-distr 7871  ax-i2m1 7872  ax-0lt1 7873  ax-0id 7875  ax-rnegex 7876  ax-cnre 7878  ax-pre-ltirr 7879  ax-pre-ltwlin 7880  ax-pre-lttrn 7881  ax-pre-apti 7882  ax-pre-ltadd 7883
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-if 3526  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-id 4276  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-riota 5807  df-ov 5854  df-oprab 5855  df-mpo 5856  df-pnf 7949  df-mnf 7950  df-xr 7951  df-ltxr 7952  df-le 7953  df-sub 8085  df-neg 8086  df-inn 8872  df-n0 9129  df-z 9206  df-uz 9481  df-shft 10772  df-clim 11235
This theorem is referenced by:  climshft2  11262  iser3shft  11302  eftlub  11646
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