Theorem climshft2 11991

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 9701	. . . . 5 |- ( ph -> K e. CC )
5	4	negcld 8571	. . . 4 |- ( ph -> -u K e. CC )
6		ovshftex 11504	. . . 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 5384	. . . . . . . 8 |- Fun _I
11		elex 2825	. . . . . . . . . 10 |- ( G e. X -> G e. _V )
12	2, 11	syl 14	. . . . . . . . 9 |- ( ph -> G e. _V )
13		dmi 4971	. . . . . . . . 9 |- dom _I = _V
14	12, 13	eleqtrrdi 2326	. . . . . . . 8 |- ( ph -> G e. dom _I )
15		funfvex 5687	. . . . . . . 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 9863	. . . . . . . . 9 |- ( k e. ( ZZ>= ` M ) -> k e. ZZ )
20	19, 1	eleq2s 2327	. . . . . . . 8 |- ( k e. Z -> k e. ZZ )
21	20	zcnd 9701	. . . . . . 7 |- ( k e. Z -> k e. CC )
22	21	adantl 277	. . . . . 6 |- ( ( ph /\ k e. Z ) -> k e. CC )
23		shftval4g 11522	. . . . . 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 5734	. . . . . . . . 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 6065	. . . . . 6 |- ( ( ph /\ k e. Z ) -> ( ( _I ` G ) shift -u K ) = ( G shift -u K ) )
29	28	fveq1d 5672	. . . . 5 |- ( ( ph /\ k e. Z ) -> ( ( ( _I ` G ) shift -u K ) ` k ) = ( ( G shift -u K ) ` k ) )
30		addcom 8410	. . . . . . 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 5677	. . . . 5 |- ( ( ph /\ k e. Z ) -> ( ( _I ` G ) ` ( K + k ) ) = ( G ` ( k + K ) ) )
33	24, 29, 32	3eqtr3d 2273	. . . 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 2265	. . 3 |- ( ( ph /\ k e. Z ) -> ( ( G shift -u K ) ` k ) = ( F ` k ) )
36	1, 7, 8, 9, 35	climeq 11984	. 2 |- ( ph -> ( ( G shift -u K ) ~~> A <-> F ~~> A ) )
37	3	znegcld 9702	. . 3 |- ( ph -> -u K e. ZZ )
38		climshft 11989	. . 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 2203   _Vcvv 2813   class class class wbr 4109   _I cid 4409   dom cdm 4749   Fun wfun 5346   ` cfv 5352  (class class class)co 6050   CCcc 8125   + caddc 8130   -ucneg 8445   ZZcz 9577   ZZ>=cuz 9853   shift cshi 11499   ~~> cli 11963

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 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-addass 8229  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-0id 8235  ax-rnegex 8236  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243

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 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-inn 9238  df-n0 9497  df-z 9578  df-uz 9854  df-shft 11500  df-clim 11964

This theorem is referenced by:  trireciplem  12186