Theorem iseqid 9467

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 8628	. . . . . 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 108	. . . . . . 7 |- ( ( ph /\ x e. ( ZZ>= ` N ) ) -> x e. ( ZZ>= ` N ) )
6	1	adantr 270	. . . . . . 7 |- ( ( ph /\ x e. ( ZZ>= ` N ) ) -> N e. ( ZZ>= ` M ) )
7		uztrn 8635	. . . . . . 7 |- ( ( x e. ( ZZ>= ` N ) /\ N e. ( ZZ>= ` M ) ) -> x e. ( ZZ>= ` M ) )
8	5, 6, 7	syl2anc 403	. . . . . 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 276	. . . . 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 9442	. . . 4 |- ( ph -> ( seq N ( .+ , F , S ) ` N ) = ( F ` N ) )
13		iseqeq1 9434	. . . . . 6 |- ( N = M -> seq N ( .+ , F , S ) = seq M ( .+ , F , S ) )
14	13	fveq1d 5200	. . . . 5 |- ( N = M -> ( seq N ( .+ , F , S ) ` N ) = ( seq M ( .+ , F , S ) ` N ) )
15	14	eqeq1d 2089	. . . 4 |- ( N = M -> ( ( seq N ( .+ , F , S ) ` N ) = ( F ` N ) <-> ( seq M ( .+ , F , S ) ` N ) = ( F ` N ) ) )
16	12, 15	syl5ibcom 153	. . 3 |- ( ph -> ( N = M -> ( seq M ( .+ , F , S ) ` N ) = ( F ` N ) ) )
17		eluzel2 8624	. . . . . . . 8 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
18	1, 17	syl 14	. . . . . . 7 |- ( ph -> M e. ZZ )
19	18	adantr 270	. . . . . 6 |- ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> M e. ZZ )
20		simpr 108	. . . . . 6 |- ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> N e. ( ZZ>= ` ( M + 1 ) ) )
21	4	adantr 270	. . . . . 6 |- ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> S e. V )
22	9	adantlr 460	. . . . . 6 |- ( ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )
23	11	adantlr 460	. . . . . 6 |- ( ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )
24	19, 20, 21, 22, 23	iseqm1 9447	. . . . 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 2434	. . . . . . . . 9 |- ( ph -> A. x e. S ( Z .+ x ) = x )
28		oveq2 5540	. . . . . . . . . . 11 |- ( x = Z -> ( Z .+ x ) = ( Z .+ Z ) )
29		id 19	. . . . . . . . . . 11 |- ( x = Z -> x = Z )
30	28, 29	eqeq12d 2095	. . . . . . . . . 10 |- ( x = Z -> ( ( Z .+ x ) = x <-> ( Z .+ Z ) = Z ) )
31	30	rspcv 2697	. . . . . . . . 9 |- ( Z e. S -> ( A. x e. S ( Z .+ x ) = x -> ( Z .+ Z ) = Z ) )
32	25, 27, 31	sylc 61	. . . . . . . 8 |- ( ph -> ( Z .+ Z ) = Z )
33	32	adantr 270	. . . . . . 7 |- ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( Z .+ Z ) = Z )
34		eluzp1m1 8642	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( N - 1 ) e. ( ZZ>= ` M ) )
35	18, 34	sylan 277	. . . . . . 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 460	. . . . . . 7 |- ( ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) /\ x e. ( M ... ( N - 1 ) ) ) -> ( F ` x ) = Z )
38	25	adantr 270	. . . . . . 7 |- ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> Z e. S )
39	33, 35, 37, 38, 21, 22, 23	iseqid3s 9466	. . . . . 6 |- ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( seq M ( .+ , F , S ) ` ( N - 1 ) ) = Z )
40	39	oveq1d 5547	. . . . 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 270	. . . . . 6 |- ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( F ` N ) e. S )
43	27	adantr 270	. . . . . 6 |- ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> A. x e. S ( Z .+ x ) = x )
44		oveq2 5540	. . . . . . . 8 |- ( x = ( F ` N ) -> ( Z .+ x ) = ( Z .+ ( F ` N ) ) )
45		id 19	. . . . . . . 8 |- ( x = ( F ` N ) -> x = ( F ` N ) )
46	44, 45	eqeq12d 2095	. . . . . . 7 |- ( x = ( F ` N ) -> ( ( Z .+ x ) = x <-> ( Z .+ ( F ` N ) ) = ( F ` N ) ) )
47	46	rspcv 2697	. . . . . 6 |- ( ( F ` N ) e. S -> ( A. x e. S ( Z .+ x ) = x -> ( Z .+ ( F ` N ) ) = ( F ` N ) ) )
48	42, 43, 47	sylc 61	. . . . 5 |- ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( Z .+ ( F ` N ) ) = ( F ` N ) )
49	24, 40, 48	3eqtrd 2117	. . . 4 |- ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( seq M ( .+ , F , S ) ` N ) = ( F ` N ) )
50	49	ex 113	. . 3 |- ( ph -> ( N e. ( ZZ>= ` ( M + 1 ) ) -> ( seq M ( .+ , F , S ) ` N ) = ( F ` N ) ) )
51		uzp1 8652	. . . 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 670	. 2 |- ( ph -> ( seq M ( .+ , F , S ) ` N ) = ( F ` N ) )
54		eqidd 2082	. 2 |- ( ( ph /\ k e. ( ZZ>= ` ( N + 1 ) ) ) -> ( F ` k ) = ( F ` k ) )
55	1, 53, 4, 9, 10, 11, 54	iseqfeq2 9449	1 |- ( ph -> ( seq M ( .+ , F , S ) |` ( ZZ>= ` N ) ) = seq N ( .+ , F , S ) )

Syntax hints:   -> wi 4   /\ wa 102   \/ wo 661   = wceq 1284   e. wcel 1433   A.wral 2348   |` cres 4365   ` cfv 4922  (class class class)co 5532   1c1 6982   + caddc 6984   - cmin 7279   ZZcz 8351   ZZ>=cuz 8619   ...cfz 9029   seqcseq 9431

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329  ax-cnex 7067  ax-resscn 7068  ax-1cn 7069  ax-1re 7070  ax-icn 7071  ax-addcl 7072  ax-addrcl 7073  ax-mulcl 7074  ax-addcom 7076  ax-addass 7078  ax-distr 7080  ax-i2m1 7081  ax-0lt1 7082  ax-0id 7084  ax-rnegex 7085  ax-cnre 7087  ax-pre-ltirr 7088  ax-pre-ltwlin 7089  ax-pre-lttrn 7090  ax-pre-ltadd 7092

This theorem depends on definitions:  df-bi 115  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-nel 2340  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-id 4048  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-riota 5488  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-frec 6001  df-pnf 7155  df-mnf 7156  df-xr 7157  df-ltxr 7158  df-le 7159  df-sub 7281  df-neg 7282  df-inn 8040  df-n0 8289  df-z 8352  df-uz 8620  df-fz 9030  df-fzo 9153  df-iseq 9432

This theorem is referenced by: (None)