Theorem serif0 10189

Description: If an infinite series converges, its underlying sequence converges to zero. (Contributed by NM, 2-Sep-2005.) (Revised by Mario Carneiro, 16-Feb-2014.)

Hypotheses

Ref	Expression
climcauc.1	⊢ 𝑍 = (ℤ≥‘𝑀)
serif0.2	⊢ (𝜑 → 𝑀 ∈ ℤ)
serif0.3	⊢ (𝜑 → 𝐹 ∈ 𝑉)
serif0.4	⊢ (𝜑 → seq𝑀( + , 𝐹, ℂ) ∈ dom ⇝ )
serif0.5	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)

Assertion

Ref	Expression
serif0	⊢ (𝜑 → 𝐹 ⇝ 0)

Distinct variable groups:   𝑘,𝐹   𝑘,𝑀   𝑘,𝑍   𝜑,𝑘   𝑘,𝑉

Proof of Theorem serif0

Dummy variables 𝑗 𝑚 𝑛 𝑥 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		serif0.2	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ)
2		serif0.4	. . . . 5 ⊢ (𝜑 → seq𝑀( + , 𝐹, ℂ) ∈ dom ⇝ )
3		climcauc.1	. . . . . 6 ⊢ 𝑍 = (ℤ≥‘𝑀)
4	3	climcaucn 10188	. . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ seq𝑀( + , 𝐹, ℂ) ∈ dom ⇝ ) → ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑚 ∈ (ℤ≥‘𝑗)((seq𝑀( + , 𝐹, ℂ)‘𝑚) ∈ ℂ ∧ (abs‘((seq𝑀( + , 𝐹, ℂ)‘𝑚) − (seq𝑀( + , 𝐹, ℂ)‘𝑗))) < 𝑥))
5	1, 2, 4	syl2anc 403	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑚 ∈ (ℤ≥‘𝑗)((seq𝑀( + , 𝐹, ℂ)‘𝑚) ∈ ℂ ∧ (abs‘((seq𝑀( + , 𝐹, ℂ)‘𝑚) − (seq𝑀( + , 𝐹, ℂ)‘𝑗))) < 𝑥))
6	3	cau3 10001	. . . 4 ⊢ (∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑚 ∈ (ℤ≥‘𝑗)((seq𝑀( + , 𝐹, ℂ)‘𝑚) ∈ ℂ ∧ (abs‘((seq𝑀( + , 𝐹, ℂ)‘𝑚) − (seq𝑀( + , 𝐹, ℂ)‘𝑗))) < 𝑥) ↔ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑚 ∈ (ℤ≥‘𝑗)((seq𝑀( + , 𝐹, ℂ)‘𝑚) ∈ ℂ ∧ ∀𝑘 ∈ (ℤ≥‘𝑚)(abs‘((seq𝑀( + , 𝐹, ℂ)‘𝑚) − (seq𝑀( + , 𝐹, ℂ)‘𝑘))) < 𝑥))
7	5, 6	sylib 120	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑚 ∈ (ℤ≥‘𝑗)((seq𝑀( + , 𝐹, ℂ)‘𝑚) ∈ ℂ ∧ ∀𝑘 ∈ (ℤ≥‘𝑚)(abs‘((seq𝑀( + , 𝐹, ℂ)‘𝑚) − (seq𝑀( + , 𝐹, ℂ)‘𝑘))) < 𝑥))
8	3	peano2uzs 8672	. . . . . . 7 ⊢ (𝑗 ∈ 𝑍 → (𝑗 + 1) ∈ 𝑍)
9	8	adantl 271	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝑗 + 1) ∈ 𝑍)
10		eluzelz 8628	. . . . . . . . . 10 ⊢ (𝑚 ∈ (ℤ≥‘𝑗) → 𝑚 ∈ ℤ)
11		uzid 8633	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℤ → 𝑚 ∈ (ℤ≥‘𝑚))
12		peano2uz 8671	. . . . . . . . . 10 ⊢ (𝑚 ∈ (ℤ≥‘𝑚) → (𝑚 + 1) ∈ (ℤ≥‘𝑚))
13		fveq2 5198	. . . . . . . . . . . . . 14 ⊢ (𝑘 = (𝑚 + 1) → (seq𝑀( + , 𝐹, ℂ)‘𝑘) = (seq𝑀( + , 𝐹, ℂ)‘(𝑚 + 1)))
14	13	oveq2d 5548	. . . . . . . . . . . . 13 ⊢ (𝑘 = (𝑚 + 1) → ((seq𝑀( + , 𝐹, ℂ)‘𝑚) − (seq𝑀( + , 𝐹, ℂ)‘𝑘)) = ((seq𝑀( + , 𝐹, ℂ)‘𝑚) − (seq𝑀( + , 𝐹, ℂ)‘(𝑚 + 1))))
15	14	fveq2d 5202	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝑚 + 1) → (abs‘((seq𝑀( + , 𝐹, ℂ)‘𝑚) − (seq𝑀( + , 𝐹, ℂ)‘𝑘))) = (abs‘((seq𝑀( + , 𝐹, ℂ)‘𝑚) − (seq𝑀( + , 𝐹, ℂ)‘(𝑚 + 1)))))
16	15	breq1d 3795	. . . . . . . . . . 11 ⊢ (𝑘 = (𝑚 + 1) → ((abs‘((seq𝑀( + , 𝐹, ℂ)‘𝑚) − (seq𝑀( + , 𝐹, ℂ)‘𝑘))) < 𝑥 ↔ (abs‘((seq𝑀( + , 𝐹, ℂ)‘𝑚) − (seq𝑀( + , 𝐹, ℂ)‘(𝑚 + 1)))) < 𝑥))
17	16	rspcv 2697	. . . . . . . . . 10 ⊢ ((𝑚 + 1) ∈ (ℤ≥‘𝑚) → (∀𝑘 ∈ (ℤ≥‘𝑚)(abs‘((seq𝑀( + , 𝐹, ℂ)‘𝑚) − (seq𝑀( + , 𝐹, ℂ)‘𝑘))) < 𝑥 → (abs‘((seq𝑀( + , 𝐹, ℂ)‘𝑚) − (seq𝑀( + , 𝐹, ℂ)‘(𝑚 + 1)))) < 𝑥))
18	10, 11, 12, 17	4syl 18	. . . . . . . . 9 ⊢ (𝑚 ∈ (ℤ≥‘𝑗) → (∀𝑘 ∈ (ℤ≥‘𝑚)(abs‘((seq𝑀( + , 𝐹, ℂ)‘𝑚) − (seq𝑀( + , 𝐹, ℂ)‘𝑘))) < 𝑥 → (abs‘((seq𝑀( + , 𝐹, ℂ)‘𝑚) − (seq𝑀( + , 𝐹, ℂ)‘(𝑚 + 1)))) < 𝑥))
19	18	adantld 272	. . . . . . . 8 ⊢ (𝑚 ∈ (ℤ≥‘𝑗) → (((seq𝑀( + , 𝐹, ℂ)‘𝑚) ∈ ℂ ∧ ∀𝑘 ∈ (ℤ≥‘𝑚)(abs‘((seq𝑀( + , 𝐹, ℂ)‘𝑚) − (seq𝑀( + , 𝐹, ℂ)‘𝑘))) < 𝑥) → (abs‘((seq𝑀( + , 𝐹, ℂ)‘𝑚) − (seq𝑀( + , 𝐹, ℂ)‘(𝑚 + 1)))) < 𝑥))
20	19	ralimia 2424	. . . . . . 7 ⊢ (∀𝑚 ∈ (ℤ≥‘𝑗)((seq𝑀( + , 𝐹, ℂ)‘𝑚) ∈ ℂ ∧ ∀𝑘 ∈ (ℤ≥‘𝑚)(abs‘((seq𝑀( + , 𝐹, ℂ)‘𝑚) − (seq𝑀( + , 𝐹, ℂ)‘𝑘))) < 𝑥) → ∀𝑚 ∈ (ℤ≥‘𝑗)(abs‘((seq𝑀( + , 𝐹, ℂ)‘𝑚) − (seq𝑀( + , 𝐹, ℂ)‘(𝑚 + 1)))) < 𝑥)
21		simpr 108	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ 𝑍)
22	21, 3	syl6eleq 2171	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ (ℤ≥‘𝑀))
23		eluzelz 8628	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (ℤ≥‘𝑀) → 𝑗 ∈ ℤ)
24	22, 23	syl 14	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ ℤ)
25		eluzp1m1 8642	. . . . . . . . . . 11 ⊢ ((𝑗 ∈ ℤ ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → (𝑘 − 1) ∈ (ℤ≥‘𝑗))
26	24, 25	sylan 277	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → (𝑘 − 1) ∈ (ℤ≥‘𝑗))
27		fveq2 5198	. . . . . . . . . . . . . 14 ⊢ (𝑚 = (𝑘 − 1) → (seq𝑀( + , 𝐹, ℂ)‘𝑚) = (seq𝑀( + , 𝐹, ℂ)‘(𝑘 − 1)))
28		oveq1 5539	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = (𝑘 − 1) → (𝑚 + 1) = ((𝑘 − 1) + 1))
29	28	fveq2d 5202	. . . . . . . . . . . . . 14 ⊢ (𝑚 = (𝑘 − 1) → (seq𝑀( + , 𝐹, ℂ)‘(𝑚 + 1)) = (seq𝑀( + , 𝐹, ℂ)‘((𝑘 − 1) + 1)))
30	27, 29	oveq12d 5550	. . . . . . . . . . . . 13 ⊢ (𝑚 = (𝑘 − 1) → ((seq𝑀( + , 𝐹, ℂ)‘𝑚) − (seq𝑀( + , 𝐹, ℂ)‘(𝑚 + 1))) = ((seq𝑀( + , 𝐹, ℂ)‘(𝑘 − 1)) − (seq𝑀( + , 𝐹, ℂ)‘((𝑘 − 1) + 1))))
31	30	fveq2d 5202	. . . . . . . . . . . 12 ⊢ (𝑚 = (𝑘 − 1) → (abs‘((seq𝑀( + , 𝐹, ℂ)‘𝑚) − (seq𝑀( + , 𝐹, ℂ)‘(𝑚 + 1)))) = (abs‘((seq𝑀( + , 𝐹, ℂ)‘(𝑘 − 1)) − (seq𝑀( + , 𝐹, ℂ)‘((𝑘 − 1) + 1)))))
32	31	breq1d 3795	. . . . . . . . . . 11 ⊢ (𝑚 = (𝑘 − 1) → ((abs‘((seq𝑀( + , 𝐹, ℂ)‘𝑚) − (seq𝑀( + , 𝐹, ℂ)‘(𝑚 + 1)))) < 𝑥 ↔ (abs‘((seq𝑀( + , 𝐹, ℂ)‘(𝑘 − 1)) − (seq𝑀( + , 𝐹, ℂ)‘((𝑘 − 1) + 1)))) < 𝑥))
33	32	rspcv 2697	. . . . . . . . . 10 ⊢ ((𝑘 − 1) ∈ (ℤ≥‘𝑗) → (∀𝑚 ∈ (ℤ≥‘𝑗)(abs‘((seq𝑀( + , 𝐹, ℂ)‘𝑚) − (seq𝑀( + , 𝐹, ℂ)‘(𝑚 + 1)))) < 𝑥 → (abs‘((seq𝑀( + , 𝐹, ℂ)‘(𝑘 − 1)) − (seq𝑀( + , 𝐹, ℂ)‘((𝑘 − 1) + 1)))) < 𝑥))
34	26, 33	syl 14	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → (∀𝑚 ∈ (ℤ≥‘𝑗)(abs‘((seq𝑀( + , 𝐹, ℂ)‘𝑚) − (seq𝑀( + , 𝐹, ℂ)‘(𝑚 + 1)))) < 𝑥 → (abs‘((seq𝑀( + , 𝐹, ℂ)‘(𝑘 − 1)) − (seq𝑀( + , 𝐹, ℂ)‘((𝑘 − 1) + 1)))) < 𝑥))
35		serif0.5	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)
36	3, 1, 35	iserf 9453	. . . . . . . . . . . . . 14 ⊢ (𝜑 → seq𝑀( + , 𝐹, ℂ):𝑍⟶ℂ)
37	36	ad2antrr 471	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → seq𝑀( + , 𝐹, ℂ):𝑍⟶ℂ)
38	3	uztrn2 8636	. . . . . . . . . . . . . . 15 ⊢ ((𝑗 ∈ 𝑍 ∧ (𝑘 − 1) ∈ (ℤ≥‘𝑗)) → (𝑘 − 1) ∈ 𝑍)
39	21, 38	sylan 277	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ (𝑘 − 1) ∈ (ℤ≥‘𝑗)) → (𝑘 − 1) ∈ 𝑍)
40	26, 39	syldan 276	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → (𝑘 − 1) ∈ 𝑍)
41	37, 40	ffvelrnd 5324	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → (seq𝑀( + , 𝐹, ℂ)‘(𝑘 − 1)) ∈ ℂ)
42	3	uztrn2 8636	. . . . . . . . . . . . . 14 ⊢ (((𝑗 + 1) ∈ 𝑍 ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → 𝑘 ∈ 𝑍)
43	9, 42	sylan 277	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → 𝑘 ∈ 𝑍)
44	37, 43	ffvelrnd 5324	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → (seq𝑀( + , 𝐹, ℂ)‘𝑘) ∈ ℂ)
45	41, 44	abssubd 10079	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → (abs‘((seq𝑀( + , 𝐹, ℂ)‘(𝑘 − 1)) − (seq𝑀( + , 𝐹, ℂ)‘𝑘))) = (abs‘((seq𝑀( + , 𝐹, ℂ)‘𝑘) − (seq𝑀( + , 𝐹, ℂ)‘(𝑘 − 1)))))
46		eluzelz 8628	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ (ℤ≥‘(𝑗 + 1)) → 𝑘 ∈ ℤ)
47	46	adantl 271	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → 𝑘 ∈ ℤ)
48	47	zcnd 8470	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → 𝑘 ∈ ℂ)
49		ax-1cn 7069	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ ℂ
50		npcan 7317	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑘 − 1) + 1) = 𝑘)
51	48, 49, 50	sylancl 404	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → ((𝑘 − 1) + 1) = 𝑘)
52	51	fveq2d 5202	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → (seq𝑀( + , 𝐹, ℂ)‘((𝑘 − 1) + 1)) = (seq𝑀( + , 𝐹, ℂ)‘𝑘))
53	52	oveq2d 5548	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → ((seq𝑀( + , 𝐹, ℂ)‘(𝑘 − 1)) − (seq𝑀( + , 𝐹, ℂ)‘((𝑘 − 1) + 1))) = ((seq𝑀( + , 𝐹, ℂ)‘(𝑘 − 1)) − (seq𝑀( + , 𝐹, ℂ)‘𝑘)))
54	53	fveq2d 5202	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → (abs‘((seq𝑀( + , 𝐹, ℂ)‘(𝑘 − 1)) − (seq𝑀( + , 𝐹, ℂ)‘((𝑘 − 1) + 1)))) = (abs‘((seq𝑀( + , 𝐹, ℂ)‘(𝑘 − 1)) − (seq𝑀( + , 𝐹, ℂ)‘𝑘))))
55	1	ad2antrr 471	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → 𝑀 ∈ ℤ)
56		eluzp1p1 8644	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ (ℤ≥‘𝑀) → (𝑗 + 1) ∈ (ℤ≥‘(𝑀 + 1)))
57	22, 56	syl 14	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝑗 + 1) ∈ (ℤ≥‘(𝑀 + 1)))
58		eqid 2081	. . . . . . . . . . . . . . . . 17 ⊢ (ℤ≥‘(𝑀 + 1)) = (ℤ≥‘(𝑀 + 1))
59	58	uztrn2 8636	. . . . . . . . . . . . . . . 16 ⊢ (((𝑗 + 1) ∈ (ℤ≥‘(𝑀 + 1)) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → 𝑘 ∈ (ℤ≥‘(𝑀 + 1)))
60	57, 59	sylan 277	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → 𝑘 ∈ (ℤ≥‘(𝑀 + 1)))
61		cnex 7097	. . . . . . . . . . . . . . . 16 ⊢ ℂ ∈ V
62	61	a1i 9	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → ℂ ∈ V)
63		simpr 108	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) ∧ 𝑎 ∈ (ℤ≥‘𝑀)) → 𝑎 ∈ (ℤ≥‘𝑀))
64	63, 3	syl6eleqr 2172	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) ∧ 𝑎 ∈ (ℤ≥‘𝑀)) → 𝑎 ∈ 𝑍)
65	35	ralrimiva 2434	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ)
66	65	ad3antrrr 475	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) ∧ 𝑎 ∈ (ℤ≥‘𝑀)) → ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ)
67		fveq2 5198	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑎 → (𝐹‘𝑘) = (𝐹‘𝑎))
68	67	eleq1d 2147	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑎 → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘𝑎) ∈ ℂ))
69	68	rspcva 2699	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 ∈ 𝑍 ∧ ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ) → (𝐹‘𝑎) ∈ ℂ)
70	64, 66, 69	syl2anc 403	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) ∧ 𝑎 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑎) ∈ ℂ)
71		addcl 7098	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 ∈ ℂ ∧ 𝑏 ∈ ℂ) → (𝑎 + 𝑏) ∈ ℂ)
72	71	adantl 271	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) ∧ (𝑎 ∈ ℂ ∧ 𝑏 ∈ ℂ)) → (𝑎 + 𝑏) ∈ ℂ)
73	55, 60, 62, 70, 72	iseqm1 9447	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → (seq𝑀( + , 𝐹, ℂ)‘𝑘) = ((seq𝑀( + , 𝐹, ℂ)‘(𝑘 − 1)) + (𝐹‘𝑘)))
74	73	oveq1d 5547	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → ((seq𝑀( + , 𝐹, ℂ)‘𝑘) − (seq𝑀( + , 𝐹, ℂ)‘(𝑘 − 1))) = (((seq𝑀( + , 𝐹, ℂ)‘(𝑘 − 1)) + (𝐹‘𝑘)) − (seq𝑀( + , 𝐹, ℂ)‘(𝑘 − 1))))
75	35	adantlr 460	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)
76	43, 75	syldan 276	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → (𝐹‘𝑘) ∈ ℂ)
77	41, 76	pncan2d 7421	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → (((seq𝑀( + , 𝐹, ℂ)‘(𝑘 − 1)) + (𝐹‘𝑘)) − (seq𝑀( + , 𝐹, ℂ)‘(𝑘 − 1))) = (𝐹‘𝑘))
78	74, 77	eqtr2d 2114	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → (𝐹‘𝑘) = ((seq𝑀( + , 𝐹, ℂ)‘𝑘) − (seq𝑀( + , 𝐹, ℂ)‘(𝑘 − 1))))
79	78	fveq2d 5202	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → (abs‘(𝐹‘𝑘)) = (abs‘((seq𝑀( + , 𝐹, ℂ)‘𝑘) − (seq𝑀( + , 𝐹, ℂ)‘(𝑘 − 1)))))
80	45, 54, 79	3eqtr4d 2123	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → (abs‘((seq𝑀( + , 𝐹, ℂ)‘(𝑘 − 1)) − (seq𝑀( + , 𝐹, ℂ)‘((𝑘 − 1) + 1)))) = (abs‘(𝐹‘𝑘)))
81	80	breq1d 3795	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → ((abs‘((seq𝑀( + , 𝐹, ℂ)‘(𝑘 − 1)) − (seq𝑀( + , 𝐹, ℂ)‘((𝑘 − 1) + 1)))) < 𝑥 ↔ (abs‘(𝐹‘𝑘)) < 𝑥))
82	34, 81	sylibd 147	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → (∀𝑚 ∈ (ℤ≥‘𝑗)(abs‘((seq𝑀( + , 𝐹, ℂ)‘𝑚) − (seq𝑀( + , 𝐹, ℂ)‘(𝑚 + 1)))) < 𝑥 → (abs‘(𝐹‘𝑘)) < 𝑥))
83	82	ralrimdva 2441	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (∀𝑚 ∈ (ℤ≥‘𝑗)(abs‘((seq𝑀( + , 𝐹, ℂ)‘𝑚) − (seq𝑀( + , 𝐹, ℂ)‘(𝑚 + 1)))) < 𝑥 → ∀𝑘 ∈ (ℤ≥‘(𝑗 + 1))(abs‘(𝐹‘𝑘)) < 𝑥))
84	20, 83	syl5 32	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (∀𝑚 ∈ (ℤ≥‘𝑗)((seq𝑀( + , 𝐹, ℂ)‘𝑚) ∈ ℂ ∧ ∀𝑘 ∈ (ℤ≥‘𝑚)(abs‘((seq𝑀( + , 𝐹, ℂ)‘𝑚) − (seq𝑀( + , 𝐹, ℂ)‘𝑘))) < 𝑥) → ∀𝑘 ∈ (ℤ≥‘(𝑗 + 1))(abs‘(𝐹‘𝑘)) < 𝑥))
85		fveq2 5198	. . . . . . . 8 ⊢ (𝑛 = (𝑗 + 1) → (ℤ≥‘𝑛) = (ℤ≥‘(𝑗 + 1)))
86	85	raleqdv 2555	. . . . . . 7 ⊢ (𝑛 = (𝑗 + 1) → (∀𝑘 ∈ (ℤ≥‘𝑛)(abs‘(𝐹‘𝑘)) < 𝑥 ↔ ∀𝑘 ∈ (ℤ≥‘(𝑗 + 1))(abs‘(𝐹‘𝑘)) < 𝑥))
87	86	rspcev 2701	. . . . . 6 ⊢ (((𝑗 + 1) ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘(𝑗 + 1))(abs‘(𝐹‘𝑘)) < 𝑥) → ∃𝑛 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑛)(abs‘(𝐹‘𝑘)) < 𝑥)
88	9, 84, 87	syl6an 1363	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (∀𝑚 ∈ (ℤ≥‘𝑗)((seq𝑀( + , 𝐹, ℂ)‘𝑚) ∈ ℂ ∧ ∀𝑘 ∈ (ℤ≥‘𝑚)(abs‘((seq𝑀( + , 𝐹, ℂ)‘𝑚) − (seq𝑀( + , 𝐹, ℂ)‘𝑘))) < 𝑥) → ∃𝑛 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑛)(abs‘(𝐹‘𝑘)) < 𝑥))
89	88	rexlimdva 2477	. . . 4 ⊢ (𝜑 → (∃𝑗 ∈ 𝑍 ∀𝑚 ∈ (ℤ≥‘𝑗)((seq𝑀( + , 𝐹, ℂ)‘𝑚) ∈ ℂ ∧ ∀𝑘 ∈ (ℤ≥‘𝑚)(abs‘((seq𝑀( + , 𝐹, ℂ)‘𝑚) − (seq𝑀( + , 𝐹, ℂ)‘𝑘))) < 𝑥) → ∃𝑛 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑛)(abs‘(𝐹‘𝑘)) < 𝑥))
90	89	ralimdv 2430	. . 3 ⊢ (𝜑 → (∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑚 ∈ (ℤ≥‘𝑗)((seq𝑀( + , 𝐹, ℂ)‘𝑚) ∈ ℂ ∧ ∀𝑘 ∈ (ℤ≥‘𝑚)(abs‘((seq𝑀( + , 𝐹, ℂ)‘𝑚) − (seq𝑀( + , 𝐹, ℂ)‘𝑘))) < 𝑥) → ∀𝑥 ∈ ℝ+ ∃𝑛 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑛)(abs‘(𝐹‘𝑘)) < 𝑥))
91	7, 90	mpd 13	. 2 ⊢ (𝜑 → ∀𝑥 ∈ ℝ+ ∃𝑛 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑛)(abs‘(𝐹‘𝑘)) < 𝑥)
92		serif0.3	. . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑉)
93		eqidd 2082	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = (𝐹‘𝑘))
94	3, 1, 92, 93, 35	clim0c 10125	. 2 ⊢ (𝜑 → (𝐹 ⇝ 0 ↔ ∀𝑥 ∈ ℝ+ ∃𝑛 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑛)(abs‘(𝐹‘𝑘)) < 𝑥))
95	91, 94	mpbird 165	1 ⊢ (𝜑 → 𝐹 ⇝ 0)

Syntax hints:   → wi 4   ∧ wa 102   = wceq 1284   ∈ wcel 1433  ∀wral 2348  ∃wrex 2349  Vcvv 2601   class class class wbr 3785  dom cdm 4363  ⟶wf 4918  ‘cfv 4922  (class class class)co 5532  ℂcc 6979  0cc0 6981  1c1 6982   + caddc 6984   < clt 7153   − cmin 7279  ℤcz 8351  ℤ_≥cuz 8619  ℝ⁺crp 8734  seqcseq 9431  abscabs 9883   ⇝ cli 10117

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329  ax-cnex 7067  ax-resscn 7068  ax-1cn 7069  ax-1re 7070  ax-icn 7071  ax-addcl 7072  ax-addrcl 7073  ax-mulcl 7074  ax-mulrcl 7075  ax-addcom 7076  ax-mulcom 7077  ax-addass 7078  ax-mulass 7079  ax-distr 7080  ax-i2m1 7081  ax-0lt1 7082  ax-1rid 7083  ax-0id 7084  ax-rnegex 7085  ax-precex 7086  ax-cnre 7087  ax-pre-ltirr 7088  ax-pre-ltwlin 7089  ax-pre-lttrn 7090  ax-pre-apti 7091  ax-pre-ltadd 7092  ax-pre-mulgt0 7093  ax-pre-mulext 7094  ax-arch 7095  ax-caucvg 7096

This theorem depends on definitions:  df-bi 115  df-dc 776  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-nel 2340  df-ral 2353  df-rex 2354  df-reu 2355  df-rmo 2356  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-if 3352  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-id 4048  df-po 4051  df-iso 4052  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-riota 5488  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-frec 6001  df-pnf 7155  df-mnf 7156  df-xr 7157  df-ltxr 7158  df-le 7159  df-sub 7281  df-neg 7282  df-reap 7675  df-ap 7682  df-div 7761  df-inn 8040  df-2 8098  df-3 8099  df-4 8100  df-n0 8289  df-z 8352  df-uz 8620  df-rp 8735  df-iseq 9432  df-iexp 9476  df-cj 9729  df-re 9730  df-im 9731  df-rsqrt 9884  df-abs 9885  df-clim 10118

This theorem is referenced by: (None)