Theorem seq3id3 10616

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 10107	. . 3 |- ( N e. ( ZZ>= ` M ) -> N e. ( M ... N ) )
3		fveqeq2 5567	. . . . 5 |- ( w = M -> ( ( seq M ( .+ , F ) ` w ) = Z <-> ( seq M ( .+ , F ) ` M ) = Z ) )
4	3	imbi2d 230	. . . 4 |- ( w = M -> ( ( ph -> ( seq M ( .+ , F ) ` w ) = Z ) <-> ( ph -> ( seq M ( .+ , F ) ` M ) = Z ) ) )
5		fveqeq2 5567	. . . . 5 |- ( w = k -> ( ( seq M ( .+ , F ) ` w ) = Z <-> ( seq M ( .+ , F ) ` k ) = Z ) )
6	5	imbi2d 230	. . . 4 |- ( w = k -> ( ( ph -> ( seq M ( .+ , F ) ` w ) = Z ) <-> ( ph -> ( seq M ( .+ , F ) ` k ) = Z ) ) )
7		fveqeq2 5567	. . . . 5 |- ( w = ( k + 1 ) -> ( ( seq M ( .+ , F ) ` w ) = Z <-> ( seq M ( .+ , F ) ` ( k + 1 ) ) = Z ) )
8	7	imbi2d 230	. . . 4 |- ( w = ( k + 1 ) -> ( ( ph -> ( seq M ( .+ , F ) ` w ) = Z ) <-> ( ph -> ( seq M ( .+ , F ) ` ( k + 1 ) ) = Z ) ) )
9		fveqeq2 5567	. . . . 5 |- ( w = N -> ( ( seq M ( .+ , F ) ` w ) = Z <-> ( seq M ( .+ , F ) ` N ) = Z ) )
10	9	imbi2d 230	. . . 4 |- ( w = N -> ( ( ph -> ( seq M ( .+ , F ) ` w ) = Z ) <-> ( ph -> ( seq M ( .+ , F ) ` N ) = Z ) ) )
11		eluzel2 9606	. . . . . . . 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 10554	. . . . . 6 |- ( ph -> ( seq M ( .+ , F ) ` M ) = ( F ` M ) )
16		iseqid3s.3	. . . . . . . 8 |- ( ( ph /\ x e. ( M ... N ) ) -> ( F ` x ) = Z )
17	16	ralrimiva 2570	. . . . . . 7 |- ( ph -> A. x e. ( M ... N ) ( F ` x ) = Z )
18		eluzfz1 10106	. . . . . . . 8 |- ( N e. ( ZZ>= ` M ) -> M e. ( M ... N ) )
19		fveqeq2 5567	. . . . . . . . 9 |- ( x = M -> ( ( F ` x ) = Z <-> ( F ` M ) = Z ) )
20	19	rspcv 2864	. . . . . . . 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 2229	. . . . 5 |- ( ph -> ( seq M ( .+ , F ) ` M ) = Z )
24	23	a1i 9	. . . 4 |- ( N e. ( ZZ>= ` M ) -> ( ph -> ( seq M ( .+ , F ) ` M ) = Z ) )
25		elfzouz 10226	. . . . . . . . . . 11 |- ( k e. ( M..^ N ) -> k e. ( ZZ>= ` M ) )
26	25	adantl 277	. . . . . . . . . 10 |- ( ( ph /\ k e. ( M..^ N ) ) -> k e. ( ZZ>= ` M ) )
27	13	adantlr 477	. . . . . . . . . 10 |- ( ( ( ph /\ k e. ( M..^ N ) ) /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )
28	14	adantlr 477	. . . . . . . . . 10 |- ( ( ( ph /\ k e. ( M..^ N ) ) /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )
29	26, 27, 28	seq3p1 10557	. . . . . . . . 9 |- ( ( ph /\ k e. ( M..^ N ) ) -> ( seq M ( .+ , F ) ` ( k + 1 ) ) = ( ( seq M ( .+ , F ) ` k ) .+ ( F ` ( k + 1 ) ) ) )
30	29	adantr 276	. . . . . . . 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 110	. . . . . . . . 9 |- ( ( ( ph /\ k e. ( M..^ N ) ) /\ ( seq M ( .+ , F ) ` k ) = Z ) -> ( seq M ( .+ , F ) ` k ) = Z )
32		fveqeq2 5567	. . . . . . . . . . 11 |- ( x = ( k + 1 ) -> ( ( F ` x ) = Z <-> ( F ` ( k + 1 ) ) = Z ) )
33	17	adantr 276	. . . . . . . . . . 11 |- ( ( ph /\ k e. ( M..^ N ) ) -> A. x e. ( M ... N ) ( F ` x ) = Z )
34		fzofzp1 10303	. . . . . . . . . . . 12 |- ( k e. ( M..^ N ) -> ( k + 1 ) e. ( M ... N ) )
35	34	adantl 277	. . . . . . . . . . 11 |- ( ( ph /\ k e. ( M..^ N ) ) -> ( k + 1 ) e. ( M ... N ) )
36	32, 33, 35	rspcdva 2873	. . . . . . . . . 10 |- ( ( ph /\ k e. ( M..^ N ) ) -> ( F ` ( k + 1 ) ) = Z )
37	36	adantr 276	. . . . . . . . 9 |- ( ( ( ph /\ k e. ( M..^ N ) ) /\ ( seq M ( .+ , F ) ` k ) = Z ) -> ( F ` ( k + 1 ) ) = Z )
38	31, 37	oveq12d 5940	. . . . . . . 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 488	. . . . . . . 8 |- ( ( ( ph /\ k e. ( M..^ N ) ) /\ ( seq M ( .+ , F ) ` k ) = Z ) -> ( Z .+ Z ) = Z )
41	30, 38, 40	3eqtrd 2233	. . . . . . 7 |- ( ( ( ph /\ k e. ( M..^ N ) ) /\ ( seq M ( .+ , F ) ` k ) = Z ) -> ( seq M ( .+ , F ) ` ( k + 1 ) ) = Z )
42	41	ex 115	. . . . . 6 |- ( ( ph /\ k e. ( M..^ N ) ) -> ( ( seq M ( .+ , F ) ` k ) = Z -> ( seq M ( .+ , F ) ` ( k + 1 ) ) = Z ) )
43	42	expcom 116	. . . . 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 10315	. . 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 104   = wceq 1364   e. wcel 2167   A.wral 2475   ` cfv 5258  (class class class)co 5922   1c1 7880   + caddc 7882   ZZcz 9326   ZZ>=cuz 9601   ...cfz 10083  ..^cfzo 10217   seqcseq 10539

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-ltadd 7995

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-iord 4401  df-on 4403  df-ilim 4404  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-frec 6449  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-inn 8991  df-n0 9250  df-z 9327  df-uz 9602  df-fz 10084  df-fzo 10218  df-seqfrec 10540

This theorem is referenced by:  seq3id  10617  ser0  10625  prodf1  11707  mulgnn0z  13279  lgsval2lem  15251