Theorem climshft2 12072

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 9769	. . . . 5 |- ( ph -> K e. CC )
5	4	negcld 8624	. . . 4 |- ( ph -> -u K e. CC )
6		ovshftex 11584	. . . 4 |- ( ( G e. X /\ -u K e. CC ) -> ( G shift -u K ) e. _V )
7	2, 5, 6	syl2anc 415	. . 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 5409	. . . . . . . 8 |- Fun _I
11		elex 2833	. . . . . . . . . 10 |- ( G e. X -> G e. _V )
12	2, 11	syl 14	. . . . . . . . 9 |- ( ph -> G e. _V )
13		dmi 4996	. . . . . . . . 9 |- dom _I = _V
14	12, 13	eleqtrrdi 2332	. . . . . . . 8 |- ( ph -> G e. dom _I )
15		funfvex 5712	. . . . . . . 8 |- ( ( Fun _I /\ G e. dom _I ) -> ( _I ` G ) e. _V )
16	10, 14, 15	sylancr 418	. . . . . . 7 |- ( ph -> ( _I ` G ) e. _V )
17	16	adantr 276	. . . . . 6 |- ( ( ph /\ k e. Z ) -> ( _I ` G ) e. _V )
18	4	adantr 276	. . . . . 6 |- ( ( ph /\ k e. Z ) -> K e. CC )
19		eluzelz 9931	. . . . . . . . 9 |- ( k e. ( ZZ>= ` M ) -> k e. ZZ )
20	19, 1	eleq2s 2333	. . . . . . . 8 |- ( k e. Z -> k e. ZZ )
21	20	zcnd 9769	. . . . . . 7 |- ( k e. Z -> k e. CC )
22	21	adantl 277	. . . . . 6 |- ( ( ph /\ k e. Z ) -> k e. CC )
23		shftval4g 11602	. . . . . 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 1278	. . . . 5 |- ( ( ph /\ k e. Z ) -> ( ( ( _I ` G ) shift -u K ) ` k ) = ( ( _I ` G ) ` ( K + k ) ) )
25		fvi 5760	. . . . . . . . 9 |- ( G e. X -> ( _I ` G ) = G )
26	2, 25	syl 14	. . . . . . . 8 |- ( ph -> ( _I ` G ) = G )
27	26	adantr 276	. . . . . . 7 |- ( ( ph /\ k e. Z ) -> ( _I ` G ) = G )
28	27	oveq1d 6100	. . . . . 6 |- ( ( ph /\ k e. Z ) -> ( ( _I ` G ) shift -u K ) = ( G shift -u K ) )
29	28	fveq1d 5697	. . . . 5 |- ( ( ph /\ k e. Z ) -> ( ( ( _I ` G ) shift -u K ) ` k ) = ( ( G shift -u K ) ` k ) )
30		addcom 8463	. . . . . . 7 |- ( ( K e. CC /\ k e. CC ) -> ( K + k ) = ( k + K ) )
31	4, 21, 30	syl2an 289	. . . . . 6 |- ( ( ph /\ k e. Z ) -> ( K + k ) = ( k + K ) )
32	27, 31	fveq12d 5702	. . . . 5 |- ( ( ph /\ k e. Z ) -> ( ( _I ` G ) ` ( K + k ) ) = ( G ` ( k + K ) ) )
33	24, 29, 32	3eqtr3d 2279	. . . 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 2271	. . 3 |- ( ( ph /\ k e. Z ) -> ( ( G shift -u K ) ` k ) = ( F ` k ) )
36	1, 7, 8, 9, 35	climeq 12065	. 2 |- ( ph -> ( ( G shift -u K ) ~~> A <-> F ~~> A ) )
37	3	znegcld 9770	. . 3 |- ( ph -> -u K e. ZZ )
38		climshft 12070	. . 3 |- ( ( -u K e. ZZ /\ G e. X ) -> ( ( G shift -u K ) ~~> A <-> G ~~> A ) )
39	37, 2, 38	syl2anc 415	. 2 |- ( ph -> ( ( G shift -u K ) ~~> A <-> G ~~> A ) )
40	36, 39	bitr3d 190	1 |- ( ph -> ( F ~~> A <-> G ~~> A ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   _Vcvv 2821   class class class wbr 4130   _I cid 4433   dom cdm 4774   Fun wfun 5371   ` cfv 5377  (class class class)co 6085   CCcc 8177   + caddc 8182   -ucneg 8498   ZZcz 9644   ZZ>=cuz 9921   shift cshi 11579   ~~> cli 12044

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-inn 9305  df-n0 9564  df-z 9645  df-uz 9922  df-shft 11580  df-clim 12045

This theorem is used by:  trireciplem  12267