Theorem iseqid3s 9562

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.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

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 9127	. . 3 |- ( N e. ( ZZ>= ` M ) -> N e. ( M ... N ) )
3		fveq2 5209	. . . . . 6 |- ( w = M -> ( seq M ( .+ , F , S ) ` w ) = ( seq M ( .+ , F , S ) ` M ) )
4	3	eqeq1d 2090	. . . . 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 5209	. . . . . 6 |- ( w = k -> ( seq M ( .+ , F , S ) ` w ) = ( seq M ( .+ , F , S ) ` k ) )
7	6	eqeq1d 2090	. . . . 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 5209	. . . . . 6 |- ( w = ( k + 1 ) -> ( seq M ( .+ , F , S ) ` w ) = ( seq M ( .+ , F , S ) ` ( k + 1 ) ) )
10	9	eqeq1d 2090	. . . . 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 5209	. . . . . 6 |- ( w = N -> ( seq M ( .+ , F , S ) ` w ) = ( seq M ( .+ , F , S ) ` N ) )
13	12	eqeq1d 2090	. . . . 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 8705	. . . . . . . 8 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
16	1, 15	syl 14	. . . . . . 7 |- ( ph -> M e. ZZ )
17		iseqid3s.f	. . . . . . 7 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )
18		iseqid3s.cl	. . . . . . 7 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )
19	16, 17, 18	iseq1 9533	. . . . . 6 |- ( ph -> ( seq M ( .+ , F , S ) ` M ) = ( F ` M ) )
20		iseqid3s.3	. . . . . . . 8 |- ( ( ph /\ x e. ( M ... N ) ) -> ( F ` x ) = Z )
21	20	ralrimiva 2435	. . . . . . 7 |- ( ph -> A. x e. ( M ... N ) ( F ` x ) = Z )
22		eluzfz1 9126	. . . . . . . 8 |- ( N e. ( ZZ>= ` M ) -> M e. ( M ... N ) )
23		fveq2 5209	. . . . . . . . . 10 |- ( x = M -> ( F ` x ) = ( F ` M ) )
24	23	eqeq1d 2090	. . . . . . . . 9 |- ( x = M -> ( ( F ` x ) = Z <-> ( F ` M ) = Z ) )
25	24	rspcv 2698	. . . . . . . 8 |- ( M e. ( M ... N ) -> ( A. x e. ( M ... N ) ( F ` x ) = Z -> ( F ` M ) = Z ) )
26	1, 22, 25	3syl 17	. . . . . . 7 |- ( ph -> ( A. x e. ( M ... N ) ( F ` x ) = Z -> ( F ` M ) = Z ) )
27	21, 26	mpd 13	. . . . . 6 |- ( ph -> ( F ` M ) = Z )
28	19, 27	eqtrd 2114	. . . . 5 |- ( ph -> ( seq M ( .+ , F , S ) ` M ) = Z )
29	28	a1i 9	. . . 4 |- ( N e. ( ZZ>= ` M ) -> ( ph -> ( seq M ( .+ , F , S ) ` M ) = Z ) )
30		elfzouz 9238	. . . . . . . . . . 11 |- ( k e. ( M..^ N ) -> k e. ( ZZ>= ` M ) )
31	30	adantl 271	. . . . . . . . . 10 |- ( ( ph /\ k e. ( M..^ N ) ) -> k e. ( ZZ>= ` M ) )
32	17	adantlr 461	. . . . . . . . . 10 |- ( ( ( ph /\ k e. ( M..^ N ) ) /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )
33	18	adantlr 461	. . . . . . . . . 10 |- ( ( ( ph /\ k e. ( M..^ N ) ) /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )
34	31, 32, 33	iseqp1 9538	. . . . . . . . 9 |- ( ( ph /\ k e. ( M..^ N ) ) -> ( seq M ( .+ , F , S ) ` ( k + 1 ) ) = ( ( seq M ( .+ , F , S ) ` k ) .+ ( F ` ( k + 1 ) ) ) )
35	34	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 ) ) ) )
36		simpr 108	. . . . . . . . 9 |- ( ( ( ph /\ k e. ( M..^ N ) ) /\ ( seq M ( .+ , F , S ) ` k ) = Z ) -> ( seq M ( .+ , F , S ) ` k ) = Z )
37		fveq2 5209	. . . . . . . . . . . 12 |- ( x = ( k + 1 ) -> ( F ` x ) = ( F ` ( k + 1 ) ) )
38	37	eqeq1d 2090	. . . . . . . . . . 11 |- ( x = ( k + 1 ) -> ( ( F ` x ) = Z <-> ( F ` ( k + 1 ) ) = Z ) )
39	21	adantr 270	. . . . . . . . . . 11 |- ( ( ph /\ k e. ( M..^ N ) ) -> A. x e. ( M ... N ) ( F ` x ) = Z )
40		fzofzp1 9313	. . . . . . . . . . . 12 |- ( k e. ( M..^ N ) -> ( k + 1 ) e. ( M ... N ) )
41	40	adantl 271	. . . . . . . . . . 11 |- ( ( ph /\ k e. ( M..^ N ) ) -> ( k + 1 ) e. ( M ... N ) )
42	38, 39, 41	rspcdva 2708	. . . . . . . . . 10 |- ( ( ph /\ k e. ( M..^ N ) ) -> ( F ` ( k + 1 ) ) = Z )
43	42	adantr 270	. . . . . . . . 9 |- ( ( ( ph /\ k e. ( M..^ N ) ) /\ ( seq M ( .+ , F , S ) ` k ) = Z ) -> ( F ` ( k + 1 ) ) = Z )
44	36, 43	oveq12d 5561	. . . . . . . 8 |- ( ( ( ph /\ k e. ( M..^ N ) ) /\ ( seq M ( .+ , F , S ) ` k ) = Z ) -> ( ( seq M ( .+ , F , S ) ` k ) .+ ( F ` ( k + 1 ) ) ) = ( Z .+ Z ) )
45		iseqid3s.1	. . . . . . . . 9 |- ( ph -> ( Z .+ Z ) = Z )
46	45	ad2antrr 472	. . . . . . . 8 |- ( ( ( ph /\ k e. ( M..^ N ) ) /\ ( seq M ( .+ , F , S ) ` k ) = Z ) -> ( Z .+ Z ) = Z )
47	35, 44, 46	3eqtrd 2118	. . . . . . 7 |- ( ( ( ph /\ k e. ( M..^ N ) ) /\ ( seq M ( .+ , F , S ) ` k ) = Z ) -> ( seq M ( .+ , F , S ) ` ( k + 1 ) ) = Z )
48	47	ex 113	. . . . . 6 |- ( ( ph /\ k e. ( M..^ N ) ) -> ( ( seq M ( .+ , F , S ) ` k ) = Z -> ( seq M ( .+ , F , S ) ` ( k + 1 ) ) = Z ) )
49	48	expcom 114	. . . . 5 |- ( k e. ( M..^ N ) -> ( ph -> ( ( seq M ( .+ , F , S ) ` k ) = Z -> ( seq M ( .+ , F , S ) ` ( k + 1 ) ) = Z ) ) )
50	49	a2d 26	. . . 4 |- ( k e. ( M..^ N ) -> ( ( ph -> ( seq M ( .+ , F , S ) ` k ) = Z ) -> ( ph -> ( seq M ( .+ , F , S ) ` ( k + 1 ) ) = Z ) ) )
51	5, 8, 11, 14, 29, 50	fzind2 9325	. . 3 |- ( N e. ( M ... N ) -> ( ph -> ( seq M ( .+ , F , S ) ` N ) = Z ) )
52	1, 2, 51	3syl 17	. 2 |- ( ph -> ( ph -> ( seq M ( .+ , F , S ) ` N ) = Z ) )
53	52	pm2.43i 48	1 |- ( ph -> ( seq M ( .+ , F , S ) ` N ) = Z )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1285   e. wcel 1434   A.wral 2349   ` cfv 4932  (class class class)co 5543   1c1 7044   + caddc 7046   ZZcz 8432   ZZ>=cuz 8700   ...cfz 9105  ..^cfzo 9229   seqcseq 9521

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 2064  ax-coll 3901  ax-sep 3904  ax-nul 3912  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-iinf 4337  ax-cnex 7129  ax-resscn 7130  ax-1cn 7131  ax-1re 7132  ax-icn 7133  ax-addcl 7134  ax-addrcl 7135  ax-mulcl 7136  ax-addcom 7138  ax-addass 7140  ax-distr 7142  ax-i2m1 7143  ax-0lt1 7144  ax-0id 7146  ax-rnegex 7147  ax-cnre 7149  ax-pre-ltirr 7150  ax-pre-ltwlin 7151  ax-pre-lttrn 7152  ax-pre-ltadd 7154

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 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-nel 2341  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-tr 3884  df-id 4056  df-iord 4129  df-on 4131  df-ilim 4132  df-suc 4134  df-iom 4340  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-riota 5499  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-1st 5798  df-2nd 5799  df-recs 5954  df-frec 6040  df-pnf 7217  df-mnf 7218  df-xr 7219  df-ltxr 7220  df-le 7221  df-sub 7348  df-neg 7349  df-inn 8107  df-n0 8356  df-z 8433  df-uz 8701  df-fz 9106  df-fzo 9230  df-iseq 9522

This theorem is referenced by:  iseqid  9563  iser0  9568