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Theorem climshft 10281
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 5550 . . . . . 6  |-  ( f  =  F  ->  (
f  shift  M )  =  ( F  shift  M ) )
21breq1d 3803 . . . . 5  |-  ( f  =  F  ->  (
( f  shift  M )  ~~>  A  <->  ( F  shift  M )  ~~>  A ) )
3 breq1 3796 . . . . 5  |-  ( f  =  F  ->  (
f  ~~>  A  <->  F  ~~>  A ) )
42, 3bibi12d 233 . . . 4  |-  ( f  =  F  ->  (
( ( f  shift  M )  ~~>  A  <->  f  ~~>  A )  <-> 
( ( F  shift  M )  ~~>  A  <->  F  ~~>  A ) ) )
54imbi2d 228 . . 3  |-  ( f  =  F  ->  (
( M  e.  ZZ  ->  ( ( f  shift  M )  ~~>  A  <->  f  ~~>  A ) )  <->  ( M  e.  ZZ  ->  ( ( F  shift  M )  ~~>  A  <->  F  ~~>  A ) ) ) )
6 znegcl 8463 . . . . . 6  |-  ( M  e.  ZZ  ->  -u M  e.  ZZ )
7 vex 2605 . . . . . . 7  |-  f  e. 
_V
8 zcn 8437 . . . . . . 7  |-  ( M  e.  ZZ  ->  M  e.  CC )
9 ovshftex 9845 . . . . . . 7  |-  ( ( f  e.  _V  /\  M  e.  CC )  ->  ( f  shift  M )  e.  _V )
107, 8, 9sylancr 405 . . . . . 6  |-  ( M  e.  ZZ  ->  (
f  shift  M )  e. 
_V )
11 climshftlemg 10279 . . . . . 6  |-  ( (
-u M  e.  ZZ  /\  ( f  shift  M )  e.  _V )  -> 
( ( f  shift  M )  ~~>  A  ->  (
( f  shift  M ) 
shift  -u M )  ~~>  A ) )
126, 10, 11syl2anc 403 . . . . 5  |-  ( M  e.  ZZ  ->  (
( f  shift  M )  ~~>  A  ->  ( (
f  shift  M )  shift  -u M )  ~~>  A ) )
13 eqid 2082 . . . . . 6  |-  ( ZZ>= `  M )  =  (
ZZ>= `  M )
148negcld 7473 . . . . . . 7  |-  ( M  e.  ZZ  ->  -u M  e.  CC )
15 ovshftex 9845 . . . . . . 7  |-  ( ( ( f  shift  M )  e.  _V  /\  -u M  e.  CC )  ->  (
( f  shift  M ) 
shift  -u M )  e. 
_V )
1610, 14, 15syl2anc 403 . . . . . 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 8711 . . . . . . 7  |-  ( k  e.  ( ZZ>= `  M
)  ->  k  e.  CC )
207shftcan1 9860 . . . . . . 7  |-  ( ( M  e.  CC  /\  k  e.  CC )  ->  ( ( ( f 
shift  M )  shift  -u M
) `  k )  =  ( f `  k ) )
218, 19, 20syl2an 283 . . . . . 6  |-  ( ( M  e.  ZZ  /\  k  e.  ( ZZ>= `  M ) )  -> 
( ( ( f 
shift  M )  shift  -u M
) `  k )  =  ( f `  k ) )
2213, 16, 17, 18, 21climeq 10276 . . . . 5  |-  ( M  e.  ZZ  ->  (
( ( f  shift  M )  shift  -u M )  ~~>  A  <->  f  ~~>  A ) )
2312, 22sylibd 147 . . . 4  |-  ( M  e.  ZZ  ->  (
( f  shift  M )  ~~>  A  ->  f  ~~>  A ) )
24 climshftlemg 10279 . . . . 5  |-  ( ( M  e.  ZZ  /\  f  e.  _V )  ->  ( f  ~~>  A  -> 
( f  shift  M )  ~~>  A ) )
257, 24mpan2 416 . . . 4  |-  ( M  e.  ZZ  ->  (
f  ~~>  A  ->  (
f  shift  M )  ~~>  A ) )
2623, 25impbid 127 . . 3  |-  ( M  e.  ZZ  ->  (
( f  shift  M )  ~~>  A  <->  f  ~~>  A ) )
275, 26vtoclg 2659 . 2  |-  ( F  e.  V  ->  ( M  e.  ZZ  ->  ( ( F  shift  M )  ~~>  A  <->  F  ~~>  A )
) )
2827impcom 123 1  |-  ( ( M  e.  ZZ  /\  F  e.  V )  ->  ( ( F  shift  M )  ~~>  A  <->  F  ~~>  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1285    e. wcel 1434   _Vcvv 2602   class class class wbr 3793   ` cfv 4932  (class class class)co 5543   CCcc 7041   -ucneg 7347   ZZcz 8432   ZZ>=cuz 8700    shift cshi 9840    ~~> cli 10255
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3901  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-cnex 7129  ax-resscn 7130  ax-1cn 7131  ax-1re 7132  ax-icn 7133  ax-addcl 7134  ax-addrcl 7135  ax-mulcl 7136  ax-addcom 7138  ax-addass 7140  ax-distr 7142  ax-i2m1 7143  ax-0lt1 7144  ax-0id 7146  ax-rnegex 7147  ax-cnre 7149  ax-pre-ltirr 7150  ax-pre-ltwlin 7151  ax-pre-lttrn 7152  ax-pre-apti 7153  ax-pre-ltadd 7154
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-nel 2341  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-if 3360  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-riota 5499  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-pnf 7217  df-mnf 7218  df-xr 7219  df-ltxr 7220  df-le 7221  df-sub 7348  df-neg 7349  df-inn 8107  df-n0 8356  df-z 8433  df-uz 8701  df-shft 9841  df-clim 10256
This theorem is referenced by:  climshft2  10283
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