Theorem seq3id3 10509

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 10034	. . 3 |- ( N e. ( ZZ>= ` M ) -> N e. ( M ... N ) )
3		fveqeq2 5526	. . . . 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 5526	. . . . 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 5526	. . . . 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 5526	. . . . 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 9535	. . . . . . . 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 10462	. . . . . 6 |- ( ph -> ( seq M ( .+ , F ) ` M ) = ( F ` M ) )
16		iseqid3s.3	. . . . . . . 8 |- ( ( ph /\ x e. ( M ... N ) ) -> ( F ` x ) = Z )
17	16	ralrimiva 2550	. . . . . . 7 |- ( ph -> A. x e. ( M ... N ) ( F ` x ) = Z )
18		eluzfz1 10033	. . . . . . . 8 |- ( N e. ( ZZ>= ` M ) -> M e. ( M ... N ) )
19		fveqeq2 5526	. . . . . . . . 9 |- ( x = M -> ( ( F ` x ) = Z <-> ( F ` M ) = Z ) )
20	19	rspcv 2839	. . . . . . . 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 2210	. . . . 5 |- ( ph -> ( seq M ( .+ , F ) ` M ) = Z )
24	23	a1i 9	. . . 4 |- ( N e. ( ZZ>= ` M ) -> ( ph -> ( seq M ( .+ , F ) ` M ) = Z ) )
25		elfzouz 10153	. . . . . . . . . . 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 10464	. . . . . . . . 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 5526	. . . . . . . . . . 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 10229	. . . . . . . . . . . 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 2848	. . . . . . . . . 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 5895	. . . . . . . 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 2214	. . . . . . 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 10241	. . 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 1353   e. wcel 2148   A.wral 2455   ` cfv 5218  (class class class)co 5877   1c1 7814   + caddc 7816   ZZcz 9255   ZZ>=cuz 9530   ...cfz 10010  ..^cfzo 10144   seqcseq 10447

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-addcom 7913  ax-addass 7915  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-0id 7921  ax-rnegex 7922  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927  ax-pre-ltadd 7929

This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-iord 4368  df-on 4370  df-ilim 4371  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-frec 6394  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-sub 8132  df-neg 8133  df-inn 8922  df-n0 9179  df-z 9256  df-uz 9531  df-fz 10011  df-fzo 10145  df-seqfrec 10448

This theorem is referenced by:  seq3id  10510  ser0  10516  prodf1  11552  mulgnn0z  13015  lgsval2lem  14450