Theorem seq3id3 10463

Description: A sequence that consists entirely of "zeroes" sums to "zero". More precisely, a constant sequence with value an element which is a .+ -idempotent sums (or " .+'s") to that element. (Contributed by Mario Carneiro, 15-Dec-2014.) (Revised by Jim Kingdon, 8-Apr-2023.)

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
seq3id3	|- ( ph -> ( seq M ( .+ , F ) ` 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 seq3id3

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 9988	. . 3 |- ( N e. ( ZZ>= ` M ) -> N e. ( M ... N ) )
3		fveqeq2 5505	. . . . 5 |- ( w = M -> ( ( seq M ( .+ , F ) ` w ) = Z <-> ( seq M ( .+ , F ) ` M ) = Z ) )
4	3	imbi2d 229	. . . 4 |- ( w = M -> ( ( ph -> ( seq M ( .+ , F ) ` w ) = Z ) <-> ( ph -> ( seq M ( .+ , F ) ` M ) = Z ) ) )
5		fveqeq2 5505	. . . . 5 |- ( w = k -> ( ( seq M ( .+ , F ) ` w ) = Z <-> ( seq M ( .+ , F ) ` k ) = Z ) )
6	5	imbi2d 229	. . . 4 |- ( w = k -> ( ( ph -> ( seq M ( .+ , F ) ` w ) = Z ) <-> ( ph -> ( seq M ( .+ , F ) ` k ) = Z ) ) )
7		fveqeq2 5505	. . . . 5 |- ( w = ( k + 1 ) -> ( ( seq M ( .+ , F ) ` w ) = Z <-> ( seq M ( .+ , F ) ` ( k + 1 ) ) = Z ) )
8	7	imbi2d 229	. . . 4 |- ( w = ( k + 1 ) -> ( ( ph -> ( seq M ( .+ , F ) ` w ) = Z ) <-> ( ph -> ( seq M ( .+ , F ) ` ( k + 1 ) ) = Z ) ) )
9		fveqeq2 5505	. . . . 5 |- ( w = N -> ( ( seq M ( .+ , F ) ` w ) = Z <-> ( seq M ( .+ , F ) ` N ) = Z ) )
10	9	imbi2d 229	. . . 4 |- ( w = N -> ( ( ph -> ( seq M ( .+ , F ) ` w ) = Z ) <-> ( ph -> ( seq M ( .+ , F ) ` N ) = Z ) ) )
11		eluzel2 9492	. . . . . . . 8 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
12	1, 11	syl 14	. . . . . . 7 |- ( ph -> M e. ZZ )
13		iseqid3s.f	. . . . . . 7 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )
14		iseqid3s.cl	. . . . . . 7 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )
15	12, 13, 14	seq3-1 10416	. . . . . 6 |- ( ph -> ( seq M ( .+ , F ) ` M ) = ( F ` M ) )
16		iseqid3s.3	. . . . . . . 8 |- ( ( ph /\ x e. ( M ... N ) ) -> ( F ` x ) = Z )
17	16	ralrimiva 2543	. . . . . . 7 |- ( ph -> A. x e. ( M ... N ) ( F ` x ) = Z )
18		eluzfz1 9987	. . . . . . . 8 |- ( N e. ( ZZ>= ` M ) -> M e. ( M ... N ) )
19		fveqeq2 5505	. . . . . . . . 9 |- ( x = M -> ( ( F ` x ) = Z <-> ( F ` M ) = Z ) )
20	19	rspcv 2830	. . . . . . . 8 |- ( M e. ( M ... N ) -> ( A. x e. ( M ... N ) ( F ` x ) = Z -> ( F ` M ) = Z ) )
21	1, 18, 20	3syl 17	. . . . . . 7 |- ( ph -> ( A. x e. ( M ... N ) ( F ` x ) = Z -> ( F ` M ) = Z ) )
22	17, 21	mpd 13	. . . . . 6 |- ( ph -> ( F ` M ) = Z )
23	15, 22	eqtrd 2203	. . . . 5 |- ( ph -> ( seq M ( .+ , F ) ` M ) = Z )
24	23	a1i 9	. . . 4 |- ( N e. ( ZZ>= ` M ) -> ( ph -> ( seq M ( .+ , F ) ` M ) = Z ) )
25		elfzouz 10107	. . . . . . . . . . 11 |- ( k e. ( M..^ N ) -> k e. ( ZZ>= ` M ) )
26	25	adantl 275	. . . . . . . . . 10 |- ( ( ph /\ k e. ( M..^ N ) ) -> k e. ( ZZ>= ` M ) )
27	13	adantlr 474	. . . . . . . . . 10 |- ( ( ( ph /\ k e. ( M..^ N ) ) /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )
28	14	adantlr 474	. . . . . . . . . 10 |- ( ( ( ph /\ k e. ( M..^ N ) ) /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )
29	26, 27, 28	seq3p1 10418	. . . . . . . . 9 |- ( ( ph /\ k e. ( M..^ N ) ) -> ( seq M ( .+ , F ) ` ( k + 1 ) ) = ( ( seq M ( .+ , F ) ` k ) .+ ( F ` ( k + 1 ) ) ) )
30	29	adantr 274	. . . . . . . 8 |- ( ( ( ph /\ k e. ( M..^ N ) ) /\ ( seq M ( .+ , F ) ` k ) = Z ) -> ( seq M ( .+ , F ) ` ( k + 1 ) ) = ( ( seq M ( .+ , F ) ` k ) .+ ( F ` ( k + 1 ) ) ) )
31		simpr 109	. . . . . . . . 9 |- ( ( ( ph /\ k e. ( M..^ N ) ) /\ ( seq M ( .+ , F ) ` k ) = Z ) -> ( seq M ( .+ , F ) ` k ) = Z )
32		fveqeq2 5505	. . . . . . . . . . 11 |- ( x = ( k + 1 ) -> ( ( F ` x ) = Z <-> ( F ` ( k + 1 ) ) = Z ) )
33	17	adantr 274	. . . . . . . . . . 11 |- ( ( ph /\ k e. ( M..^ N ) ) -> A. x e. ( M ... N ) ( F ` x ) = Z )
34		fzofzp1 10183	. . . . . . . . . . . 12 |- ( k e. ( M..^ N ) -> ( k + 1 ) e. ( M ... N ) )
35	34	adantl 275	. . . . . . . . . . 11 |- ( ( ph /\ k e. ( M..^ N ) ) -> ( k + 1 ) e. ( M ... N ) )
36	32, 33, 35	rspcdva 2839	. . . . . . . . . 10 |- ( ( ph /\ k e. ( M..^ N ) ) -> ( F ` ( k + 1 ) ) = Z )
37	36	adantr 274	. . . . . . . . 9 |- ( ( ( ph /\ k e. ( M..^ N ) ) /\ ( seq M ( .+ , F ) ` k ) = Z ) -> ( F ` ( k + 1 ) ) = Z )
38	31, 37	oveq12d 5871	. . . . . . . 8 |- ( ( ( ph /\ k e. ( M..^ N ) ) /\ ( seq M ( .+ , F ) ` k ) = Z ) -> ( ( seq M ( .+ , F ) ` k ) .+ ( F ` ( k + 1 ) ) ) = ( Z .+ Z ) )
39		iseqid3s.1	. . . . . . . . 9 |- ( ph -> ( Z .+ Z ) = Z )
40	39	ad2antrr 485	. . . . . . . 8 |- ( ( ( ph /\ k e. ( M..^ N ) ) /\ ( seq M ( .+ , F ) ` k ) = Z ) -> ( Z .+ Z ) = Z )
41	30, 38, 40	3eqtrd 2207	. . . . . . 7 |- ( ( ( ph /\ k e. ( M..^ N ) ) /\ ( seq M ( .+ , F ) ` k ) = Z ) -> ( seq M ( .+ , F ) ` ( k + 1 ) ) = Z )
42	41	ex 114	. . . . . 6 |- ( ( ph /\ k e. ( M..^ N ) ) -> ( ( seq M ( .+ , F ) ` k ) = Z -> ( seq M ( .+ , F ) ` ( k + 1 ) ) = Z ) )
43	42	expcom 115	. . . . 5 |- ( k e. ( M..^ N ) -> ( ph -> ( ( seq M ( .+ , F ) ` k ) = Z -> ( seq M ( .+ , F ) ` ( k + 1 ) ) = Z ) ) )
44	43	a2d 26	. . . 4 |- ( k e. ( M..^ N ) -> ( ( ph -> ( seq M ( .+ , F ) ` k ) = Z ) -> ( ph -> ( seq M ( .+ , F ) ` ( k + 1 ) ) = Z ) ) )
45	4, 6, 8, 10, 24, 44	fzind2 10195	. . 3 |- ( N e. ( M ... N ) -> ( ph -> ( seq M ( .+ , F ) ` N ) = Z ) )
46	1, 2, 45	3syl 17	. 2 |- ( ph -> ( ph -> ( seq M ( .+ , F ) ` N ) = Z ) )
47	46	pm2.43i 49	1 |- ( ph -> ( seq M ( .+ , F ) ` N ) = Z )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   e. wcel 2141   A.wral 2448   ` cfv 5198  (class class class)co 5853   1c1 7775   + caddc 7777   ZZcz 9212   ZZ>=cuz 9487   ...cfz 9965  ..^cfzo 10098   seqcseq 10401

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-addcom 7874  ax-addass 7876  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-0id 7882  ax-rnegex 7883  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-ltadd 7890

This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-frec 6370  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-inn 8879  df-n0 9136  df-z 9213  df-uz 9488  df-fz 9966  df-fzo 10099  df-seqfrec 10402

This theorem is referenced by:  seq3id  10464  ser0  10470  prodf1  11505  lgsval2lem  13705