Theorem iseqid 9247

Description: Discard the first few terms of a sequence that starts with all zeroes (or whatever the identity Z is for operation .+). (Contributed by Mario Carneiro, 13-Jul-2013.) (Revised by Mario Carneiro, 27-May-2014.)

Hypotheses

Ref	Expression
iseqid.1	|- ( ( ph /\ x e. S ) -> ( Z .+ x ) = x )
iseqid.2	|- ( ph -> Z e. S )
iseqid.3	|- ( ph -> N e. ( ZZ>= ` M ) )
iseqid.4	|- ( ph -> ( F ` N ) e. S )
iseqid.5	|- ( ( ph /\ x e. ( M ... ( N - 1 ) ) ) -> ( F ` x ) = Z )
iseqid.s	|- ( ph -> S e. V )
iseqid.f	|- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )
iseqid.cl	|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )

Assertion

Ref	Expression
iseqid	|- ( ph -> ( seq M ( .+ , F , S ) |` ( ZZ>= ` N ) ) = seq N ( .+ , F , S ) )

Distinct variable groups:   x, .+ , y   x, F, y   x, M, y   x, N, y   x, S, y   x, Z, y   ph, x, y

Allowed substitution hints:   V( x, y)

Proof of Theorem iseqid

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iseqid.3	. 2 |- ( ph -> N e. ( ZZ>= ` M ) )
2		eluzelz 8482	. . . . . 6 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
3	1, 2	syl 14	. . . . 5 |- ( ph -> N e. ZZ )
4		iseqid.s	. . . . 5 |- ( ph -> S e. V )
5		simpr 103	. . . . . . 7 |- ( ( ph /\ x e. ( ZZ>= ` N ) ) -> x e. ( ZZ>= ` N ) )
6	1	adantr 261	. . . . . . 7 |- ( ( ph /\ x e. ( ZZ>= ` N ) ) -> N e. ( ZZ>= ` M ) )
7		uztrn 8489	. . . . . . 7 |- ( ( x e. ( ZZ>= ` N ) /\ N e. ( ZZ>= ` M ) ) -> x e. ( ZZ>= ` M ) )
8	5, 6, 7	syl2anc 391	. . . . . 6 |- ( ( ph /\ x e. ( ZZ>= ` N ) ) -> x e. ( ZZ>= ` M ) )
9		iseqid.f	. . . . . 6 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )
10	8, 9	syldan 266	. . . . 5 |- ( ( ph /\ x e. ( ZZ>= ` N ) ) -> ( F ` x ) e. S )
11		iseqid.cl	. . . . 5 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )
12	3, 4, 10, 11	iseq1 9222	. . . 4 |- ( ph -> ( seq N ( .+ , F , S ) ` N ) = ( F ` N ) )
13		iseqeq1 9214	. . . . . 6 |- ( N = M -> seq N ( .+ , F , S ) = seq M ( .+ , F , S ) )
14	13	fveq1d 5180	. . . . 5 |- ( N = M -> ( seq N ( .+ , F , S ) ` N ) = ( seq M ( .+ , F , S ) ` N ) )
15	14	eqeq1d 2048	. . . 4 |- ( N = M -> ( ( seq N ( .+ , F , S ) ` N ) = ( F ` N ) <-> ( seq M ( .+ , F , S ) ` N ) = ( F ` N ) ) )
16	12, 15	syl5ibcom 144	. . 3 |- ( ph -> ( N = M -> ( seq M ( .+ , F , S ) ` N ) = ( F ` N ) ) )
17		eluzel2 8478	. . . . . . . 8 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
18	1, 17	syl 14	. . . . . . 7 |- ( ph -> M e. ZZ )
19	18	adantr 261	. . . . . 6 |- ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> M e. ZZ )
20		simpr 103	. . . . . 6 |- ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> N e. ( ZZ>= ` ( M + 1 ) ) )
21	4	adantr 261	. . . . . 6 |- ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> S e. V )
22	9	adantlr 446	. . . . . 6 |- ( ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )
23	11	adantlr 446	. . . . . 6 |- ( ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )
24	19, 20, 21, 22, 23	iseqm1 9227	. . . . 5 |- ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( seq M ( .+ , F , S ) ` N ) = ( ( seq M ( .+ , F , S ) ` ( N - 1 ) ) .+ ( F ` N ) ) )
25		iseqid.2	. . . . . . . . 9 |- ( ph -> Z e. S )
26		iseqid.1	. . . . . . . . . 10 |- ( ( ph /\ x e. S ) -> ( Z .+ x ) = x )
27	26	ralrimiva 2392	. . . . . . . . 9 |- ( ph -> A. x e. S ( Z .+ x ) = x )
28		oveq2 5520	. . . . . . . . . . 11 |- ( x = Z -> ( Z .+ x ) = ( Z .+ Z ) )
29		id 19	. . . . . . . . . . 11 |- ( x = Z -> x = Z )
30	28, 29	eqeq12d 2054	. . . . . . . . . 10 |- ( x = Z -> ( ( Z .+ x ) = x <-> ( Z .+ Z ) = Z ) )
31	30	rspcv 2652	. . . . . . . . 9 |- ( Z e. S -> ( A. x e. S ( Z .+ x ) = x -> ( Z .+ Z ) = Z ) )
32	25, 27, 31	sylc 56	. . . . . . . 8 |- ( ph -> ( Z .+ Z ) = Z )
33	32	adantr 261	. . . . . . 7 |- ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( Z .+ Z ) = Z )
34		eluzp1m1 8496	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( N - 1 ) e. ( ZZ>= ` M ) )
35	18, 34	sylan 267	. . . . . . 7 |- ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( N - 1 ) e. ( ZZ>= ` M ) )
36		iseqid.5	. . . . . . . 8 |- ( ( ph /\ x e. ( M ... ( N - 1 ) ) ) -> ( F ` x ) = Z )
37	36	adantlr 446	. . . . . . 7 |- ( ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) /\ x e. ( M ... ( N - 1 ) ) ) -> ( F ` x ) = Z )
38	25	adantr 261	. . . . . . 7 |- ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> Z e. S )
39	33, 35, 37, 38, 21, 22, 23	iseqid3s 9246	. . . . . 6 |- ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( seq M ( .+ , F , S ) ` ( N - 1 ) ) = Z )
40	39	oveq1d 5527	. . . . 5 |- ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( ( seq M ( .+ , F , S ) ` ( N - 1 ) ) .+ ( F ` N ) ) = ( Z .+ ( F ` N ) ) )
41		iseqid.4	. . . . . . 7 |- ( ph -> ( F ` N ) e. S )
42	41	adantr 261	. . . . . 6 |- ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( F ` N ) e. S )
43	27	adantr 261	. . . . . 6 |- ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> A. x e. S ( Z .+ x ) = x )
44		oveq2 5520	. . . . . . . 8 |- ( x = ( F ` N ) -> ( Z .+ x ) = ( Z .+ ( F ` N ) ) )
45		id 19	. . . . . . . 8 |- ( x = ( F ` N ) -> x = ( F ` N ) )
46	44, 45	eqeq12d 2054	. . . . . . 7 |- ( x = ( F ` N ) -> ( ( Z .+ x ) = x <-> ( Z .+ ( F ` N ) ) = ( F ` N ) ) )
47	46	rspcv 2652	. . . . . 6 |- ( ( F ` N ) e. S -> ( A. x e. S ( Z .+ x ) = x -> ( Z .+ ( F ` N ) ) = ( F ` N ) ) )
48	42, 43, 47	sylc 56	. . . . 5 |- ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( Z .+ ( F ` N ) ) = ( F ` N ) )
49	24, 40, 48	3eqtrd 2076	. . . 4 |- ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( seq M ( .+ , F , S ) ` N ) = ( F ` N ) )
50	49	ex 108	. . 3 |- ( ph -> ( N e. ( ZZ>= ` ( M + 1 ) ) -> ( seq M ( .+ , F , S ) ` N ) = ( F ` N ) ) )
51		uzp1 8506	. . . 4 |- ( N e. ( ZZ>= ` M ) -> ( N = M \/ N e. ( ZZ>= ` ( M + 1 ) ) ) )
52	1, 51	syl 14	. . 3 |- ( ph -> ( N = M \/ N e. ( ZZ>= ` ( M + 1 ) ) ) )
53	16, 50, 52	mpjaod 638	. 2 |- ( ph -> ( seq M ( .+ , F , S ) ` N ) = ( F ` N ) )
54		eqidd 2041	. 2 |- ( ( ph /\ k e. ( ZZ>= ` ( N + 1 ) ) ) -> ( F ` k ) = ( F ` k ) )
55	1, 53, 4, 9, 10, 11, 54	iseqfeq2 9229	1 |- ( ph -> ( seq M ( .+ , F , S ) |` ( ZZ>= ` N ) ) = seq N ( .+ , F , S ) )

Syntax hints:   -> wi 4   /\ wa 97   \/ wo 629   = wceq 1243   e. wcel 1393   A.wral 2306   |` cres 4347   ` cfv 4902  (class class class)co 5512   1c1 6890   + caddc 6892   - cmin 7182   ZZcz 8245   ZZ>=cuz 8473   ...cfz 8874   seqcseq 9211

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311  ax-cnex 6975  ax-resscn 6976  ax-1cn 6977  ax-1re 6978  ax-icn 6979  ax-addcl 6980  ax-addrcl 6981  ax-mulcl 6982  ax-addcom 6984  ax-addass 6986  ax-distr 6988  ax-i2m1 6989  ax-0id 6992  ax-rnegex 6993  ax-cnre 6995  ax-pre-ltirr 6996  ax-pre-ltwlin 6997  ax-pre-lttrn 6998  ax-pre-ltadd 7000

This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-nel 2207  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-riota 5468  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-frec 5978  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-pli 6403  df-mi 6404  df-lti 6405  df-plpq 6442  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-plqqs 6447  df-mqqs 6448  df-1nqqs 6449  df-rq 6450  df-ltnqqs 6451  df-enq0 6522  df-nq0 6523  df-0nq0 6524  df-plq0 6525  df-mq0 6526  df-inp 6564  df-i1p 6565  df-iplp 6566  df-iltp 6568  df-enr 6811  df-nr 6812  df-ltr 6815  df-0r 6816  df-1r 6817  df-0 6896  df-1 6897  df-r 6899  df-lt 6902  df-pnf 7062  df-mnf 7063  df-xr 7064  df-ltxr 7065  df-le 7066  df-sub 7184  df-neg 7185  df-inn 7915  df-n0 8182  df-z 8246  df-uz 8474  df-fz 8875  df-fzo 9000  df-iseq 9212

This theorem is referenced by: (None)