Theorem climshft2 11306

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 9371	. . . . 5 |- ( ph -> K e. CC )
5	4	negcld 8250	. . . 4 |- ( ph -> -u K e. CC )
6		ovshftex 10820	. . . 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 5246	. . . . . . . 8 |- Fun _I
11		elex 2748	. . . . . . . . . 10 |- ( G e. X -> G e. _V )
12	2, 11	syl 14	. . . . . . . . 9 |- ( ph -> G e. _V )
13		dmi 4840	. . . . . . . . 9 |- dom _I = _V
14	12, 13	eleqtrrdi 2271	. . . . . . . 8 |- ( ph -> G e. dom _I )
15		funfvex 5530	. . . . . . . 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 9532	. . . . . . . . 9 |- ( k e. ( ZZ>= ` M ) -> k e. ZZ )
20	19, 1	eleq2s 2272	. . . . . . . 8 |- ( k e. Z -> k e. ZZ )
21	20	zcnd 9371	. . . . . . 7 |- ( k e. Z -> k e. CC )
22	21	adantl 277	. . . . . 6 |- ( ( ph /\ k e. Z ) -> k e. CC )
23		shftval4g 10838	. . . . . 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 1238	. . . . 5 |- ( ( ph /\ k e. Z ) -> ( ( ( _I ` G ) shift -u K ) ` k ) = ( ( _I ` G ) ` ( K + k ) ) )
25		fvi 5571	. . . . . . . . 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 5886	. . . . . 6 |- ( ( ph /\ k e. Z ) -> ( ( _I ` G ) shift -u K ) = ( G shift -u K ) )
29	28	fveq1d 5515	. . . . 5 |- ( ( ph /\ k e. Z ) -> ( ( ( _I ` G ) shift -u K ) ` k ) = ( ( G shift -u K ) ` k ) )
30		addcom 8089	. . . . . . 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 5520	. . . . 5 |- ( ( ph /\ k e. Z ) -> ( ( _I ` G ) ` ( K + k ) ) = ( G ` ( k + K ) ) )
33	24, 29, 32	3eqtr3d 2218	. . . 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 2210	. . 3 |- ( ( ph /\ k e. Z ) -> ( ( G shift -u K ) ` k ) = ( F ` k ) )
36	1, 7, 8, 9, 35	climeq 11299	. 2 |- ( ph -> ( ( G shift -u K ) ~~> A <-> F ~~> A ) )
37	3	znegcld 9372	. . 3 |- ( ph -> -u K e. ZZ )
38		climshft 11304	. . 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 1353   e. wcel 2148   _Vcvv 2737   class class class wbr 4002   _I cid 4287   dom cdm 4625   Fun wfun 5208   ` cfv 5214  (class class class)co 5871   CCcc 7805   + caddc 7810   -ucneg 8124   ZZcz 9248   ZZ>=cuz 9523   shift cshi 10815   ~~> cli 11278

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4117  ax-sep 4120  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-setind 4535  ax-cnex 7898  ax-resscn 7899  ax-1cn 7900  ax-1re 7901  ax-icn 7902  ax-addcl 7903  ax-addrcl 7904  ax-mulcl 7905  ax-addcom 7907  ax-addass 7909  ax-distr 7911  ax-i2m1 7912  ax-0lt1 7913  ax-0id 7915  ax-rnegex 7916  ax-cnre 7918  ax-pre-ltirr 7919  ax-pre-ltwlin 7920  ax-pre-lttrn 7921  ax-pre-apti 7922  ax-pre-ltadd 7923

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-if 3535  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4003  df-opab 4064  df-mpt 4065  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-ima 4638  df-iota 5176  df-fun 5216  df-fn 5217  df-f 5218  df-f1 5219  df-fo 5220  df-f1o 5221  df-fv 5222  df-riota 5827  df-ov 5874  df-oprab 5875  df-mpo 5876  df-pnf 7989  df-mnf 7990  df-xr 7991  df-ltxr 7992  df-le 7993  df-sub 8125  df-neg 8126  df-inn 8915  df-n0 9172  df-z 9249  df-uz 9524  df-shft 10816  df-clim 11279

This theorem is referenced by:  trireciplem  11500