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Theorem climshft 11615
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 5951 . . . . . 6  |-  ( f  =  F  ->  (
f  shift  M )  =  ( F  shift  M ) )
21breq1d 4054 . . . . 5  |-  ( f  =  F  ->  (
( f  shift  M )  ~~>  A  <->  ( F  shift  M )  ~~>  A ) )
3 breq1 4047 . . . . 5  |-  ( f  =  F  ->  (
f  ~~>  A  <->  F  ~~>  A ) )
42, 3bibi12d 235 . . . 4  |-  ( f  =  F  ->  (
( ( f  shift  M )  ~~>  A  <->  f  ~~>  A )  <-> 
( ( F  shift  M )  ~~>  A  <->  F  ~~>  A ) ) )
54imbi2d 230 . . 3  |-  ( f  =  F  ->  (
( M  e.  ZZ  ->  ( ( f  shift  M )  ~~>  A  <->  f  ~~>  A ) )  <->  ( M  e.  ZZ  ->  ( ( F  shift  M )  ~~>  A  <->  F  ~~>  A ) ) ) )
6 znegcl 9403 . . . . . 6  |-  ( M  e.  ZZ  ->  -u M  e.  ZZ )
7 vex 2775 . . . . . . 7  |-  f  e. 
_V
8 zcn 9377 . . . . . . 7  |-  ( M  e.  ZZ  ->  M  e.  CC )
9 ovshftex 11130 . . . . . . 7  |-  ( ( f  e.  _V  /\  M  e.  CC )  ->  ( f  shift  M )  e.  _V )
107, 8, 9sylancr 414 . . . . . 6  |-  ( M  e.  ZZ  ->  (
f  shift  M )  e. 
_V )
11 climshftlemg 11613 . . . . . 6  |-  ( (
-u M  e.  ZZ  /\  ( f  shift  M )  e.  _V )  -> 
( ( f  shift  M )  ~~>  A  ->  (
( f  shift  M ) 
shift  -u M )  ~~>  A ) )
126, 10, 11syl2anc 411 . . . . 5  |-  ( M  e.  ZZ  ->  (
( f  shift  M )  ~~>  A  ->  ( (
f  shift  M )  shift  -u M )  ~~>  A ) )
13 eqid 2205 . . . . . 6  |-  ( ZZ>= `  M )  =  (
ZZ>= `  M )
148negcld 8370 . . . . . . 7  |-  ( M  e.  ZZ  ->  -u M  e.  CC )
15 ovshftex 11130 . . . . . . 7  |-  ( ( ( f  shift  M )  e.  _V  /\  -u M  e.  CC )  ->  (
( f  shift  M ) 
shift  -u M )  e. 
_V )
1610, 14, 15syl2anc 411 . . . . . 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 9659 . . . . . . 7  |-  ( k  e.  ( ZZ>= `  M
)  ->  k  e.  CC )
207shftcan1 11145 . . . . . . 7  |-  ( ( M  e.  CC  /\  k  e.  CC )  ->  ( ( ( f 
shift  M )  shift  -u M
) `  k )  =  ( f `  k ) )
218, 19, 20syl2an 289 . . . . . 6  |-  ( ( M  e.  ZZ  /\  k  e.  ( ZZ>= `  M ) )  -> 
( ( ( f 
shift  M )  shift  -u M
) `  k )  =  ( f `  k ) )
2213, 16, 17, 18, 21climeq 11610 . . . . 5  |-  ( M  e.  ZZ  ->  (
( ( f  shift  M )  shift  -u M )  ~~>  A  <->  f  ~~>  A ) )
2312, 22sylibd 149 . . . 4  |-  ( M  e.  ZZ  ->  (
( f  shift  M )  ~~>  A  ->  f  ~~>  A ) )
24 climshftlemg 11613 . . . . 5  |-  ( ( M  e.  ZZ  /\  f  e.  _V )  ->  ( f  ~~>  A  -> 
( f  shift  M )  ~~>  A ) )
257, 24mpan2 425 . . . 4  |-  ( M  e.  ZZ  ->  (
f  ~~>  A  ->  (
f  shift  M )  ~~>  A ) )
2623, 25impbid 129 . . 3  |-  ( M  e.  ZZ  ->  (
( f  shift  M )  ~~>  A  <->  f  ~~>  A ) )
275, 26vtoclg 2833 . 2  |-  ( F  e.  V  ->  ( M  e.  ZZ  ->  ( ( F  shift  M )  ~~>  A  <->  F  ~~>  A )
) )
2827impcom 125 1  |-  ( ( M  e.  ZZ  /\  F  e.  V )  ->  ( ( F  shift  M )  ~~>  A  <->  F  ~~>  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1373    e. wcel 2176   _Vcvv 2772   class class class wbr 4044   ` cfv 5271  (class class class)co 5944   CCcc 7923   -ucneg 8244   ZZcz 9372   ZZ>=cuz 9648    shift cshi 11125    ~~> cli 11589
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-addcom 8025  ax-addass 8027  ax-distr 8029  ax-i2m1 8030  ax-0lt1 8031  ax-0id 8033  ax-rnegex 8034  ax-cnre 8036  ax-pre-ltirr 8037  ax-pre-ltwlin 8038  ax-pre-lttrn 8039  ax-pre-apti 8040  ax-pre-ltadd 8041
This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-if 3572  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-pnf 8109  df-mnf 8110  df-xr 8111  df-ltxr 8112  df-le 8113  df-sub 8245  df-neg 8246  df-inn 9037  df-n0 9296  df-z 9373  df-uz 9649  df-shft 11126  df-clim 11590
This theorem is referenced by:  climshft2  11617  iser3shft  11657  eftlub  12001
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