Theorem seq3id 10962

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 9931	. . . . . 6 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
3	1, 2	syl 14	. . . . 5 |- ( ph -> N e. ZZ )
4		simpr 110	. . . . . . 7 |- ( ( ph /\ x e. ( ZZ>= ` N ) ) -> x e. ( ZZ>= ` N ) )
5	1	adantr 276	. . . . . . 7 |- ( ( ph /\ x e. ( ZZ>= ` N ) ) -> N e. ( ZZ>= ` M ) )
6		uztrn 9939	. . . . . . 7 |- ( ( x e. ( ZZ>= ` N ) /\ N e. ( ZZ>= ` M ) ) -> x e. ( ZZ>= ` M ) )
7	4, 5, 6	syl2anc 415	. . . . . 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 282	. . . . 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 10899	. . . 4 |- ( ph -> ( seq N ( .+ , F ) ` N ) = ( F ` N ) )
12		seqeq1 10887	. . . . . 6 |- ( N = M -> seq N ( .+ , F ) = seq M ( .+ , F ) )
13	12	fveq1d 5697	. . . . 5 |- ( N = M -> ( seq N ( .+ , F ) ` N ) = ( seq M ( .+ , F ) ` N ) )
14	13	eqeq1d 2247	. . . 4 |- ( N = M -> ( ( seq N ( .+ , F ) ` N ) = ( F ` N ) <-> ( seq M ( .+ , F ) ` N ) = ( F ` N ) ) )
15	11, 14	syl5ibcom 155	. . 3 |- ( ph -> ( N = M -> ( seq M ( .+ , F ) ` N ) = ( F ` N ) ) )
16		eluzel2 9926	. . . . . . . 8 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
17	1, 16	syl 14	. . . . . . 7 |- ( ph -> M e. ZZ )
18	17	adantr 276	. . . . . 6 |- ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> M e. ZZ )
19		simpr 110	. . . . . 6 |- ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> N e. ( ZZ>= ` ( M + 1 ) ) )
20	8	adantlr 481	. . . . . 6 |- ( ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )
21	10	adantlr 481	. . . . . 6 |- ( ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )
22	18, 19, 20, 21	seq3m1 10910	. . . . 5 |- ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( seq M ( .+ , F ) ` N ) = ( ( seq M ( .+ , F ) ` ( N - 1 ) ) .+ ( F ` N ) ) )
23		oveq2 6093	. . . . . . . . . 10 |- ( x = Z -> ( Z .+ x ) = ( Z .+ Z ) )
24		id 19	. . . . . . . . . 10 |- ( x = Z -> x = Z )
25	23, 24	eqeq12d 2253	. . . . . . . . 9 |- ( x = Z -> ( ( Z .+ x ) = x <-> ( Z .+ Z ) = Z ) )
26		iseqid.1	. . . . . . . . . 10 |- ( ( ph /\ x e. S ) -> ( Z .+ x ) = x )
27	26	ralrimiva 2623	. . . . . . . . 9 |- ( ph -> A. x e. S ( Z .+ x ) = x )
28		iseqid.2	. . . . . . . . 9 |- ( ph -> Z e. S )
29	25, 27, 28	rspcdva 2934	. . . . . . . 8 |- ( ph -> ( Z .+ Z ) = Z )
30	29	adantr 276	. . . . . . 7 |- ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( Z .+ Z ) = Z )
31		eluzp1m1 9946	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( N - 1 ) e. ( ZZ>= ` M ) )
32	17, 31	sylan 283	. . . . . . 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 481	. . . . . . 7 |- ( ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) /\ x e. ( M ... ( N - 1 ) ) ) -> ( F ` x ) = Z )
35	28	adantr 276	. . . . . . 7 |- ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> Z e. S )
36	30, 32, 34, 35, 20, 21	seq3id3 10961	. . . . . 6 |- ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( seq M ( .+ , F ) ` ( N - 1 ) ) = Z )
37	36	oveq1d 6100	. . . . 5 |- ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( ( seq M ( .+ , F ) ` ( N - 1 ) ) .+ ( F ` N ) ) = ( Z .+ ( F ` N ) ) )
38		oveq2 6093	. . . . . . 7 |- ( x = ( F ` N ) -> ( Z .+ x ) = ( Z .+ ( F ` N ) ) )
39		id 19	. . . . . . 7 |- ( x = ( F ` N ) -> x = ( F ` N ) )
40	38, 39	eqeq12d 2253	. . . . . 6 |- ( x = ( F ` N ) -> ( ( Z .+ x ) = x <-> ( Z .+ ( F ` N ) ) = ( F ` N ) ) )
41	27	adantr 276	. . . . . 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 276	. . . . . 6 |- ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( F ` N ) e. S )
44	40, 41, 43	rspcdva 2934	. . . . 5 |- ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( Z .+ ( F ` N ) ) = ( F ` N ) )
45	22, 37, 44	3eqtrd 2275	. . . 4 |- ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( seq M ( .+ , F ) ` N ) = ( F ` N ) )
46	45	ex 115	. . 3 |- ( ph -> ( N e. ( ZZ>= ` ( M + 1 ) ) -> ( seq M ( .+ , F ) ` N ) = ( F ` N ) ) )
47		uzp1 9956	. . . 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 730	. 2 |- ( ph -> ( seq M ( .+ , F ) ` N ) = ( F ` N ) )
50		eqidd 2239	. 2 |- ( ( ph /\ k e. ( ZZ>= ` ( N + 1 ) ) ) -> ( F ` k ) = ( F ` k ) )
51	1, 49, 8, 9, 10, 50	seq3feq2 10913	1 |- ( ph -> ( seq M ( .+ , F ) |` ( ZZ>= ` N ) ) = seq N ( .+ , F ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   \/ wo 720   = wceq 1402   e. wcel 2209   A.wral 2528   |` cres 4776   ` cfv 5377  (class class class)co 6085   1c1 8180   + caddc 8182   - cmin 8497   ZZcz 9644   ZZ>=cuz 9921   ...cfz 10411   seqcseq 10884

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-ltadd 8295

This proof depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-inn 9305  df-n0 9564  df-z 9645  df-uz 9922  df-fz 10412  df-fzo 10550  df-seqfrec 10885

This theorem is used by:  seq3coll  11294  sumrbdclem  12144  prodrbdclem  12338