Theorem climshft 11730

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 5974	. . . . . 6 |- ( f = F -> ( f shift M ) = ( F shift M ) )
2	1	breq1d 4069	. . . . 5 |- ( f = F -> ( ( f shift M ) ~~> A <-> ( F shift M ) ~~> A ) )
3		breq1 4062	. . . . 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 9438	. . . . . 6 |- ( M e. ZZ -> -u M e. ZZ )
7		vex 2779	. . . . . . 7 |- f e. _V
8		zcn 9412	. . . . . . 7 |- ( M e. ZZ -> M e. CC )
9		ovshftex 11245	. . . . . . 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 11728	. . . . . 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 2207	. . . . . 6 |- ( ZZ>= ` M ) = ( ZZ>= ` M )
14	8	negcld 8405	. . . . . . 7 |- ( M e. ZZ -> -u M e. CC )
15		ovshftex 11245	. . . . . . 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 9694	. . . . . . 7 |- ( k e. ( ZZ>= ` M ) -> k e. CC )
20	7	shftcan1 11260	. . . . . . 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 11725	. . . . 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 11728	. . . . 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 2838	. 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 1373   e. wcel 2178   _Vcvv 2776   class class class wbr 4059   ` cfv 5290  (class class class)co 5967   CCcc 7958   -ucneg 8279   ZZcz 9407   ZZ>=cuz 9683   shift cshi 11240   ~~> cli 11704

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-addcom 8060  ax-addass 8062  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-0id 8068  ax-rnegex 8069  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-apti 8075  ax-pre-ltadd 8076

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-if 3580  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-sub 8280  df-neg 8281  df-inn 9072  df-n0 9331  df-z 9408  df-uz 9684  df-shft 11241  df-clim 11705

This theorem is referenced by:  climshft2  11732  iser3shft  11772  eftlub  12116