Theorem ulmshft 24144

Description: A sequence of functions converges iff the shifted sequence converges. (Contributed by Mario Carneiro, 24-Mar-2015.)

Hypotheses

Ref	Expression
ulmshft.z	|- Z = ( ZZ>= ` M )
ulmshft.w	|- W = ( ZZ>= ` ( M + K ) )
ulmshft.m	|- ( ph -> M e. ZZ )
ulmshft.k	|- ( ph -> K e. ZZ )
ulmshft.f	|- ( ph -> F : Z --> ( CC ^m S ) )
ulmshft.h	|- ( ph -> H = ( n e. W |-> ( F ` ( n - K ) ) ) )

Assertion

Ref	Expression
ulmshft	|- ( ph -> ( F ( ~~> u ` S ) G <-> H ( ~~> u ` S ) G ) )

Distinct variable groups:   ph, n   n, W   n, F   n, K   S, n

Allowed substitution hints:   G( n)   H( n)   M( n)   Z( n)

Proof of Theorem ulmshft

Dummy variable m is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ulmshft.z	. . 3 |- Z = ( ZZ>= ` M )
2		ulmshft.w	. . 3 |- W = ( ZZ>= ` ( M + K ) )
3		ulmshft.m	. . 3 |- ( ph -> M e. ZZ )
4		ulmshft.k	. . 3 |- ( ph -> K e. ZZ )
5		ulmshft.f	. . 3 |- ( ph -> F : Z --> ( CC ^m S ) )
6		ulmshft.h	. . 3 |- ( ph -> H = ( n e. W |-> ( F ` ( n - K ) ) ) )
7	1, 2, 3, 4, 5, 6	ulmshftlem 24143	. 2 |- ( ph -> ( F ( ~~> u ` S ) G -> H ( ~~> u ` S ) G ) )
8		eqid 2622	. . 3 |- ( ZZ>= ` ( ( M + K ) + -u K ) ) = ( ZZ>= ` ( ( M + K ) + -u K ) )
9	3, 4	zaddcld 11486	. . 3 |- ( ph -> ( M + K ) e. ZZ )
10	4	znegcld 11484	. . 3 |- ( ph -> -u K e. ZZ )
11	5	adantr 481	. . . . . 6 |- ( ( ph /\ n e. W ) -> F : Z --> ( CC ^m S ) )
12	3	adantr 481	. . . . . . . 8 |- ( ( ph /\ n e. W ) -> M e. ZZ )
13	4	adantr 481	. . . . . . . 8 |- ( ( ph /\ n e. W ) -> K e. ZZ )
14		simpr 477	. . . . . . . . 9 |- ( ( ph /\ n e. W ) -> n e. W )
15	14, 2	syl6eleq 2711	. . . . . . . 8 |- ( ( ph /\ n e. W ) -> n e. ( ZZ>= ` ( M + K ) ) )
16		eluzsub 11717	. . . . . . . 8 |- ( ( M e. ZZ /\ K e. ZZ /\ n e. ( ZZ>= ` ( M + K ) ) ) -> ( n - K ) e. ( ZZ>= ` M ) )
17	12, 13, 15, 16	syl3anc 1326	. . . . . . 7 |- ( ( ph /\ n e. W ) -> ( n - K ) e. ( ZZ>= ` M ) )
18	17, 1	syl6eleqr 2712	. . . . . 6 |- ( ( ph /\ n e. W ) -> ( n - K ) e. Z )
19	11, 18	ffvelrnd 6360	. . . . 5 |- ( ( ph /\ n e. W ) -> ( F ` ( n - K ) ) e. ( CC ^m S ) )
20		eqid 2622	. . . . 5 |- ( n e. W |-> ( F ` ( n - K ) ) ) = ( n e. W |-> ( F ` ( n - K ) ) )
21	19, 20	fmptd 6385	. . . 4 |- ( ph -> ( n e. W |-> ( F ` ( n - K ) ) ) : W --> ( CC ^m S ) )
22	6	feq1d 6030	. . . 4 |- ( ph -> ( H : W --> ( CC ^m S ) <-> ( n e. W |-> ( F ` ( n - K ) ) ) : W --> ( CC ^m S ) ) )
23	21, 22	mpbird 247	. . 3 |- ( ph -> H : W --> ( CC ^m S ) )
24		simpr 477	. . . . . . . . . . 11 |- ( ( ph /\ m e. Z ) -> m e. Z )
25	24, 1	syl6eleq 2711	. . . . . . . . . 10 |- ( ( ph /\ m e. Z ) -> m e. ( ZZ>= ` M ) )
26		eluzelz 11697	. . . . . . . . . 10 |- ( m e. ( ZZ>= ` M ) -> m e. ZZ )
27	25, 26	syl 17	. . . . . . . . 9 |- ( ( ph /\ m e. Z ) -> m e. ZZ )
28	27	zcnd 11483	. . . . . . . 8 |- ( ( ph /\ m e. Z ) -> m e. CC )
29	4	zcnd 11483	. . . . . . . . 9 |- ( ph -> K e. CC )
30	29	adantr 481	. . . . . . . 8 |- ( ( ph /\ m e. Z ) -> K e. CC )
31	28, 30	subnegd 10399	. . . . . . 7 |- ( ( ph /\ m e. Z ) -> ( m - -u K ) = ( m + K ) )
32	31	fveq2d 6195	. . . . . 6 |- ( ( ph /\ m e. Z ) -> ( H ` ( m - -u K ) ) = ( H ` ( m + K ) ) )
33	6	adantr 481	. . . . . . 7 |- ( ( ph /\ m e. Z ) -> H = ( n e. W |-> ( F ` ( n - K ) ) ) )
34	33	fveq1d 6193	. . . . . 6 |- ( ( ph /\ m e. Z ) -> ( H ` ( m + K ) ) = ( ( n e. W |-> ( F ` ( n - K ) ) ) ` ( m + K ) ) )
35	4	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ m e. Z ) -> K e. ZZ )
36		eluzadd 11716	. . . . . . . . . 10 |- ( ( m e. ( ZZ>= ` M ) /\ K e. ZZ ) -> ( m + K ) e. ( ZZ>= ` ( M + K ) ) )
37	25, 35, 36	syl2anc 693	. . . . . . . . 9 |- ( ( ph /\ m e. Z ) -> ( m + K ) e. ( ZZ>= ` ( M + K ) ) )
38	37, 2	syl6eleqr 2712	. . . . . . . 8 |- ( ( ph /\ m e. Z ) -> ( m + K ) e. W )
39		oveq1 6657	. . . . . . . . . 10 |- ( n = ( m + K ) -> ( n - K ) = ( ( m + K ) - K ) )
40	39	fveq2d 6195	. . . . . . . . 9 |- ( n = ( m + K ) -> ( F ` ( n - K ) ) = ( F ` ( ( m + K ) - K ) ) )
41		fvex 6201	. . . . . . . . 9 |- ( F ` ( ( m + K ) - K ) ) e. _V
42	40, 20, 41	fvmpt 6282	. . . . . . . 8 |- ( ( m + K ) e. W -> ( ( n e. W |-> ( F ` ( n - K ) ) ) ` ( m + K ) ) = ( F ` ( ( m + K ) - K ) ) )
43	38, 42	syl 17	. . . . . . 7 |- ( ( ph /\ m e. Z ) -> ( ( n e. W |-> ( F ` ( n - K ) ) ) ` ( m + K ) ) = ( F ` ( ( m + K ) - K ) ) )
44	28, 30	pncand 10393	. . . . . . . 8 |- ( ( ph /\ m e. Z ) -> ( ( m + K ) - K ) = m )
45	44	fveq2d 6195	. . . . . . 7 |- ( ( ph /\ m e. Z ) -> ( F ` ( ( m + K ) - K ) ) = ( F ` m ) )
46	43, 45	eqtrd 2656	. . . . . 6 |- ( ( ph /\ m e. Z ) -> ( ( n e. W |-> ( F ` ( n - K ) ) ) ` ( m + K ) ) = ( F ` m ) )
47	32, 34, 46	3eqtrd 2660	. . . . 5 |- ( ( ph /\ m e. Z ) -> ( H ` ( m - -u K ) ) = ( F ` m ) )
48	47	mpteq2dva 4744	. . . 4 |- ( ph -> ( m e. Z |-> ( H ` ( m - -u K ) ) ) = ( m e. Z |-> ( F ` m ) ) )
49	3	zcnd 11483	. . . . . . . . 9 |- ( ph -> M e. CC )
50	10	zcnd 11483	. . . . . . . . 9 |- ( ph -> -u K e. CC )
51	49, 29, 50	addassd 10062	. . . . . . . 8 |- ( ph -> ( ( M + K ) + -u K ) = ( M + ( K + -u K ) ) )
52	29	negidd 10382	. . . . . . . . 9 |- ( ph -> ( K + -u K ) = 0 )
53	52	oveq2d 6666	. . . . . . . 8 |- ( ph -> ( M + ( K + -u K ) ) = ( M + 0 ) )
54	49	addid1d 10236	. . . . . . . 8 |- ( ph -> ( M + 0 ) = M )
55	51, 53, 54	3eqtrd 2660	. . . . . . 7 |- ( ph -> ( ( M + K ) + -u K ) = M )
56	55	fveq2d 6195	. . . . . 6 |- ( ph -> ( ZZ>= ` ( ( M + K ) + -u K ) ) = ( ZZ>= ` M ) )
57	56, 1	syl6eqr 2674	. . . . 5 |- ( ph -> ( ZZ>= ` ( ( M + K ) + -u K ) ) = Z )
58	57	mpteq1d 4738	. . . 4 |- ( ph -> ( m e. ( ZZ>= ` ( ( M + K ) + -u K ) ) |-> ( H ` ( m - -u K ) ) ) = ( m e. Z |-> ( H ` ( m - -u K ) ) ) )
59	5	feqmptd 6249	. . . 4 |- ( ph -> F = ( m e. Z |-> ( F ` m ) ) )
60	48, 58, 59	3eqtr4rd 2667	. . 3 |- ( ph -> F = ( m e. ( ZZ>= ` ( ( M + K ) + -u K ) ) |-> ( H ` ( m - -u K ) ) ) )
61	2, 8, 9, 10, 23, 60	ulmshftlem 24143	. 2 |- ( ph -> ( H ( ~~> u ` S ) G -> F ( ~~> u ` S ) G ) )
62	7, 61	impbid 202	1 |- ( ph -> ( F ( ~~> u ` S ) G <-> H ( ~~> u ` S ) G ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   class class class wbr 4653   |-> cmpt 4729   -->wf 5884   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   CCcc 9934   0cc0 9936   + caddc 9939   - cmin 10266   -ucneg 10267   ZZcz 11377   ZZ>=cuz 11687   ~~> uculm 24130

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-ulm 24131

This theorem is referenced by:  pserdvlem2  24182