Theorem climshft2 10967

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

Step	Hyp	Ref	Expression
1		climshft2.1	. . 3 |- Z = ( ZZ>= ` M )
2		climshft2.6	. . . 4 |- ( ph -> G e. X )
3		climshft2.3	. . . . . 6 |- ( ph -> K e. ZZ )
4	3	zcnd 9078	. . . . 5 |- ( ph -> K e. CC )
5	4	negcld 7983	. . . 4 |- ( ph -> -u K e. CC )
6		ovshftex 10484	. . . 4 |- ( ( G e. X /\ -u K e. CC ) -> ( G shift -u K ) e. _V )
7	2, 5, 6	syl2anc 406	. . 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 5113	. . . . . . . 8 |- Fun _I
11		elex 2668	. . . . . . . . . 10 |- ( G e. X -> G e. _V )
12	2, 11	syl 14	. . . . . . . . 9 |- ( ph -> G e. _V )
13		dmi 4714	. . . . . . . . 9 |- dom _I = _V
14	12, 13	syl6eleqr 2208	. . . . . . . 8 |- ( ph -> G e. dom _I )
15		funfvex 5392	. . . . . . . 8 |- ( ( Fun _I /\ G e. dom _I ) -> ( _I ` G ) e. _V )
16	10, 14, 15	sylancr 408	. . . . . . 7 |- ( ph -> ( _I ` G ) e. _V )
17	16	adantr 272	. . . . . 6 |- ( ( ph /\ k e. Z ) -> ( _I ` G ) e. _V )
18	4	adantr 272	. . . . . 6 |- ( ( ph /\ k e. Z ) -> K e. CC )
19		eluzelz 9237	. . . . . . . . 9 |- ( k e. ( ZZ>= ` M ) -> k e. ZZ )
20	19, 1	eleq2s 2209	. . . . . . . 8 |- ( k e. Z -> k e. ZZ )
21	20	zcnd 9078	. . . . . . 7 |- ( k e. Z -> k e. CC )
22	21	adantl 273	. . . . . 6 |- ( ( ph /\ k e. Z ) -> k e. CC )
23		shftval4g 10502	. . . . . 6 |- ( ( ( _I ` G ) e. _V /\ K e. CC /\ k e. CC ) -> ( ( ( _I ` G ) shift -u K ) ` k ) = ( ( _I ` G ) ` ( K + k ) ) )
24	17, 18, 22, 23	syl3anc 1199	. . . . 5 |- ( ( ph /\ k e. Z ) -> ( ( ( _I ` G ) shift -u K ) ` k ) = ( ( _I ` G ) ` ( K + k ) ) )
25		fvi 5432	. . . . . . . . 9 |- ( G e. X -> ( _I ` G ) = G )
26	2, 25	syl 14	. . . . . . . 8 |- ( ph -> ( _I ` G ) = G )
27	26	adantr 272	. . . . . . 7 |- ( ( ph /\ k e. Z ) -> ( _I ` G ) = G )
28	27	oveq1d 5743	. . . . . 6 |- ( ( ph /\ k e. Z ) -> ( ( _I ` G ) shift -u K ) = ( G shift -u K ) )
29	28	fveq1d 5377	. . . . 5 |- ( ( ph /\ k e. Z ) -> ( ( ( _I ` G ) shift -u K ) ` k ) = ( ( G shift -u K ) ` k ) )
30		addcom 7822	. . . . . . 7 |- ( ( K e. CC /\ k e. CC ) -> ( K + k ) = ( k + K ) )
31	4, 21, 30	syl2an 285	. . . . . 6 |- ( ( ph /\ k e. Z ) -> ( K + k ) = ( k + K ) )
32	27, 31	fveq12d 5382	. . . . 5 |- ( ( ph /\ k e. Z ) -> ( ( _I ` G ) ` ( K + k ) ) = ( G ` ( k + K ) ) )
33	24, 29, 32	3eqtr3d 2155	. . . 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 ) )
35	33, 34	eqtrd 2147	. . 3 |- ( ( ph /\ k e. Z ) -> ( ( G shift -u K ) ` k ) = ( F ` k ) )
36	1, 7, 8, 9, 35	climeq 10960	. 2 |- ( ph -> ( ( G shift -u K ) ~~> A <-> F ~~> A ) )
37	3	znegcld 9079	. . 3 |- ( ph -> -u K e. ZZ )
38		climshft 10965	. . 3 |- ( ( -u K e. ZZ /\ G e. X ) -> ( ( G shift -u K ) ~~> A <-> G ~~> A ) )
39	37, 2, 38	syl2anc 406	. 2 |- ( ph -> ( ( G shift -u K ) ~~> A <-> G ~~> A ) )
40	36, 39	bitr3d 189	1 |- ( ph -> ( F ~~> A <-> G ~~> A ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1314   e. wcel 1463   _Vcvv 2657   class class class wbr 3895   _I cid 4170   dom cdm 4499   Fun wfun 5075   ` cfv 5081  (class class class)co 5728   CCcc 7545   + caddc 7550   -ucneg 7857   ZZcz 8958   ZZ>=cuz 9228   shift cshi 10479   ~~> cli 10939

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4003  ax-sep 4006  ax-pow 4058  ax-pr 4091  ax-un 4315  ax-setind 4412  ax-cnex 7636  ax-resscn 7637  ax-1cn 7638  ax-1re 7639  ax-icn 7640  ax-addcl 7641  ax-addrcl 7642  ax-mulcl 7643  ax-addcom 7645  ax-addass 7647  ax-distr 7649  ax-i2m1 7650  ax-0lt1 7651  ax-0id 7653  ax-rnegex 7654  ax-cnre 7656  ax-pre-ltirr 7657  ax-pre-ltwlin 7658  ax-pre-lttrn 7659  ax-pre-apti 7660  ax-pre-ltadd 7661

This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ne 2283  df-nel 2378  df-ral 2395  df-rex 2396  df-reu 2397  df-rab 2399  df-v 2659  df-sbc 2879  df-csb 2972  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050  df-if 3441  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-int 3738  df-iun 3781  df-br 3896  df-opab 3950  df-mpt 3951  df-id 4175  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512  df-iota 5046  df-fun 5083  df-fn 5084  df-f 5085  df-f1 5086  df-fo 5087  df-f1o 5088  df-fv 5089  df-riota 5684  df-ov 5731  df-oprab 5732  df-mpo 5733  df-pnf 7726  df-mnf 7727  df-xr 7728  df-ltxr 7729  df-le 7730  df-sub 7858  df-neg 7859  df-inn 8631  df-n0 8882  df-z 8959  df-uz 9229  df-shft 10480  df-clim 10940

This theorem is referenced by:  trireciplem  11161