Theorem climshft 12014

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 6065	. . . . . 6 |- ( f = F -> ( f shift M ) = ( F shift M ) )
2	1	breq1d 4124	. . . . 5 |- ( f = F -> ( ( f shift M ) ~~> A <-> ( F shift M ) ~~> A ) )
3		breq1 4117	. . . . 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 9625	. . . . . 6 |- ( M e. ZZ -> -u M e. ZZ )
7		vex 2818	. . . . . . 7 |- f e. _V
8		zcn 9599	. . . . . . 7 |- ( M e. ZZ -> M e. CC )
9		ovshftex 11529	. . . . . . 7 |- ( ( f e. _V /\ M e. CC ) -> ( f shift M ) e. _V )
10	7, 8, 9	sylancr 414	. . . . . 6 |- ( M e. ZZ -> ( f shift M ) e. _V )
11		climshftlemg 12012	. . . . . 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 411	. . . . 5 |- ( M e. ZZ -> ( ( f shift M ) ~~> A -> ( ( f shift M ) shift -u M ) ~~> A ) )
13		eqid 2234	. . . . . 6 |- ( ZZ>= ` M ) = ( ZZ>= ` M )
14	8	negcld 8587	. . . . . . 7 |- ( M e. ZZ -> -u M e. CC )
15		ovshftex 11529	. . . . . . 7 |- ( ( ( f shift M ) e. _V /\ -u M e. CC ) -> ( ( f shift M ) shift -u M ) e. _V )
16	10, 14, 15	syl2anc 411	. . . . . 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 9883	. . . . . . 7 |- ( k e. ( ZZ>= ` M ) -> k e. CC )
20	7	shftcan1 11544	. . . . . . 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 12009	. . . . 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 12012	. . . . 5 |- ( ( M e. ZZ /\ f e. _V ) -> ( f ~~> A -> ( f shift M ) ~~> A ) )
25	7, 24	mpan2 425	. . . 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 2877	. 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 ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2205   _Vcvv 2815   class class class wbr 4114   ` cfv 5357  (class class class)co 6058   CCcc 8141   -ucneg 8461   ZZcz 9594   ZZ>=cuz 9871   shift cshi 11524   ~~> cli 11988

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-inn 9255  df-n0 9514  df-z 9595  df-uz 9872  df-shft 11525  df-clim 11989

This theorem is referenced by:  climshft2  12016  iser3shft  12056  eftlub  12401