Theorem climshft 12070

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.

Step	Hyp	Ref	Expression
1		oveq1 6092	. . . . . 6 |- ( f = F -> ( f shift M ) = ( F shift M ) )
2	1	breq1d 4140	. . . . 5 |- ( f = F -> ( ( f shift M ) ~~> A <-> ( F shift M ) ~~> A ) )
3		breq1 4133	. . . . 5 |- ( f = F -> ( f ~~> A <-> F ~~> A ) )
4	2, 3	bibi12d 235	. . . 4 |- ( f = F -> ( ( ( f shift M ) ~~> A <-> f ~~> A ) <-> ( ( F shift M ) ~~> A <-> F ~~> A ) ) )
5	4	imbi2d 230	. . 3 |- ( f = F -> ( ( M e. ZZ -> ( ( f shift M ) ~~> A <-> f ~~> A ) ) <-> ( M e. ZZ -> ( ( F shift M ) ~~> A <-> F ~~> A ) ) ) )
6		znegcl 9675	. . . . . 6 |- ( M e. ZZ -> -u M e. ZZ )
7		vex 2824	. . . . . . 7 |- f e. _V
8		zcn 9649	. . . . . . 7 |- ( M e. ZZ -> M e. CC )
9		ovshftex 11584	. . . . . . 7 |- ( ( f e. _V /\ M e. CC ) -> ( f shift M ) e. _V )
10	7, 8, 9	sylancr 418	. . . . . 6 |- ( M e. ZZ -> ( f shift M ) e. _V )
11		climshftlemg 12068	. . . . . 6 |- ( ( -u M e. ZZ /\ ( f shift M ) e. _V ) -> ( ( f shift M ) ~~> A -> ( ( f shift M ) shift -u M ) ~~> A ) )
12	6, 10, 11	syl2anc 415	. . . . 5 |- ( M e. ZZ -> ( ( f shift M ) ~~> A -> ( ( f shift M ) shift -u M ) ~~> A ) )
13		eqid 2238	. . . . . 6 |- ( ZZ>= ` M ) = ( ZZ>= ` M )
14	8	negcld 8624	. . . . . . 7 |- ( M e. ZZ -> -u M e. CC )
15		ovshftex 11584	. . . . . . 7 |- ( ( ( f shift M ) e. _V /\ -u M e. CC ) -> ( ( f shift M ) shift -u M ) e. _V )
16	10, 14, 15	syl2anc 415	. . . . . 6 |- ( M e. ZZ -> ( ( f shift M ) shift -u M ) e. _V )
17	7	a1i 9	. . . . . 6 |- ( M e. ZZ -> f e. _V )
18		id 19	. . . . . 6 |- ( M e. ZZ -> M e. ZZ )
19		eluzelcn 9933	. . . . . . 7 |- ( k e. ( ZZ>= ` M ) -> k e. CC )
20	7	shftcan1 11599	. . . . . . 7 |- ( ( M e. CC /\ k e. CC ) -> ( ( ( f shift M ) shift -u M ) ` k ) = ( f ` k ) )
21	8, 19, 20	syl2an 289	. . . . . 6 |- ( ( M e. ZZ /\ k e. ( ZZ>= ` M ) ) -> ( ( ( f shift M ) shift -u M ) ` k ) = ( f ` k ) )
22	13, 16, 17, 18, 21	climeq 12065	. . . . 5 |- ( M e. ZZ -> ( ( ( f shift M ) shift -u M ) ~~> A <-> f ~~> A ) )
23	12, 22	sylibd 149	. . . 4 |- ( M e. ZZ -> ( ( f shift M ) ~~> A -> f ~~> A ) )
24		climshftlemg 12068	. . . . 5 |- ( ( M e. ZZ /\ f e. _V ) -> ( f ~~> A -> ( f shift M ) ~~> A ) )
25	7, 24	mpan2 429	. . . 4 |- ( M e. ZZ -> ( f ~~> A -> ( f shift M ) ~~> A ) )
26	23, 25	impbid 129	. . 3 |- ( M e. ZZ -> ( ( f shift M ) ~~> A <-> f ~~> A ) )
27	5, 26	vtoclg 2883	. 2 |- ( F e. V -> ( M e. ZZ -> ( ( F shift M ) ~~> A <-> F ~~> A ) ) )
28	27	impcom 125	1 |- ( ( M e. ZZ /\ F e. V ) -> ( ( F shift M ) ~~> A <-> F ~~> 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   ` cfv 5377  (class class class)co 6085   CCcc 8177   -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:  climshft2  12072  iser3shft  12112  eftlub  12457