Theorem seqid 12846

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
seqid.1	|- ( ( ph /\ x e. S ) -> ( Z .+ x ) = x )
seqid.2	|- ( ph -> Z e. S )
seqid.3	|- ( ph -> N e. ( ZZ>= ` M ) )
seqid.4	|- ( ph -> ( F ` N ) e. S )
seqid.5	|- ( ( ph /\ x e. ( M ... ( N - 1 ) ) ) -> ( F ` x ) = Z )

Assertion

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

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

Proof of Theorem seqid

Step	Hyp	Ref	Expression
1		seqid.3	. 2 |- ( ph -> N e. ( ZZ>= ` M ) )
2		eluzelz 11697	. . . . 5 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
3		seq1 12814	. . . . 5 |- ( N e. ZZ -> ( seq N ( .+ , F ) ` N ) = ( F ` N ) )
4	1, 2, 3	3syl 18	. . . 4 |- ( ph -> ( seq N ( .+ , F ) ` N ) = ( F ` N ) )
5		seqeq1 12804	. . . . . 6 |- ( N = M -> seq N ( .+ , F ) = seq M ( .+ , F ) )
6	5	fveq1d 6193	. . . . 5 |- ( N = M -> ( seq N ( .+ , F ) ` N ) = ( seq M ( .+ , F ) ` N ) )
7	6	eqeq1d 2624	. . . 4 |- ( N = M -> ( ( seq N ( .+ , F ) ` N ) = ( F ` N ) <-> ( seq M ( .+ , F ) ` N ) = ( F ` N ) ) )
8	4, 7	syl5ibcom 235	. . 3 |- ( ph -> ( N = M -> ( seq M ( .+ , F ) ` N ) = ( F ` N ) ) )
9		eluzel2 11692	. . . . . . 7 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
10	1, 9	syl 17	. . . . . 6 |- ( ph -> M e. ZZ )
11		seqm1 12818	. . . . . 6 |- ( ( M e. ZZ /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( seq M ( .+ , F ) ` N ) = ( ( seq M ( .+ , F ) ` ( N - 1 ) ) .+ ( F ` N ) ) )
12	10, 11	sylan 488	. . . . 5 |- ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( seq M ( .+ , F ) ` N ) = ( ( seq M ( .+ , F ) ` ( N - 1 ) ) .+ ( F ` N ) ) )
13		seqid.2	. . . . . . . . 9 |- ( ph -> Z e. S )
14		seqid.1	. . . . . . . . . 10 |- ( ( ph /\ x e. S ) -> ( Z .+ x ) = x )
15	14	ralrimiva 2966	. . . . . . . . 9 |- ( ph -> A. x e. S ( Z .+ x ) = x )
16		oveq2 6658	. . . . . . . . . . 11 |- ( x = Z -> ( Z .+ x ) = ( Z .+ Z ) )
17		id 22	. . . . . . . . . . 11 |- ( x = Z -> x = Z )
18	16, 17	eqeq12d 2637	. . . . . . . . . 10 |- ( x = Z -> ( ( Z .+ x ) = x <-> ( Z .+ Z ) = Z ) )
19	18	rspcv 3305	. . . . . . . . 9 |- ( Z e. S -> ( A. x e. S ( Z .+ x ) = x -> ( Z .+ Z ) = Z ) )
20	13, 15, 19	sylc 65	. . . . . . . 8 |- ( ph -> ( Z .+ Z ) = Z )
21	20	adantr 481	. . . . . . 7 |- ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( Z .+ Z ) = Z )
22		eluzp1m1 11711	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( N - 1 ) e. ( ZZ>= ` M ) )
23	10, 22	sylan 488	. . . . . . 7 |- ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( N - 1 ) e. ( ZZ>= ` M ) )
24		seqid.5	. . . . . . . 8 |- ( ( ph /\ x e. ( M ... ( N - 1 ) ) ) -> ( F ` x ) = Z )
25	24	adantlr 751	. . . . . . 7 |- ( ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) /\ x e. ( M ... ( N - 1 ) ) ) -> ( F ` x ) = Z )
26	21, 23, 25	seqid3 12845	. . . . . 6 |- ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( seq M ( .+ , F ) ` ( N - 1 ) ) = Z )
27	26	oveq1d 6665	. . . . 5 |- ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( ( seq M ( .+ , F ) ` ( N - 1 ) ) .+ ( F ` N ) ) = ( Z .+ ( F ` N ) ) )
28		seqid.4	. . . . . . 7 |- ( ph -> ( F ` N ) e. S )
29	28	adantr 481	. . . . . 6 |- ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( F ` N ) e. S )
30	15	adantr 481	. . . . . 6 |- ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> A. x e. S ( Z .+ x ) = x )
31		oveq2 6658	. . . . . . . 8 |- ( x = ( F ` N ) -> ( Z .+ x ) = ( Z .+ ( F ` N ) ) )
32		id 22	. . . . . . . 8 |- ( x = ( F ` N ) -> x = ( F ` N ) )
33	31, 32	eqeq12d 2637	. . . . . . 7 |- ( x = ( F ` N ) -> ( ( Z .+ x ) = x <-> ( Z .+ ( F ` N ) ) = ( F ` N ) ) )
34	33	rspcv 3305	. . . . . 6 |- ( ( F ` N ) e. S -> ( A. x e. S ( Z .+ x ) = x -> ( Z .+ ( F ` N ) ) = ( F ` N ) ) )
35	29, 30, 34	sylc 65	. . . . 5 |- ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( Z .+ ( F ` N ) ) = ( F ` N ) )
36	12, 27, 35	3eqtrd 2660	. . . 4 |- ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( seq M ( .+ , F ) ` N ) = ( F ` N ) )
37	36	ex 450	. . 3 |- ( ph -> ( N e. ( ZZ>= ` ( M + 1 ) ) -> ( seq M ( .+ , F ) ` N ) = ( F ` N ) ) )
38		uzp1 11721	. . . 4 |- ( N e. ( ZZ>= ` M ) -> ( N = M \/ N e. ( ZZ>= ` ( M + 1 ) ) ) )
39	1, 38	syl 17	. . 3 |- ( ph -> ( N = M \/ N e. ( ZZ>= ` ( M + 1 ) ) ) )
40	8, 37, 39	mpjaod 396	. 2 |- ( ph -> ( seq M ( .+ , F ) ` N ) = ( F ` N ) )
41		eqidd 2623	. 2 |- ( ( ph /\ x e. ( ZZ>= ` ( N + 1 ) ) ) -> ( F ` x ) = ( F ` x ) )
42	1, 40, 41	seqfeq2 12824	1 |- ( ph -> ( seq M ( .+ , F ) |` ( ZZ>= ` N ) ) = seq N ( .+ , F ) )

Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   |` cres 5116   ` cfv 5888  (class class class)co 6650   1c1 9937   + caddc 9939   - cmin 10266   ZZcz 11377   ZZ>=cuz 11687   ...cfz 12326   seqcseq 12801

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-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-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  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-fz 12327  df-seq 12802

This theorem is referenced by:  seqcoll  13248  sumrblem  14442  prodrblem  14659  logtayl  24406  leibpilem2  24668