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	⊢ (𝜑 → (𝑍 + 𝑍) = 𝑍)
iseqid3s.2	⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
iseqid3s.3	⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘𝑥) = 𝑍)
iseqid3s.z	⊢ (𝜑 → 𝑍 ∈ 𝑆)
iseqid3s.f	⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆)
iseqid3s.cl	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)

Assertion

Ref	Expression
seq3id3	⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = 𝑍)

Distinct variable groups:   𝑥,𝑦, +   𝑥,𝐹,𝑦   𝑥,𝑀,𝑦   𝜑,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 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
2		eluzfz2 9988	. . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁))
3		fveqeq2 5505	. . . . 5 ⊢ (𝑤 = 𝑀 → ((seq𝑀( + , 𝐹)‘𝑤) = 𝑍 ↔ (seq𝑀( + , 𝐹)‘𝑀) = 𝑍))
4	3	imbi2d 229	. . . 4 ⊢ (𝑤 = 𝑀 → ((𝜑 → (seq𝑀( + , 𝐹)‘𝑤) = 𝑍) ↔ (𝜑 → (seq𝑀( + , 𝐹)‘𝑀) = 𝑍)))
5		fveqeq2 5505	. . . . 5 ⊢ (𝑤 = 𝑘 → ((seq𝑀( + , 𝐹)‘𝑤) = 𝑍 ↔ (seq𝑀( + , 𝐹)‘𝑘) = 𝑍))
6	5	imbi2d 229	. . . 4 ⊢ (𝑤 = 𝑘 → ((𝜑 → (seq𝑀( + , 𝐹)‘𝑤) = 𝑍) ↔ (𝜑 → (seq𝑀( + , 𝐹)‘𝑘) = 𝑍)))
7		fveqeq2 5505	. . . . 5 ⊢ (𝑤 = (𝑘 + 1) → ((seq𝑀( + , 𝐹)‘𝑤) = 𝑍 ↔ (seq𝑀( + , 𝐹)‘(𝑘 + 1)) = 𝑍))
8	7	imbi2d 229	. . . 4 ⊢ (𝑤 = (𝑘 + 1) → ((𝜑 → (seq𝑀( + , 𝐹)‘𝑤) = 𝑍) ↔ (𝜑 → (seq𝑀( + , 𝐹)‘(𝑘 + 1)) = 𝑍)))
9		fveqeq2 5505	. . . . 5 ⊢ (𝑤 = 𝑁 → ((seq𝑀( + , 𝐹)‘𝑤) = 𝑍 ↔ (seq𝑀( + , 𝐹)‘𝑁) = 𝑍))
10	9	imbi2d 229	. . . 4 ⊢ (𝑤 = 𝑁 → ((𝜑 → (seq𝑀( + , 𝐹)‘𝑤) = 𝑍) ↔ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = 𝑍)))
11		eluzel2 9492	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ)
12	1, 11	syl 14	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℤ)
13		iseqid3s.f	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆)
14		iseqid3s.cl	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
15	12, 13, 14	seq3-1 10416	. . . . . 6 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹‘𝑀))
16		iseqid3s.3	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘𝑥) = 𝑍)
17	16	ralrimiva 2543	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ (𝑀...𝑁)(𝐹‘𝑥) = 𝑍)
18		eluzfz1 9987	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁))
19		fveqeq2 5505	. . . . . . . . 9 ⊢ (𝑥 = 𝑀 → ((𝐹‘𝑥) = 𝑍 ↔ (𝐹‘𝑀) = 𝑍))
20	19	rspcv 2830	. . . . . . . 8 ⊢ (𝑀 ∈ (𝑀...𝑁) → (∀𝑥 ∈ (𝑀...𝑁)(𝐹‘𝑥) = 𝑍 → (𝐹‘𝑀) = 𝑍))
21	1, 18, 20	3syl 17	. . . . . . 7 ⊢ (𝜑 → (∀𝑥 ∈ (𝑀...𝑁)(𝐹‘𝑥) = 𝑍 → (𝐹‘𝑀) = 𝑍))
22	17, 21	mpd 13	. . . . . 6 ⊢ (𝜑 → (𝐹‘𝑀) = 𝑍)
23	15, 22	eqtrd 2203	. . . . 5 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑀) = 𝑍)
24	23	a1i 9	. . . 4 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝜑 → (seq𝑀( + , 𝐹)‘𝑀) = 𝑍))
25		elfzouz 10107	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (𝑀..^𝑁) → 𝑘 ∈ (ℤ≥‘𝑀))
26	25	adantl 275	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑘 ∈ (ℤ≥‘𝑀))
27	13	adantlr 474	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆)
28	14	adantlr 474	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
29	26, 27, 28	seq3p1 10418	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (seq𝑀( + , 𝐹)‘(𝑘 + 1)) = ((seq𝑀( + , 𝐹)‘𝑘) + (𝐹‘(𝑘 + 1))))
30	29	adantr 274	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ (seq𝑀( + , 𝐹)‘𝑘) = 𝑍) → (seq𝑀( + , 𝐹)‘(𝑘 + 1)) = ((seq𝑀( + , 𝐹)‘𝑘) + (𝐹‘(𝑘 + 1))))
31		simpr 109	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ (seq𝑀( + , 𝐹)‘𝑘) = 𝑍) → (seq𝑀( + , 𝐹)‘𝑘) = 𝑍)
32		fveqeq2 5505	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑘 + 1) → ((𝐹‘𝑥) = 𝑍 ↔ (𝐹‘(𝑘 + 1)) = 𝑍))
33	17	adantr 274	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ∀𝑥 ∈ (𝑀...𝑁)(𝐹‘𝑥) = 𝑍)
34		fzofzp1 10183	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (𝑀..^𝑁) → (𝑘 + 1) ∈ (𝑀...𝑁))
35	34	adantl 275	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘 + 1) ∈ (𝑀...𝑁))
36	32, 33, 35	rspcdva 2839	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝐹‘(𝑘 + 1)) = 𝑍)
37	36	adantr 274	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ (seq𝑀( + , 𝐹)‘𝑘) = 𝑍) → (𝐹‘(𝑘 + 1)) = 𝑍)
38	31, 37	oveq12d 5871	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ (seq𝑀( + , 𝐹)‘𝑘) = 𝑍) → ((seq𝑀( + , 𝐹)‘𝑘) + (𝐹‘(𝑘 + 1))) = (𝑍 + 𝑍))
39		iseqid3s.1	. . . . . . . . 9 ⊢ (𝜑 → (𝑍 + 𝑍) = 𝑍)
40	39	ad2antrr 485	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ (seq𝑀( + , 𝐹)‘𝑘) = 𝑍) → (𝑍 + 𝑍) = 𝑍)
41	30, 38, 40	3eqtrd 2207	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ (seq𝑀( + , 𝐹)‘𝑘) = 𝑍) → (seq𝑀( + , 𝐹)‘(𝑘 + 1)) = 𝑍)
42	41	ex 114	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ((seq𝑀( + , 𝐹)‘𝑘) = 𝑍 → (seq𝑀( + , 𝐹)‘(𝑘 + 1)) = 𝑍))
43	42	expcom 115	. . . . 5 ⊢ (𝑘 ∈ (𝑀..^𝑁) → (𝜑 → ((seq𝑀( + , 𝐹)‘𝑘) = 𝑍 → (seq𝑀( + , 𝐹)‘(𝑘 + 1)) = 𝑍)))
44	43	a2d 26	. . . 4 ⊢ (𝑘 ∈ (𝑀..^𝑁) → ((𝜑 → (seq𝑀( + , 𝐹)‘𝑘) = 𝑍) → (𝜑 → (seq𝑀( + , 𝐹)‘(𝑘 + 1)) = 𝑍)))
45	4, 6, 8, 10, 24, 44	fzind2 10195	. . 3 ⊢ (𝑁 ∈ (𝑀...𝑁) → (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = 𝑍))
46	1, 2, 45	3syl 17	. 2 ⊢ (𝜑 → (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = 𝑍))
47	46	pm2.43i 49	1 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = 𝑍)

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1348   ∈ wcel 2141  ∀wral 2448  ‘cfv 5198  (class class class)co 5853  1c1 7775   + caddc 7777  ℤcz 9212  ℤ_≥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