Theorem seq3id3 10280

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 9812	. . 3 |- ( N e. ( ZZ>= ` M ) -> N e. ( M ... N ) )
3		fveqeq2 5430	. . . . 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 5430	. . . . 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 5430	. . . . 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 5430	. . . . 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 9331	. . . . . . . 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 10233	. . . . . 6 |- ( ph -> ( seq M ( .+ , F ) ` M ) = ( F ` M ) )
16		iseqid3s.3	. . . . . . . 8 |- ( ( ph /\ x e. ( M ... N ) ) -> ( F ` x ) = Z )
17	16	ralrimiva 2505	. . . . . . 7 |- ( ph -> A. x e. ( M ... N ) ( F ` x ) = Z )
18		eluzfz1 9811	. . . . . . . 8 |- ( N e. ( ZZ>= ` M ) -> M e. ( M ... N ) )
19		fveqeq2 5430	. . . . . . . . 9 |- ( x = M -> ( ( F ` x ) = Z <-> ( F ` M ) = Z ) )
20	19	rspcv 2785	. . . . . . . 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 2172	. . . . 5 |- ( ph -> ( seq M ( .+ , F ) ` M ) = Z )
24	23	a1i 9	. . . 4 |- ( N e. ( ZZ>= ` M ) -> ( ph -> ( seq M ( .+ , F ) ` M ) = Z ) )
25		elfzouz 9928	. . . . . . . . . . 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 468	. . . . . . . . . 10 |- ( ( ( ph /\ k e. ( M..^ N ) ) /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )
28	14	adantlr 468	. . . . . . . . . 10 |- ( ( ( ph /\ k e. ( M..^ N ) ) /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )
29	26, 27, 28	seq3p1 10235	. . . . . . . . 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 5430	. . . . . . . . . . 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 10004	. . . . . . . . . . . 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 2794	. . . . . . . . . 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 5792	. . . . . . . 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 479	. . . . . . . 8 |- ( ( ( ph /\ k e. ( M..^ N ) ) /\ ( seq M ( .+ , F ) ` k ) = Z ) -> ( Z .+ Z ) = Z )
41	30, 38, 40	3eqtrd 2176	. . . . . . 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 10016	. . 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 1331   e. wcel 1480   A.wral 2416   ` cfv 5123  (class class class)co 5774   1c1 7621   + caddc 7623   ZZcz 9054   ZZ>=cuz 9326   ...cfz 9790  ..^cfzo 9919   seqcseq 10218

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-addcom 7720  ax-addass 7722  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-0id 7728  ax-rnegex 7729  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-ltadd 7736

This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-frec 6288  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-sub 7935  df-neg 7936  df-inn 8721  df-n0 8978  df-z 9055  df-uz 9327  df-fz 9791  df-fzo 9920  df-seqfrec 10219

This theorem is referenced by:  seq3id  10281  ser0  10287  prodf1  11311