Theorem climshft2 12016

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 9719	. . . . 5 |- ( ph -> K e. CC )
5	4	negcld 8587	. . . 4 |- ( ph -> -u K e. CC )
6		ovshftex 11529	. . . 4 |- ( ( G e. X /\ -u K e. CC ) -> ( G shift -u K ) e. _V )
7	2, 5, 6	syl2anc 411	. . 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 5389	. . . . . . . 8 |- Fun _I
11		elex 2827	. . . . . . . . . 10 |- ( G e. X -> G e. _V )
12	2, 11	syl 14	. . . . . . . . 9 |- ( ph -> G e. _V )
13		dmi 4976	. . . . . . . . 9 |- dom _I = _V
14	12, 13	eleqtrrdi 2328	. . . . . . . 8 |- ( ph -> G e. dom _I )
15		funfvex 5692	. . . . . . . 8 |- ( ( Fun _I /\ G e. dom _I ) -> ( _I ` G ) e. _V )
16	10, 14, 15	sylancr 414	. . . . . . 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 9881	. . . . . . . . 9 |- ( k e. ( ZZ>= ` M ) -> k e. ZZ )
20	19, 1	eleq2s 2329	. . . . . . . 8 |- ( k e. Z -> k e. ZZ )
21	20	zcnd 9719	. . . . . . 7 |- ( k e. Z -> k e. CC )
22	21	adantl 277	. . . . . 6 |- ( ( ph /\ k e. Z ) -> k e. CC )
23		shftval4g 11547	. . . . . 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 1274	. . . . 5 |- ( ( ph /\ k e. Z ) -> ( ( ( _I ` G ) shift -u K ) ` k ) = ( ( _I ` G ) ` ( K + k ) ) )
25		fvi 5739	. . . . . . . . 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 6073	. . . . . 6 |- ( ( ph /\ k e. Z ) -> ( ( _I ` G ) shift -u K ) = ( G shift -u K ) )
29	28	fveq1d 5677	. . . . 5 |- ( ( ph /\ k e. Z ) -> ( ( ( _I ` G ) shift -u K ) ` k ) = ( ( G shift -u K ) ` k ) )
30		addcom 8426	. . . . . . 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 5682	. . . . 5 |- ( ( ph /\ k e. Z ) -> ( ( _I ` G ) ` ( K + k ) ) = ( G ` ( k + K ) ) )
33	24, 29, 32	3eqtr3d 2275	. . . 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 2267	. . 3 |- ( ( ph /\ k e. Z ) -> ( ( G shift -u K ) ` k ) = ( F ` k ) )
36	1, 7, 8, 9, 35	climeq 12009	. 2 |- ( ph -> ( ( G shift -u K ) ~~> A <-> F ~~> A ) )
37	3	znegcld 9720	. . 3 |- ( ph -> -u K e. ZZ )
38		climshft 12014	. . 3 |- ( ( -u K e. ZZ /\ G e. X ) -> ( ( G shift -u K ) ~~> A <-> G ~~> A ) )
39	37, 2, 38	syl2anc 411	. 2 |- ( ph -> ( ( G shift -u K ) ~~> A <-> G ~~> A ) )
40	36, 39	bitr3d 190	1 |- ( ph -> ( F ~~> A <-> G ~~> A ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2205   _Vcvv 2815   class class class wbr 4114   _I cid 4414   dom cdm 4754   Fun wfun 5351   ` cfv 5357  (class class class)co 6058   CCcc 8141   + caddc 8146   -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:  trireciplem  12211