ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  climshft2 Unicode version

Theorem climshft2 11043
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 9142 . . . . 5  |-  ( ph  ->  K  e.  CC )
54negcld 8028 . . . 4  |-  ( ph  -> 
-u K  e.  CC )
6 ovshftex 10559 . . . 4  |-  ( ( G  e.  X  /\  -u K  e.  CC )  ->  ( G  shift  -u K )  e.  _V )
72, 5, 6syl2anc 408 . . 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 5125 . . . . . . . 8  |-  Fun  _I
11 elex 2671 . . . . . . . . . 10  |-  ( G  e.  X  ->  G  e.  _V )
122, 11syl 14 . . . . . . . . 9  |-  ( ph  ->  G  e.  _V )
13 dmi 4724 . . . . . . . . 9  |-  dom  _I  =  _V
1412, 13eleqtrrdi 2211 . . . . . . . 8  |-  ( ph  ->  G  e.  dom  _I  )
15 funfvex 5406 . . . . . . . 8  |-  ( ( Fun  _I  /\  G  e.  dom  _I  )  -> 
(  _I  `  G
)  e.  _V )
1610, 14, 15sylancr 410 . . . . . . 7  |-  ( ph  ->  (  _I  `  G
)  e.  _V )
1716adantr 274 . . . . . 6  |-  ( (
ph  /\  k  e.  Z )  ->  (  _I  `  G )  e. 
_V )
184adantr 274 . . . . . 6  |-  ( (
ph  /\  k  e.  Z )  ->  K  e.  CC )
19 eluzelz 9303 . . . . . . . . 9  |-  ( k  e.  ( ZZ>= `  M
)  ->  k  e.  ZZ )
2019, 1eleq2s 2212 . . . . . . . 8  |-  ( k  e.  Z  ->  k  e.  ZZ )
2120zcnd 9142 . . . . . . 7  |-  ( k  e.  Z  ->  k  e.  CC )
2221adantl 275 . . . . . 6  |-  ( (
ph  /\  k  e.  Z )  ->  k  e.  CC )
23 shftval4g 10577 . . . . . 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 1201 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  (
( (  _I  `  G )  shift  -u K
) `  k )  =  ( (  _I 
`  G ) `  ( K  +  k
) ) )
25 fvi 5446 . . . . . . . . 9  |-  ( G  e.  X  ->  (  _I  `  G )  =  G )
262, 25syl 14 . . . . . . . 8  |-  ( ph  ->  (  _I  `  G
)  =  G )
2726adantr 274 . . . . . . 7  |-  ( (
ph  /\  k  e.  Z )  ->  (  _I  `  G )  =  G )
2827oveq1d 5757 . . . . . 6  |-  ( (
ph  /\  k  e.  Z )  ->  (
(  _I  `  G
)  shift  -u K )  =  ( G  shift  -u K
) )
2928fveq1d 5391 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  (
( (  _I  `  G )  shift  -u K
) `  k )  =  ( ( G 
shift  -u K ) `  k ) )
30 addcom 7867 . . . . . . 7  |-  ( ( K  e.  CC  /\  k  e.  CC )  ->  ( K  +  k )  =  ( k  +  K ) )
314, 21, 30syl2an 287 . . . . . 6  |-  ( (
ph  /\  k  e.  Z )  ->  ( K  +  k )  =  ( k  +  K ) )
3227, 31fveq12d 5396 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  (
(  _I  `  G
) `  ( K  +  k ) )  =  ( G `  ( k  +  K
) ) )
3324, 29, 323eqtr3d 2158 . . . 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 2150 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  (
( G  shift  -u K
) `  k )  =  ( F `  k ) )
361, 7, 8, 9, 35climeq 11036 . 2  |-  ( ph  ->  ( ( G  shift  -u K )  ~~>  A  <->  F  ~~>  A ) )
373znegcld 9143 . . 3  |-  ( ph  -> 
-u K  e.  ZZ )
38 climshft 11041 . . 3  |-  ( (
-u K  e.  ZZ  /\  G  e.  X )  ->  ( ( G 
shift  -u K )  ~~>  A  <->  G  ~~>  A ) )
3937, 2, 38syl2anc 408 . 2  |-  ( ph  ->  ( ( G  shift  -u K )  ~~>  A  <->  G  ~~>  A ) )
4036, 39bitr3d 189 1  |-  ( ph  ->  ( F  ~~>  A  <->  G  ~~>  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1316    e. wcel 1465   _Vcvv 2660   class class class wbr 3899    _I cid 4180   dom cdm 4509   Fun wfun 5087   ` cfv 5093  (class class class)co 5742   CCcc 7586    + caddc 7591   -ucneg 7902   ZZcz 9022   ZZ>=cuz 9294    shift cshi 10554    ~~> cli 11015
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-1re 7682  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-addcom 7688  ax-addass 7690  ax-distr 7692  ax-i2m1 7693  ax-0lt1 7694  ax-0id 7696  ax-rnegex 7697  ax-cnre 7699  ax-pre-ltirr 7700  ax-pre-ltwlin 7701  ax-pre-lttrn 7702  ax-pre-apti 7703  ax-pre-ltadd 7704
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-if 3445  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-pnf 7770  df-mnf 7771  df-xr 7772  df-ltxr 7773  df-le 7774  df-sub 7903  df-neg 7904  df-inn 8689  df-n0 8946  df-z 9023  df-uz 9295  df-shft 10555  df-clim 11016
This theorem is referenced by:  trireciplem  11237
  Copyright terms: Public domain W3C validator