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Theorem climshft2 10283
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 8551 . . . . 5  |-  ( ph  ->  K  e.  CC )
54negcld 7473 . . . 4  |-  ( ph  -> 
-u K  e.  CC )
6 ovshftex 9845 . . . 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 4962 . . . . . . . 8  |-  Fun  _I
11 elex 2611 . . . . . . . . . 10  |-  ( G  e.  X  ->  G  e.  _V )
122, 11syl 14 . . . . . . . . 9  |-  ( ph  ->  G  e.  _V )
13 dmi 4578 . . . . . . . . 9  |-  dom  _I  =  _V
1412, 13syl6eleqr 2173 . . . . . . . 8  |-  ( ph  ->  G  e.  dom  _I  )
15 funfvex 5223 . . . . . . . 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 8709 . . . . . . . . 9  |-  ( k  e.  ( ZZ>= `  M
)  ->  k  e.  ZZ )
2019, 1eleq2s 2174 . . . . . . . 8  |-  ( k  e.  Z  ->  k  e.  ZZ )
2120zcnd 8551 . . . . . . 7  |-  ( k  e.  Z  ->  k  e.  CC )
2221adantl 271 . . . . . 6  |-  ( (
ph  /\  k  e.  Z )  ->  k  e.  CC )
23 shftval4g 9863 . . . . . 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 1170 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  (
( (  _I  `  G )  shift  -u K
) `  k )  =  ( (  _I 
`  G ) `  ( K  +  k
) ) )
25 fvi 5262 . . . . . . . . 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 5558 . . . . . 6  |-  ( (
ph  /\  k  e.  Z )  ->  (
(  _I  `  G
)  shift  -u K )  =  ( G  shift  -u K
) )
2928fveq1d 5211 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  (
( (  _I  `  G )  shift  -u K
) `  k )  =  ( ( G 
shift  -u K ) `  k ) )
30 addcom 7312 . . . . . . 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 5215 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  (
(  _I  `  G
) `  ( K  +  k ) )  =  ( G `  ( k  +  K
) ) )
3324, 29, 323eqtr3d 2122 . . . 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 2114 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  (
( G  shift  -u K
) `  k )  =  ( F `  k ) )
361, 7, 8, 9, 35climeq 10276 . 2  |-  ( ph  ->  ( ( G  shift  -u K )  ~~>  A  <->  F  ~~>  A ) )
373znegcld 8552 . . 3  |-  ( ph  -> 
-u K  e.  ZZ )
38 climshft 10281 . . 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 1285    e. wcel 1434   _Vcvv 2602   class class class wbr 3793    _I cid 4051   dom cdm 4371   Fun wfun 4926   ` cfv 4932  (class class class)co 5543   CCcc 7041    + caddc 7046   -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: (None)
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