Theorem serif0 9871

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 9870	. . . . 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 391	. . . 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 9711	. . . 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 127	. . 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 8527	. . . . . . 7 ⊢ (𝑗 ∈ 𝑍 → (𝑗 + 1) ∈ 𝑍)
9	8	adantl 262	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝑗 + 1) ∈ 𝑍)
10		eluzelz 8482	. . . . . . . . . 10 ⊢ (𝑚 ∈ (ℤ≥‘𝑗) → 𝑚 ∈ ℤ)
11		uzid 8487	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℤ → 𝑚 ∈ (ℤ≥‘𝑚))
12		peano2uz 8526	. . . . . . . . . 10 ⊢ (𝑚 ∈ (ℤ≥‘𝑚) → (𝑚 + 1) ∈ (ℤ≥‘𝑚))
13		fveq2 5178	. . . . . . . . . . . . . 14 ⊢ (𝑘 = (𝑚 + 1) → (seq𝑀( + , 𝐹, ℂ)‘𝑘) = (seq𝑀( + , 𝐹, ℂ)‘(𝑚 + 1)))
14	13	oveq2d 5528	. . . . . . . . . . . . 13 ⊢ (𝑘 = (𝑚 + 1) → ((seq𝑀( + , 𝐹, ℂ)‘𝑚) − (seq𝑀( + , 𝐹, ℂ)‘𝑘)) = ((seq𝑀( + , 𝐹, ℂ)‘𝑚) − (seq𝑀( + , 𝐹, ℂ)‘(𝑚 + 1))))
15	14	fveq2d 5182	. . . . . . . . . . . 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 3774	. . . . . . . . . . 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 2652	. . . . . . . . . 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 263	. . . . . . . 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 2382	. . . . . . 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 103	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ 𝑍)
22	21, 3	syl6eleq 2130	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ (ℤ≥‘𝑀))
23		eluzelz 8482	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (ℤ≥‘𝑀) → 𝑗 ∈ ℤ)
24	22, 23	syl 14	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ ℤ)
25		eluzp1m1 8496	. . . . . . . . . . 11 ⊢ ((𝑗 ∈ ℤ ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → (𝑘 − 1) ∈ (ℤ≥‘𝑗))
26	24, 25	sylan 267	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → (𝑘 − 1) ∈ (ℤ≥‘𝑗))
27		fveq2 5178	. . . . . . . . . . . . . 14 ⊢ (𝑚 = (𝑘 − 1) → (seq𝑀( + , 𝐹, ℂ)‘𝑚) = (seq𝑀( + , 𝐹, ℂ)‘(𝑘 − 1)))
28		oveq1 5519	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = (𝑘 − 1) → (𝑚 + 1) = ((𝑘 − 1) + 1))
29	28	fveq2d 5182	. . . . . . . . . . . . . 14 ⊢ (𝑚 = (𝑘 − 1) → (seq𝑀( + , 𝐹, ℂ)‘(𝑚 + 1)) = (seq𝑀( + , 𝐹, ℂ)‘((𝑘 − 1) + 1)))
30	27, 29	oveq12d 5530	. . . . . . . . . . . . 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 5182	. . . . . . . . . . . 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 3774	. . . . . . . . . . 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 2652	. . . . . . . . . 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 9233	. . . . . . . . . . . . . 14 ⊢ (𝜑 → seq𝑀( + , 𝐹, ℂ):𝑍⟶ℂ)
37	36	ad2antrr 457	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → seq𝑀( + , 𝐹, ℂ):𝑍⟶ℂ)
38	3	uztrn2 8490	. . . . . . . . . . . . . . 15 ⊢ ((𝑗 ∈ 𝑍 ∧ (𝑘 − 1) ∈ (ℤ≥‘𝑗)) → (𝑘 − 1) ∈ 𝑍)
39	21, 38	sylan 267	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ (𝑘 − 1) ∈ (ℤ≥‘𝑗)) → (𝑘 − 1) ∈ 𝑍)
40	26, 39	syldan 266	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → (𝑘 − 1) ∈ 𝑍)
41	37, 40	ffvelrnd 5303	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → (seq𝑀( + , 𝐹, ℂ)‘(𝑘 − 1)) ∈ ℂ)
42	3	uztrn2 8490	. . . . . . . . . . . . . 14 ⊢ (((𝑗 + 1) ∈ 𝑍 ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → 𝑘 ∈ 𝑍)
43	9, 42	sylan 267	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → 𝑘 ∈ 𝑍)
44	37, 43	ffvelrnd 5303	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → (seq𝑀( + , 𝐹, ℂ)‘𝑘) ∈ ℂ)
45	41, 44	abssubd 9789	. . . . . . . . . . 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 8482	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ (ℤ≥‘(𝑗 + 1)) → 𝑘 ∈ ℤ)
47	46	adantl 262	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → 𝑘 ∈ ℤ)
48	47	zcnd 8361	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → 𝑘 ∈ ℂ)
49		ax-1cn 6977	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ ℂ
50		npcan 7220	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑘 − 1) + 1) = 𝑘)
51	48, 49, 50	sylancl 392	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → ((𝑘 − 1) + 1) = 𝑘)
52	51	fveq2d 5182	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → (seq𝑀( + , 𝐹, ℂ)‘((𝑘 − 1) + 1)) = (seq𝑀( + , 𝐹, ℂ)‘𝑘))
53	52	oveq2d 5528	. . . . . . . . . . . 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 5182	. . . . . . . . . . 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 457	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → 𝑀 ∈ ℤ)
56		eluzp1p1 8498	. . . . . . . . . . . . . . . . 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 2040	. . . . . . . . . . . . . . . . 17 ⊢ (ℤ≥‘(𝑀 + 1)) = (ℤ≥‘(𝑀 + 1))
59	58	uztrn2 8490	. . . . . . . . . . . . . . . 16 ⊢ (((𝑗 + 1) ∈ (ℤ≥‘(𝑀 + 1)) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → 𝑘 ∈ (ℤ≥‘(𝑀 + 1)))
60	57, 59	sylan 267	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → 𝑘 ∈ (ℤ≥‘(𝑀 + 1)))
61		cnex 7005	. . . . . . . . . . . . . . . 16 ⊢ ℂ ∈ V
62	61	a1i 9	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → ℂ ∈ V)
63		simpr 103	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) ∧ 𝑎 ∈ (ℤ≥‘𝑀)) → 𝑎 ∈ (ℤ≥‘𝑀))
64	63, 3	syl6eleqr 2131	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) ∧ 𝑎 ∈ (ℤ≥‘𝑀)) → 𝑎 ∈ 𝑍)
65	35	ralrimiva 2392	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ)
66	65	ad3antrrr 461	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) ∧ 𝑎 ∈ (ℤ≥‘𝑀)) → ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ)
67		fveq2 5178	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑎 → (𝐹‘𝑘) = (𝐹‘𝑎))
68	67	eleq1d 2106	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑎 → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘𝑎) ∈ ℂ))
69	68	rspcva 2654	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 ∈ 𝑍 ∧ ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ) → (𝐹‘𝑎) ∈ ℂ)
70	64, 66, 69	syl2anc 391	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) ∧ 𝑎 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑎) ∈ ℂ)
71		addcl 7006	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 ∈ ℂ ∧ 𝑏 ∈ ℂ) → (𝑎 + 𝑏) ∈ ℂ)
72	71	adantl 262	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) ∧ (𝑎 ∈ ℂ ∧ 𝑏 ∈ ℂ)) → (𝑎 + 𝑏) ∈ ℂ)
73	55, 60, 62, 70, 72	iseqm1 9227	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → (seq𝑀( + , 𝐹, ℂ)‘𝑘) = ((seq𝑀( + , 𝐹, ℂ)‘(𝑘 − 1)) + (𝐹‘𝑘)))
74	73	oveq1d 5527	. . . . . . . . . . . . 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 446	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)
76	43, 75	syldan 266	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → (𝐹‘𝑘) ∈ ℂ)
77	41, 76	pncan2d 7324	. . . . . . . . . . . . 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 2073	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → (𝐹‘𝑘) = ((seq𝑀( + , 𝐹, ℂ)‘𝑘) − (seq𝑀( + , 𝐹, ℂ)‘(𝑘 − 1))))
79	78	fveq2d 5182	. . . . . . . . . . 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 2082	. . . . . . . . . 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 3774	. . . . . . . . 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 138	. . . . . . . 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 2399	. . . . . . 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 28	. . . . . 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 5178	. . . . . . . 8 ⊢ (𝑛 = (𝑗 + 1) → (ℤ≥‘𝑛) = (ℤ≥‘(𝑗 + 1)))
86	85	raleqdv 2511	. . . . . . 7 ⊢ (𝑛 = (𝑗 + 1) → (∀𝑘 ∈ (ℤ≥‘𝑛)(abs‘(𝐹‘𝑘)) < 𝑥 ↔ ∀𝑘 ∈ (ℤ≥‘(𝑗 + 1))(abs‘(𝐹‘𝑘)) < 𝑥))
87	86	rspcev 2656	. . . . . 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 1323	. . . . 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 2433	. . . 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 2388	. . 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 2041	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = (𝐹‘𝑘))
94	3, 1, 92, 93, 35	clim0c 9807	. 2 ⊢ (𝜑 → (𝐹 ⇝ 0 ↔ ∀𝑥 ∈ ℝ+ ∃𝑛 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑛)(abs‘(𝐹‘𝑘)) < 𝑥))
95	91, 94	mpbird 156	1 ⊢ (𝜑 → 𝐹 ⇝ 0)

Syntax hints:   → wi 4   ∧ wa 97   = wceq 1243   ∈ wcel 1393  ∀wral 2306  ∃wrex 2307  Vcvv 2557   class class class wbr 3764  dom cdm 4345  ⟶wf 4898  ‘cfv 4902  (class class class)co 5512  ℂcc 6887  0cc0 6889  1c1 6890   + caddc 6892   < clt 7060   − cmin 7182  ℤcz 8245  ℤ_≥cuz 8473  ℝ⁺crp 8583  seqcseq 9211  abscabs 9595   ⇝ cli 9799

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311  ax-cnex 6975  ax-resscn 6976  ax-1cn 6977  ax-1re 6978  ax-icn 6979  ax-addcl 6980  ax-addrcl 6981  ax-mulcl 6982  ax-mulrcl 6983  ax-addcom 6984  ax-mulcom 6985  ax-addass 6986  ax-mulass 6987  ax-distr 6988  ax-i2m1 6989  ax-1rid 6991  ax-0id 6992  ax-rnegex 6993  ax-precex 6994  ax-cnre 6995  ax-pre-ltirr 6996  ax-pre-ltwlin 6997  ax-pre-lttrn 6998  ax-pre-apti 6999  ax-pre-ltadd 7000  ax-pre-mulgt0 7001  ax-pre-mulext 7002  ax-arch 7003  ax-caucvg 7004

This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-nel 2207  df-ral 2311  df-rex 2312  df-reu 2313  df-rmo 2314  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-if 3332  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-riota 5468  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-frec 5978  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-pli 6403  df-mi 6404  df-lti 6405  df-plpq 6442  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-plqqs 6447  df-mqqs 6448  df-1nqqs 6449  df-rq 6450  df-ltnqqs 6451  df-enq0 6522  df-nq0 6523  df-0nq0 6524  df-plq0 6525  df-mq0 6526  df-inp 6564  df-i1p 6565  df-iplp 6566  df-iltp 6568  df-enr 6811  df-nr 6812  df-ltr 6815  df-0r 6816  df-1r 6817  df-0 6896  df-1 6897  df-r 6899  df-lt 6902  df-pnf 7062  df-mnf 7063  df-xr 7064  df-ltxr 7065  df-le 7066  df-sub 7184  df-neg 7185  df-reap 7566  df-ap 7573  df-div 7652  df-inn 7915  df-2 7973  df-3 7974  df-4 7975  df-n0 8182  df-z 8246  df-uz 8474  df-rp 8584  df-iseq 9212  df-iexp 9255  df-cj 9442  df-re 9443  df-im 9444  df-rsqrt 9596  df-abs 9597  df-clim 9800

This theorem is referenced by: (None)