Theorem iseqid3s 9466

Description: A sequence that consists of zeroes up to N sums to zero at N. In this case by "zero" we mean whatever the identity Z is for the operation .+). (Contributed by Jim Kingdon, 18-Aug-2021.)

Hypotheses

Ref	Expression
iseqid3s.1	|- ( ph -> ( Z .+ Z ) = Z )
iseqid3s.2	|- ( ph -> N e. ( ZZ>= ` M ) )
iseqid3s.3	|- ( ( ph /\ x e. ( M ... N ) ) -> ( F ` x ) = Z )
iseqid3s.z	|- ( ph -> Z e. S )
iseqid3s.s	|- ( ph -> S e. V )
iseqid3s.f	|- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )
iseqid3s.cl	|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )

Assertion

Ref	Expression
iseqid3s	|- ( ph -> ( seq M ( .+ , F , S ) ` N ) = Z )

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

Allowed substitution hints:   V( x, y)

Proof of Theorem iseqid3s

Dummy variables k w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		iseqid3s.2	. . 3 |- ( ph -> N e. ( ZZ>= ` M ) )
2		eluzfz2 9051	. . 3 |- ( N e. ( ZZ>= ` M ) -> N e. ( M ... N ) )
3		fveq2 5198	. . . . . 6 |- ( w = M -> ( seq M ( .+ , F , S ) ` w ) = ( seq M ( .+ , F , S ) ` M ) )
4	3	eqeq1d 2089	. . . . 5 |- ( w = M -> ( ( seq M ( .+ , F , S ) ` w ) = Z <-> ( seq M ( .+ , F , S ) ` M ) = Z ) )
5	4	imbi2d 228	. . . 4 |- ( w = M -> ( ( ph -> ( seq M ( .+ , F , S ) ` w ) = Z ) <-> ( ph -> ( seq M ( .+ , F , S ) ` M ) = Z ) ) )
6		fveq2 5198	. . . . . 6 |- ( w = k -> ( seq M ( .+ , F , S ) ` w ) = ( seq M ( .+ , F , S ) ` k ) )
7	6	eqeq1d 2089	. . . . 5 |- ( w = k -> ( ( seq M ( .+ , F , S ) ` w ) = Z <-> ( seq M ( .+ , F , S ) ` k ) = Z ) )
8	7	imbi2d 228	. . . 4 |- ( w = k -> ( ( ph -> ( seq M ( .+ , F , S ) ` w ) = Z ) <-> ( ph -> ( seq M ( .+ , F , S ) ` k ) = Z ) ) )
9		fveq2 5198	. . . . . 6 |- ( w = ( k + 1 ) -> ( seq M ( .+ , F , S ) ` w ) = ( seq M ( .+ , F , S ) ` ( k + 1 ) ) )
10	9	eqeq1d 2089	. . . . 5 |- ( w = ( k + 1 ) -> ( ( seq M ( .+ , F , S ) ` w ) = Z <-> ( seq M ( .+ , F , S ) ` ( k + 1 ) ) = Z ) )
11	10	imbi2d 228	. . . 4 |- ( w = ( k + 1 ) -> ( ( ph -> ( seq M ( .+ , F , S ) ` w ) = Z ) <-> ( ph -> ( seq M ( .+ , F , S ) ` ( k + 1 ) ) = Z ) ) )
12		fveq2 5198	. . . . . 6 |- ( w = N -> ( seq M ( .+ , F , S ) ` w ) = ( seq M ( .+ , F , S ) ` N ) )
13	12	eqeq1d 2089	. . . . 5 |- ( w = N -> ( ( seq M ( .+ , F , S ) ` w ) = Z <-> ( seq M ( .+ , F , S ) ` N ) = Z ) )
14	13	imbi2d 228	. . . 4 |- ( w = N -> ( ( ph -> ( seq M ( .+ , F , S ) ` w ) = Z ) <-> ( ph -> ( seq M ( .+ , F , S ) ` N ) = Z ) ) )
15		eluzel2 8624	. . . . . . . 8 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
16	1, 15	syl 14	. . . . . . 7 |- ( ph -> M e. ZZ )
17		iseqid3s.s	. . . . . . 7 |- ( ph -> S e. V )
18		iseqid3s.f	. . . . . . 7 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )
19		iseqid3s.cl	. . . . . . 7 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )
20	16, 17, 18, 19	iseq1 9442	. . . . . 6 |- ( ph -> ( seq M ( .+ , F , S ) ` M ) = ( F ` M ) )
21		iseqid3s.3	. . . . . . . 8 |- ( ( ph /\ x e. ( M ... N ) ) -> ( F ` x ) = Z )
22	21	ralrimiva 2434	. . . . . . 7 |- ( ph -> A. x e. ( M ... N ) ( F ` x ) = Z )
23		eluzfz1 9050	. . . . . . . 8 |- ( N e. ( ZZ>= ` M ) -> M e. ( M ... N ) )
24		fveq2 5198	. . . . . . . . . 10 |- ( x = M -> ( F ` x ) = ( F ` M ) )
25	24	eqeq1d 2089	. . . . . . . . 9 |- ( x = M -> ( ( F ` x ) = Z <-> ( F ` M ) = Z ) )
26	25	rspcv 2697	. . . . . . . 8 |- ( M e. ( M ... N ) -> ( A. x e. ( M ... N ) ( F ` x ) = Z -> ( F ` M ) = Z ) )
27	1, 23, 26	3syl 17	. . . . . . 7 |- ( ph -> ( A. x e. ( M ... N ) ( F ` x ) = Z -> ( F ` M ) = Z ) )
28	22, 27	mpd 13	. . . . . 6 |- ( ph -> ( F ` M ) = Z )
29	20, 28	eqtrd 2113	. . . . 5 |- ( ph -> ( seq M ( .+ , F , S ) ` M ) = Z )
30	29	a1i 9	. . . 4 |- ( N e. ( ZZ>= ` M ) -> ( ph -> ( seq M ( .+ , F , S ) ` M ) = Z ) )
31		elfzouz 9161	. . . . . . . . . . 11 |- ( k e. ( M..^ N ) -> k e. ( ZZ>= ` M ) )
32	31	adantl 271	. . . . . . . . . 10 |- ( ( ph /\ k e. ( M..^ N ) ) -> k e. ( ZZ>= ` M ) )
33	17	adantr 270	. . . . . . . . . 10 |- ( ( ph /\ k e. ( M..^ N ) ) -> S e. V )
34	18	adantlr 460	. . . . . . . . . 10 |- ( ( ( ph /\ k e. ( M..^ N ) ) /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )
35	19	adantlr 460	. . . . . . . . . 10 |- ( ( ( ph /\ k e. ( M..^ N ) ) /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )
36	32, 33, 34, 35	iseqp1 9445	. . . . . . . . 9 |- ( ( ph /\ k e. ( M..^ N ) ) -> ( seq M ( .+ , F , S ) ` ( k + 1 ) ) = ( ( seq M ( .+ , F , S ) ` k ) .+ ( F ` ( k + 1 ) ) ) )
37	36	adantr 270	. . . . . . . 8 |- ( ( ( ph /\ k e. ( M..^ N ) ) /\ ( seq M ( .+ , F , S ) ` k ) = Z ) -> ( seq M ( .+ , F , S ) ` ( k + 1 ) ) = ( ( seq M ( .+ , F , S ) ` k ) .+ ( F ` ( k + 1 ) ) ) )
38		simpr 108	. . . . . . . . 9 |- ( ( ( ph /\ k e. ( M..^ N ) ) /\ ( seq M ( .+ , F , S ) ` k ) = Z ) -> ( seq M ( .+ , F , S ) ` k ) = Z )
39		fzofzp1 9236	. . . . . . . . . . . 12 |- ( k e. ( M..^ N ) -> ( k + 1 ) e. ( M ... N ) )
40	39	adantl 271	. . . . . . . . . . 11 |- ( ( ph /\ k e. ( M..^ N ) ) -> ( k + 1 ) e. ( M ... N ) )
41	22	adantr 270	. . . . . . . . . . 11 |- ( ( ph /\ k e. ( M..^ N ) ) -> A. x e. ( M ... N ) ( F ` x ) = Z )
42		fveq2 5198	. . . . . . . . . . . . 13 |- ( x = ( k + 1 ) -> ( F ` x ) = ( F ` ( k + 1 ) ) )
43	42	eqeq1d 2089	. . . . . . . . . . . 12 |- ( x = ( k + 1 ) -> ( ( F ` x ) = Z <-> ( F ` ( k + 1 ) ) = Z ) )
44	43	rspcv 2697	. . . . . . . . . . 11 |- ( ( k + 1 ) e. ( M ... N ) -> ( A. x e. ( M ... N ) ( F ` x ) = Z -> ( F ` ( k + 1 ) ) = Z ) )
45	40, 41, 44	sylc 61	. . . . . . . . . 10 |- ( ( ph /\ k e. ( M..^ N ) ) -> ( F ` ( k + 1 ) ) = Z )
46	45	adantr 270	. . . . . . . . 9 |- ( ( ( ph /\ k e. ( M..^ N ) ) /\ ( seq M ( .+ , F , S ) ` k ) = Z ) -> ( F ` ( k + 1 ) ) = Z )
47	38, 46	oveq12d 5550	. . . . . . . 8 |- ( ( ( ph /\ k e. ( M..^ N ) ) /\ ( seq M ( .+ , F , S ) ` k ) = Z ) -> ( ( seq M ( .+ , F , S ) ` k ) .+ ( F ` ( k + 1 ) ) ) = ( Z .+ Z ) )
48		iseqid3s.1	. . . . . . . . 9 |- ( ph -> ( Z .+ Z ) = Z )
49	48	ad2antrr 471	. . . . . . . 8 |- ( ( ( ph /\ k e. ( M..^ N ) ) /\ ( seq M ( .+ , F , S ) ` k ) = Z ) -> ( Z .+ Z ) = Z )
50	37, 47, 49	3eqtrd 2117	. . . . . . 7 |- ( ( ( ph /\ k e. ( M..^ N ) ) /\ ( seq M ( .+ , F , S ) ` k ) = Z ) -> ( seq M ( .+ , F , S ) ` ( k + 1 ) ) = Z )
51	50	ex 113	. . . . . 6 |- ( ( ph /\ k e. ( M..^ N ) ) -> ( ( seq M ( .+ , F , S ) ` k ) = Z -> ( seq M ( .+ , F , S ) ` ( k + 1 ) ) = Z ) )
52	51	expcom 114	. . . . 5 |- ( k e. ( M..^ N ) -> ( ph -> ( ( seq M ( .+ , F , S ) ` k ) = Z -> ( seq M ( .+ , F , S ) ` ( k + 1 ) ) = Z ) ) )
53	52	a2d 26	. . . 4 |- ( k e. ( M..^ N ) -> ( ( ph -> ( seq M ( .+ , F , S ) ` k ) = Z ) -> ( ph -> ( seq M ( .+ , F , S ) ` ( k + 1 ) ) = Z ) ) )
54	5, 8, 11, 14, 30, 53	fzind2 9248	. . 3 |- ( N e. ( M ... N ) -> ( ph -> ( seq M ( .+ , F , S ) ` N ) = Z ) )
55	1, 2, 54	3syl 17	. 2 |- ( ph -> ( ph -> ( seq M ( .+ , F , S ) ` N ) = Z ) )
56	55	pm2.43i 48	1 |- ( ph -> ( seq M ( .+ , F , S ) ` N ) = Z )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   e. wcel 1433   A.wral 2348   ` cfv 4922  (class class class)co 5532   1c1 6982   + caddc 6984   ZZcz 8351   ZZ>=cuz 8619   ...cfz 9029  ..^cfzo 9152   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:  iseqid  9467  iser0  9471