Theorem seq3id 10443

Description: Discarding the first few terms of a sequence that starts with all zeroes (or any element which is a left-identity for .+) has no effect on its sum. (Contributed by Mario Carneiro, 13-Jul-2013.) (Revised by Jim Kingdon, 8-Apr-2023.)

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
seq3id	|- ( ph -> ( seq M ( .+ , F ) |` ( ZZ>= ` N ) ) = seq N ( .+ , F ) )

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 seq3id

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iseqid.3	. 2 |- ( ph -> N e. ( ZZ>= ` M ) )
2		eluzelz 9475	. . . . . 6 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
3	1, 2	syl 14	. . . . 5 |- ( ph -> N e. ZZ )
4		simpr 109	. . . . . . 7 |- ( ( ph /\ x e. ( ZZ>= ` N ) ) -> x e. ( ZZ>= ` N ) )
5	1	adantr 274	. . . . . . 7 |- ( ( ph /\ x e. ( ZZ>= ` N ) ) -> N e. ( ZZ>= ` M ) )
6		uztrn 9482	. . . . . . 7 |- ( ( x e. ( ZZ>= ` N ) /\ N e. ( ZZ>= ` M ) ) -> x e. ( ZZ>= ` M ) )
7	4, 5, 6	syl2anc 409	. . . . . 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 280	. . . . 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	seq3-1 10395	. . . 4 |- ( ph -> ( seq N ( .+ , F ) ` N ) = ( F ` N ) )
12		seqeq1 10383	. . . . . 6 |- ( N = M -> seq N ( .+ , F ) = seq M ( .+ , F ) )
13	12	fveq1d 5488	. . . . 5 |- ( N = M -> ( seq N ( .+ , F ) ` N ) = ( seq M ( .+ , F ) ` N ) )
14	13	eqeq1d 2174	. . . 4 |- ( N = M -> ( ( seq N ( .+ , F ) ` N ) = ( F ` N ) <-> ( seq M ( .+ , F ) ` N ) = ( F ` N ) ) )
15	11, 14	syl5ibcom 154	. . 3 |- ( ph -> ( N = M -> ( seq M ( .+ , F ) ` N ) = ( F ` N ) ) )
16		eluzel2 9471	. . . . . . . 8 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
17	1, 16	syl 14	. . . . . . 7 |- ( ph -> M e. ZZ )
18	17	adantr 274	. . . . . 6 |- ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> M e. ZZ )
19		simpr 109	. . . . . 6 |- ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> N e. ( ZZ>= ` ( M + 1 ) ) )
20	8	adantlr 469	. . . . . 6 |- ( ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )
21	10	adantlr 469	. . . . . 6 |- ( ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )
22	18, 19, 20, 21	seq3m1 10403	. . . . 5 |- ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( seq M ( .+ , F ) ` N ) = ( ( seq M ( .+ , F ) ` ( N - 1 ) ) .+ ( F ` N ) ) )
23		oveq2 5850	. . . . . . . . . 10 |- ( x = Z -> ( Z .+ x ) = ( Z .+ Z ) )
24		id 19	. . . . . . . . . 10 |- ( x = Z -> x = Z )
25	23, 24	eqeq12d 2180	. . . . . . . . 9 |- ( x = Z -> ( ( Z .+ x ) = x <-> ( Z .+ Z ) = Z ) )
26		iseqid.1	. . . . . . . . . 10 |- ( ( ph /\ x e. S ) -> ( Z .+ x ) = x )
27	26	ralrimiva 2539	. . . . . . . . 9 |- ( ph -> A. x e. S ( Z .+ x ) = x )
28		iseqid.2	. . . . . . . . 9 |- ( ph -> Z e. S )
29	25, 27, 28	rspcdva 2835	. . . . . . . 8 |- ( ph -> ( Z .+ Z ) = Z )
30	29	adantr 274	. . . . . . 7 |- ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( Z .+ Z ) = Z )
31		eluzp1m1 9489	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( N - 1 ) e. ( ZZ>= ` M ) )
32	17, 31	sylan 281	. . . . . . 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 469	. . . . . . 7 |- ( ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) /\ x e. ( M ... ( N - 1 ) ) ) -> ( F ` x ) = Z )
35	28	adantr 274	. . . . . . 7 |- ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> Z e. S )
36	30, 32, 34, 35, 20, 21	seq3id3 10442	. . . . . 6 |- ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( seq M ( .+ , F ) ` ( N - 1 ) ) = Z )
37	36	oveq1d 5857	. . . . 5 |- ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( ( seq M ( .+ , F ) ` ( N - 1 ) ) .+ ( F ` N ) ) = ( Z .+ ( F ` N ) ) )
38		oveq2 5850	. . . . . . 7 |- ( x = ( F ` N ) -> ( Z .+ x ) = ( Z .+ ( F ` N ) ) )
39		id 19	. . . . . . 7 |- ( x = ( F ` N ) -> x = ( F ` N ) )
40	38, 39	eqeq12d 2180	. . . . . 6 |- ( x = ( F ` N ) -> ( ( Z .+ x ) = x <-> ( Z .+ ( F ` N ) ) = ( F ` N ) ) )
41	27	adantr 274	. . . . . 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 274	. . . . . 6 |- ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( F ` N ) e. S )
44	40, 41, 43	rspcdva 2835	. . . . 5 |- ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( Z .+ ( F ` N ) ) = ( F ` N ) )
45	22, 37, 44	3eqtrd 2202	. . . 4 |- ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( seq M ( .+ , F ) ` N ) = ( F ` N ) )
46	45	ex 114	. . 3 |- ( ph -> ( N e. ( ZZ>= ` ( M + 1 ) ) -> ( seq M ( .+ , F ) ` N ) = ( F ` N ) ) )
47		uzp1 9499	. . . 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 708	. 2 |- ( ph -> ( seq M ( .+ , F ) ` N ) = ( F ` N ) )
50		eqidd 2166	. 2 |- ( ( ph /\ k e. ( ZZ>= ` ( N + 1 ) ) ) -> ( F ` k ) = ( F ` k ) )
51	1, 49, 8, 9, 10, 50	seq3feq2 10405	1 |- ( ph -> ( seq M ( .+ , F ) |` ( ZZ>= ` N ) ) = seq N ( .+ , F ) )

Syntax hints:   -> wi 4   /\ wa 103   \/ wo 698   = wceq 1343   e. wcel 2136   A.wral 2444   |` cres 4606   ` cfv 5188  (class class class)co 5842   1c1 7754   + caddc 7756   - cmin 8069   ZZcz 9191   ZZ>=cuz 9466   ...cfz 9944   seqcseq 10380

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-addcom 7853  ax-addass 7855  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-0id 7861  ax-rnegex 7862  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-ltadd 7869

This theorem depends on definitions:  df-bi 116  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-iord 4344  df-on 4346  df-ilim 4347  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-frec 6359  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-inn 8858  df-n0 9115  df-z 9192  df-uz 9467  df-fz 9945  df-fzo 10078  df-seqfrec 10381

This theorem is referenced by:  seq3coll  10755  sumrbdclem  11318  prodrbdclem  11512