Theorem iseqid 9633

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.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

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 8779	. . . . . 6 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
3	1, 2	syl 14	. . . . 5 |- ( ph -> N e. ZZ )
4		simpr 108	. . . . . . 7 |- ( ( ph /\ x e. ( ZZ>= ` N ) ) -> x e. ( ZZ>= ` N ) )
5	1	adantr 270	. . . . . . 7 |- ( ( ph /\ x e. ( ZZ>= ` N ) ) -> N e. ( ZZ>= ` M ) )
6		uztrn 8786	. . . . . . 7 |- ( ( x e. ( ZZ>= ` N ) /\ N e. ( ZZ>= ` M ) ) -> x e. ( ZZ>= ` M ) )
7	4, 5, 6	syl2anc 403	. . . . . 6 |- ( ( ph /\ x e. ( ZZ>= ` N ) ) -> x e. ( ZZ>= ` M ) )
8		iseqid.f	. . . . . 6 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )
9	7, 8	syldan 276	. . . . 5 |- ( ( ph /\ x e. ( ZZ>= ` N ) ) -> ( F ` x ) e. S )
10		iseqid.cl	. . . . 5 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )
11	3, 9, 10	iseq1 9603	. . . 4 |- ( ph -> ( seq N ( .+ , F , S ) ` N ) = ( F ` N ) )
12		iseqeq1 9594	. . . . . 6 |- ( N = M -> seq N ( .+ , F , S ) = seq M ( .+ , F , S ) )
13	12	fveq1d 5232	. . . . 5 |- ( N = M -> ( seq N ( .+ , F , S ) ` N ) = ( seq M ( .+ , F , S ) ` N ) )
14	13	eqeq1d 2091	. . . 4 |- ( N = M -> ( ( seq N ( .+ , F , S ) ` N ) = ( F ` N ) <-> ( seq M ( .+ , F , S ) ` N ) = ( F ` N ) ) )
15	11, 14	syl5ibcom 153	. . 3 |- ( ph -> ( N = M -> ( seq M ( .+ , F , S ) ` N ) = ( F ` N ) ) )
16		eluzel2 8775	. . . . . . . 8 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
17	1, 16	syl 14	. . . . . . 7 |- ( ph -> M e. ZZ )
18	17	adantr 270	. . . . . 6 |- ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> M e. ZZ )
19		simpr 108	. . . . . 6 |- ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> N e. ( ZZ>= ` ( M + 1 ) ) )
20	8	adantlr 461	. . . . . 6 |- ( ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )
21	10	adantlr 461	. . . . . 6 |- ( ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )
22	18, 19, 20, 21	iseqm1 9613	. . . . 5 |- ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( seq M ( .+ , F , S ) ` N ) = ( ( seq M ( .+ , F , S ) ` ( N - 1 ) ) .+ ( F ` N ) ) )
23		oveq2 5572	. . . . . . . . . 10 |- ( x = Z -> ( Z .+ x ) = ( Z .+ Z ) )
24		id 19	. . . . . . . . . 10 |- ( x = Z -> x = Z )
25	23, 24	eqeq12d 2097	. . . . . . . . 9 |- ( x = Z -> ( ( Z .+ x ) = x <-> ( Z .+ Z ) = Z ) )
26		iseqid.1	. . . . . . . . . 10 |- ( ( ph /\ x e. S ) -> ( Z .+ x ) = x )
27	26	ralrimiva 2439	. . . . . . . . 9 |- ( ph -> A. x e. S ( Z .+ x ) = x )
28		iseqid.2	. . . . . . . . 9 |- ( ph -> Z e. S )
29	25, 27, 28	rspcdva 2715	. . . . . . . 8 |- ( ph -> ( Z .+ Z ) = Z )
30	29	adantr 270	. . . . . . 7 |- ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( Z .+ Z ) = Z )
31		eluzp1m1 8793	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( N - 1 ) e. ( ZZ>= ` M ) )
32	17, 31	sylan 277	. . . . . . 7 |- ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( N - 1 ) e. ( ZZ>= ` M ) )
33		iseqid.5	. . . . . . . 8 |- ( ( ph /\ x e. ( M ... ( N - 1 ) ) ) -> ( F ` x ) = Z )
34	33	adantlr 461	. . . . . . 7 |- ( ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) /\ x e. ( M ... ( N - 1 ) ) ) -> ( F ` x ) = Z )
35	28	adantr 270	. . . . . . 7 |- ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> Z e. S )
36	30, 32, 34, 35, 20, 21	iseqid3s 9632	. . . . . 6 |- ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( seq M ( .+ , F , S ) ` ( N - 1 ) ) = Z )
37	36	oveq1d 5579	. . . . 5 |- ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( ( seq M ( .+ , F , S ) ` ( N - 1 ) ) .+ ( F ` N ) ) = ( Z .+ ( F ` N ) ) )
38		oveq2 5572	. . . . . . 7 |- ( x = ( F ` N ) -> ( Z .+ x ) = ( Z .+ ( F ` N ) ) )
39		id 19	. . . . . . 7 |- ( x = ( F ` N ) -> x = ( F ` N ) )
40	38, 39	eqeq12d 2097	. . . . . 6 |- ( x = ( F ` N ) -> ( ( Z .+ x ) = x <-> ( Z .+ ( F ` N ) ) = ( F ` N ) ) )
41	27	adantr 270	. . . . . 6 |- ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> A. x e. S ( Z .+ x ) = x )
42		iseqid.4	. . . . . . 7 |- ( ph -> ( F ` N ) e. S )
43	42	adantr 270	. . . . . 6 |- ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( F ` N ) e. S )
44	40, 41, 43	rspcdva 2715	. . . . 5 |- ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( Z .+ ( F ` N ) ) = ( F ` N ) )
45	22, 37, 44	3eqtrd 2119	. . . 4 |- ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( seq M ( .+ , F , S ) ` N ) = ( F ` N ) )
46	45	ex 113	. . 3 |- ( ph -> ( N e. ( ZZ>= ` ( M + 1 ) ) -> ( seq M ( .+ , F , S ) ` N ) = ( F ` N ) ) )
47		uzp1 8803	. . . 4 |- ( N e. ( ZZ>= ` M ) -> ( N = M \/ N e. ( ZZ>= ` ( M + 1 ) ) ) )
48	1, 47	syl 14	. . 3 |- ( ph -> ( N = M \/ N e. ( ZZ>= ` ( M + 1 ) ) ) )
49	15, 46, 48	mpjaod 671	. 2 |- ( ph -> ( seq M ( .+ , F , S ) ` N ) = ( F ` N ) )
50		eqidd 2084	. 2 |- ( ( ph /\ k e. ( ZZ>= ` ( N + 1 ) ) ) -> ( F ` k ) = ( F ` k ) )
51	1, 49, 8, 9, 10, 50	iseqfeq2 9615	1 |- ( ph -> ( seq M ( .+ , F , S ) |` ( ZZ>= ` N ) ) = seq N ( .+ , F , S ) )

Syntax hints:   -> wi 4   /\ wa 102   \/ wo 662   = wceq 1285   e. wcel 1434   A.wral 2353   |` cres 4393   ` cfv 4952  (class class class)co 5564   1c1 7114   + caddc 7116   - cmin 7416   ZZcz 8502   ZZ>=cuz 8770   ...cfz 9175   seqcseq 9591

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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3913  ax-sep 3916  ax-nul 3924  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-iinf 4357  ax-cnex 7199  ax-resscn 7200  ax-1cn 7201  ax-1re 7202  ax-icn 7203  ax-addcl 7204  ax-addrcl 7205  ax-mulcl 7206  ax-addcom 7208  ax-addass 7210  ax-distr 7212  ax-i2m1 7213  ax-0lt1 7214  ax-0id 7216  ax-rnegex 7217  ax-cnre 7219  ax-pre-ltirr 7220  ax-pre-ltwlin 7221  ax-pre-lttrn 7222  ax-pre-ltadd 7224

This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-tr 3896  df-id 4076  df-iord 4149  df-on 4151  df-ilim 4152  df-suc 4154  df-iom 4360  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960  df-riota 5520  df-ov 5567  df-oprab 5568  df-mpt2 5569  df-1st 5819  df-2nd 5820  df-recs 5975  df-frec 6061  df-pnf 7287  df-mnf 7288  df-xr 7289  df-ltxr 7290  df-le 7291  df-sub 7418  df-neg 7419  df-inn 8177  df-n0 8426  df-z 8503  df-uz 8771  df-fz 9176  df-fzo 9300  df-iseq 9592

This theorem is referenced by:  isumrblem  10418