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Theorem caucvgb 15319

Description: A function is convergent if and only if it is Cauchy. Theorem 12-5.3 of [Gleason] p. 180. (Contributed by Mario Carneiro, 15-Feb-2014.)

**Hypothesis**
Ref	Expression
caucvgb.1	⊢ 𝑍 = (ℤ_≥‘𝑀)

**Assertion**
Ref	Expression
caucvgb	⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ 𝑉) → (𝐹 ∈ dom ⇝ ↔ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥)))

Distinct variable groups: 𝑗,𝑘,𝑥,𝐹 𝑗,𝑀,𝑘,𝑥 𝑗,𝑍,𝑘,𝑥 𝑘,𝑉 Allowed substitution hints: 𝑉(𝑥,𝑗)

Proof of Theorem caucvgb Dummy variables 𝑖 𝑚 𝑛 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eldm2g 5797	. . . 4 ⊢ (𝐹 ∈ dom ⇝ → (𝐹 ∈ dom ⇝ ↔ ∃𝑚⟨𝐹, 𝑚⟩ ∈ ⇝ ))
2	1	ibi 266	. . 3 ⊢ (𝐹 ∈ dom ⇝ → ∃𝑚⟨𝐹, 𝑚⟩ ∈ ⇝ )
3		df-br 5071	. . . . 5 ⊢ (𝐹 ⇝ 𝑚 ↔ ⟨𝐹, 𝑚⟩ ∈ ⇝ )
4		caucvgb.1	. . . . . . . 8 ⊢ 𝑍 = (ℤ_≥‘𝑀)
5		simpll 763	. . . . . . . 8 ⊢ (((𝑀 ∈ ℤ ∧ 𝐹 ∈ 𝑉) ∧ 𝐹 ⇝ 𝑚) → 𝑀 ∈ ℤ)
6		1rp 12663	. . . . . . . . 9 ⊢ 1 ∈ ℝ⁺
7	6	a1i 11	. . . . . . . 8 ⊢ (((𝑀 ∈ ℤ ∧ 𝐹 ∈ 𝑉) ∧ 𝐹 ⇝ 𝑚) → 1 ∈ ℝ⁺)
8		eqidd 2739	. . . . . . . 8 ⊢ ((((𝑀 ∈ ℤ ∧ 𝐹 ∈ 𝑉) ∧ 𝐹 ⇝ 𝑚) ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = (𝐹‘𝑘))
9		simpr 484	. . . . . . . 8 ⊢ (((𝑀 ∈ ℤ ∧ 𝐹 ∈ 𝑉) ∧ 𝐹 ⇝ 𝑚) → 𝐹 ⇝ 𝑚)
10	4, 5, 7, 8, 9	climi 15147	. . . . . . 7 ⊢ (((𝑀 ∈ ℤ ∧ 𝐹 ∈ 𝑉) ∧ 𝐹 ⇝ 𝑚) → ∃𝑛 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑛)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝑚)) < 1))
11		simpl 482	. . . . . . . . 9 ⊢ (((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝑚)) < 1) → (𝐹‘𝑘) ∈ ℂ)
12	11	ralimi 3086	. . . . . . . 8 ⊢ (∀𝑘 ∈ (ℤ_≥‘𝑛)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝑚)) < 1) → ∀𝑘 ∈ (ℤ_≥‘𝑛)(𝐹‘𝑘) ∈ ℂ)
13	12	reximi 3174	. . . . . . 7 ⊢ (∃𝑛 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑛)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝑚)) < 1) → ∃𝑛 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑛)(𝐹‘𝑘) ∈ ℂ)
14	10, 13	syl 17	. . . . . 6 ⊢ (((𝑀 ∈ ℤ ∧ 𝐹 ∈ 𝑉) ∧ 𝐹 ⇝ 𝑚) → ∃𝑛 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑛)(𝐹‘𝑘) ∈ ℂ)
15	14	ex 412	. . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ 𝑉) → (𝐹 ⇝ 𝑚 → ∃𝑛 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑛)(𝐹‘𝑘) ∈ ℂ))
16	3, 15	syl5bir 242	. . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ 𝑉) → (⟨𝐹, 𝑚⟩ ∈ ⇝ → ∃𝑛 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑛)(𝐹‘𝑘) ∈ ℂ))
17	16	exlimdv 1937	. . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ 𝑉) → (∃𝑚⟨𝐹, 𝑚⟩ ∈ ⇝ → ∃𝑛 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑛)(𝐹‘𝑘) ∈ ℂ))
18	2, 17	syl5 34	. 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ 𝑉) → (𝐹 ∈ dom ⇝ → ∃𝑛 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑛)(𝐹‘𝑘) ∈ ℂ))
19		fveq2 6756	. . . . . . 7 ⊢ (𝑗 = 𝑛 → (ℤ_≥‘𝑗) = (ℤ_≥‘𝑛))
20	19	raleqdv 3339	. . . . . 6 ⊢ (𝑗 = 𝑛 → (∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) ∈ ℂ ↔ ∀𝑘 ∈ (ℤ_≥‘𝑛)(𝐹‘𝑘) ∈ ℂ))
21	20	cbvrexvw 3373	. . . . 5 ⊢ (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) ∈ ℂ ↔ ∃𝑛 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑛)(𝐹‘𝑘) ∈ ℂ)
22	21	a1i 11	. . . 4 ⊢ (𝑥 = 1 → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) ∈ ℂ ↔ ∃𝑛 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑛)(𝐹‘𝑘) ∈ ℂ))
23		simpl 482	. . . . . . 7 ⊢ (((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥) → (𝐹‘𝑘) ∈ ℂ)
24	23	ralimi 3086	. . . . . 6 ⊢ (∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥) → ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) ∈ ℂ)
25	24	reximi 3174	. . . . 5 ⊢ (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) ∈ ℂ)
26	25	ralimi 3086	. . . 4 ⊢ (∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥) → ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) ∈ ℂ)
27	6	a1i 11	. . . 4 ⊢ (∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥) → 1 ∈ ℝ⁺)
28	22, 26, 27	rspcdva 3554	. . 3 ⊢ (∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥) → ∃𝑛 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑛)(𝐹‘𝑘) ∈ ℂ)
29	28	a1i 11	. 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ 𝑉) → (∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥) → ∃𝑛 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑛)(𝐹‘𝑘) ∈ ℂ))
30		eluzelz 12521	. . . . . . . . . 10 ⊢ (𝑛 ∈ (ℤ_≥‘𝑀) → 𝑛 ∈ ℤ)
31	30, 4	eleq2s 2857	. . . . . . . . 9 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ ℤ)
32		eqid 2738	. . . . . . . . . 10 ⊢ (ℤ_≥‘𝑛) = (ℤ_≥‘𝑛)
33	32	climcau 15310	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℤ ∧ 𝐹 ∈ dom ⇝ ) → ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ (ℤ_≥‘𝑛)∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥)
34	31, 33	sylan 579	. . . . . . . 8 ⊢ ((𝑛 ∈ 𝑍 ∧ 𝐹 ∈ dom ⇝ ) → ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ (ℤ_≥‘𝑛)∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥)
35	32	r19.29uz 14990	. . . . . . . . . 10 ⊢ ((∀𝑘 ∈ (ℤ_≥‘𝑛)(𝐹‘𝑘) ∈ ℂ ∧ ∃𝑗 ∈ (ℤ_≥‘𝑛)∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥) → ∃𝑗 ∈ (ℤ_≥‘𝑛)∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥))
36	35	ex 412	. . . . . . . . 9 ⊢ (∀𝑘 ∈ (ℤ_≥‘𝑛)(𝐹‘𝑘) ∈ ℂ → (∃𝑗 ∈ (ℤ_≥‘𝑛)∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥 → ∃𝑗 ∈ (ℤ_≥‘𝑛)∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥)))
37	36	ralimdv 3103	. . . . . . . 8 ⊢ (∀𝑘 ∈ (ℤ_≥‘𝑛)(𝐹‘𝑘) ∈ ℂ → (∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ (ℤ_≥‘𝑛)∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥 → ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ (ℤ_≥‘𝑛)∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥)))
38	34, 37	mpan9 506	. . . . . . 7 ⊢ (((𝑛 ∈ 𝑍 ∧ 𝐹 ∈ dom ⇝ ) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑛)(𝐹‘𝑘) ∈ ℂ) → ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ (ℤ_≥‘𝑛)∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥))
39	38	an32s 648	. . . . . 6 ⊢ (((𝑛 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑛)(𝐹‘𝑘) ∈ ℂ) ∧ 𝐹 ∈ dom ⇝ ) → ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ (ℤ_≥‘𝑛)∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥))
40	39	adantll 710	. . . . 5 ⊢ ((((𝑀 ∈ ℤ ∧ 𝐹 ∈ 𝑉) ∧ (𝑛 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑛)(𝐹‘𝑘) ∈ ℂ)) ∧ 𝐹 ∈ dom ⇝ ) → ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ (ℤ_≥‘𝑛)∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥))
41		simplrr 774	. . . . . . . 8 ⊢ (((𝐹 ∈ 𝑉 ∧ (𝑛 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑛)(𝐹‘𝑘) ∈ ℂ)) ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ (ℤ_≥‘𝑛)∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥)) → ∀𝑘 ∈ (ℤ_≥‘𝑛)(𝐹‘𝑘) ∈ ℂ)
42		fveq2 6756	. . . . . . . . . 10 ⊢ (𝑘 = 𝑚 → (𝐹‘𝑘) = (𝐹‘𝑚))
43	42	eleq1d 2823	. . . . . . . . 9 ⊢ (𝑘 = 𝑚 → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘𝑚) ∈ ℂ))
44	43	rspccva 3551	. . . . . . . 8 ⊢ ((∀𝑘 ∈ (ℤ_≥‘𝑛)(𝐹‘𝑘) ∈ ℂ ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → (𝐹‘𝑚) ∈ ℂ)
45	41, 44	sylan 579	. . . . . . 7 ⊢ ((((𝐹 ∈ 𝑉 ∧ (𝑛 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑛)(𝐹‘𝑘) ∈ ℂ)) ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ (ℤ_≥‘𝑛)∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥)) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → (𝐹‘𝑚) ∈ ℂ)
46		simpr 484	. . . . . . . . . . . . 13 ⊢ (((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥) → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥)
47	46	ralimi 3086	. . . . . . . . . . . 12 ⊢ (∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥) → ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥)
48	42	fvoveq1d 7277	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑚 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) = (abs‘((𝐹‘𝑚) − (𝐹‘𝑗))))
49	48	breq1d 5080	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑚 → ((abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥 ↔ (abs‘((𝐹‘𝑚) − (𝐹‘𝑗))) < 𝑥))
50	49	cbvralvw 3372	. . . . . . . . . . . 12 ⊢ (∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥 ↔ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑚) − (𝐹‘𝑗))) < 𝑥)
51	47, 50	sylib 217	. . . . . . . . . . 11 ⊢ (∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥) → ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑚) − (𝐹‘𝑗))) < 𝑥)
52	51	reximi 3174	. . . . . . . . . 10 ⊢ (∃𝑗 ∈ (ℤ_≥‘𝑛)∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥) → ∃𝑗 ∈ (ℤ_≥‘𝑛)∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑚) − (𝐹‘𝑗))) < 𝑥)
53	52	ralimi 3086	. . . . . . . . 9 ⊢ (∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ (ℤ_≥‘𝑛)∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥) → ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ (ℤ_≥‘𝑛)∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑚) − (𝐹‘𝑗))) < 𝑥)
54	53	adantl 481	. . . . . . . 8 ⊢ (((𝐹 ∈ 𝑉 ∧ (𝑛 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑛)(𝐹‘𝑘) ∈ ℂ)) ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ (ℤ_≥‘𝑛)∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥)) → ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ (ℤ_≥‘𝑛)∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑚) − (𝐹‘𝑗))) < 𝑥)
55		fveq2 6756	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑖 → (ℤ_≥‘𝑗) = (ℤ_≥‘𝑖))
56		fveq2 6756	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑖 → (𝐹‘𝑗) = (𝐹‘𝑖))
57	56	oveq2d 7271	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑖 → ((𝐹‘𝑚) − (𝐹‘𝑗)) = ((𝐹‘𝑚) − (𝐹‘𝑖)))
58	57	fveq2d 6760	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑖 → (abs‘((𝐹‘𝑚) − (𝐹‘𝑗))) = (abs‘((𝐹‘𝑚) − (𝐹‘𝑖))))
59	58	breq1d 5080	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑖 → ((abs‘((𝐹‘𝑚) − (𝐹‘𝑗))) < 𝑥 ↔ (abs‘((𝐹‘𝑚) − (𝐹‘𝑖))) < 𝑥))
60	55, 59	raleqbidv 3327	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑖 → (∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑚) − (𝐹‘𝑗))) < 𝑥 ↔ ∀𝑚 ∈ (ℤ_≥‘𝑖)(abs‘((𝐹‘𝑚) − (𝐹‘𝑖))) < 𝑥))
61	60	cbvrexvw 3373	. . . . . . . . . 10 ⊢ (∃𝑗 ∈ (ℤ_≥‘𝑛)∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑚) − (𝐹‘𝑗))) < 𝑥 ↔ ∃𝑖 ∈ (ℤ_≥‘𝑛)∀𝑚 ∈ (ℤ_≥‘𝑖)(abs‘((𝐹‘𝑚) − (𝐹‘𝑖))) < 𝑥)
62		breq2 5074	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → ((abs‘((𝐹‘𝑚) − (𝐹‘𝑖))) < 𝑥 ↔ (abs‘((𝐹‘𝑚) − (𝐹‘𝑖))) < 𝑦))
63	62	rexralbidv 3229	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (∃𝑖 ∈ (ℤ_≥‘𝑛)∀𝑚 ∈ (ℤ_≥‘𝑖)(abs‘((𝐹‘𝑚) − (𝐹‘𝑖))) < 𝑥 ↔ ∃𝑖 ∈ (ℤ_≥‘𝑛)∀𝑚 ∈ (ℤ_≥‘𝑖)(abs‘((𝐹‘𝑚) − (𝐹‘𝑖))) < 𝑦))
64	61, 63	syl5bb 282	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (∃𝑗 ∈ (ℤ_≥‘𝑛)∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑚) − (𝐹‘𝑗))) < 𝑥 ↔ ∃𝑖 ∈ (ℤ_≥‘𝑛)∀𝑚 ∈ (ℤ_≥‘𝑖)(abs‘((𝐹‘𝑚) − (𝐹‘𝑖))) < 𝑦))
65	64	cbvralvw 3372	. . . . . . . 8 ⊢ (∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ (ℤ_≥‘𝑛)∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑚) − (𝐹‘𝑗))) < 𝑥 ↔ ∀𝑦 ∈ ℝ⁺ ∃𝑖 ∈ (ℤ_≥‘𝑛)∀𝑚 ∈ (ℤ_≥‘𝑖)(abs‘((𝐹‘𝑚) − (𝐹‘𝑖))) < 𝑦)
66	54, 65	sylib 217	. . . . . . 7 ⊢ (((𝐹 ∈ 𝑉 ∧ (𝑛 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑛)(𝐹‘𝑘) ∈ ℂ)) ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ (ℤ_≥‘𝑛)∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥)) → ∀𝑦 ∈ ℝ⁺ ∃𝑖 ∈ (ℤ_≥‘𝑛)∀𝑚 ∈ (ℤ_≥‘𝑖)(abs‘((𝐹‘𝑚) − (𝐹‘𝑖))) < 𝑦)
67		simpll 763	. . . . . . 7 ⊢ (((𝐹 ∈ 𝑉 ∧ (𝑛 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑛)(𝐹‘𝑘) ∈ ℂ)) ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ (ℤ_≥‘𝑛)∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥)) → 𝐹 ∈ 𝑉)
68	32, 45, 66, 67	caucvg 15318	. . . . . 6 ⊢ (((𝐹 ∈ 𝑉 ∧ (𝑛 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑛)(𝐹‘𝑘) ∈ ℂ)) ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ (ℤ_≥‘𝑛)∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥)) → 𝐹 ∈ dom ⇝ )
69	68	adantlll 714	. . . . 5 ⊢ ((((𝑀 ∈ ℤ ∧ 𝐹 ∈ 𝑉) ∧ (𝑛 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑛)(𝐹‘𝑘) ∈ ℂ)) ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ (ℤ_≥‘𝑛)∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥)) → 𝐹 ∈ dom ⇝ )
70	40, 69	impbida 797	. . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝐹 ∈ 𝑉) ∧ (𝑛 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑛)(𝐹‘𝑘) ∈ ℂ)) → (𝐹 ∈ dom ⇝ ↔ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ (ℤ_≥‘𝑛)∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥)))
71	4, 32	cau4 14996	. . . . 5 ⊢ (𝑛 ∈ 𝑍 → (∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥) ↔ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ (ℤ_≥‘𝑛)∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥)))
72	71	ad2antrl 724	. . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝐹 ∈ 𝑉) ∧ (𝑛 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑛)(𝐹‘𝑘) ∈ ℂ)) → (∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥) ↔ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ (ℤ_≥‘𝑛)∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥)))
73	70, 72	bitr4d 281	. . 3 ⊢ (((𝑀 ∈ ℤ ∧ 𝐹 ∈ 𝑉) ∧ (𝑛 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑛)(𝐹‘𝑘) ∈ ℂ)) → (𝐹 ∈ dom ⇝ ↔ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥)))
74	73	rexlimdvaa 3213	. 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ 𝑉) → (∃𝑛 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑛)(𝐹‘𝑘) ∈ ℂ → (𝐹 ∈ dom ⇝ ↔ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥))))
75	18, 29, 74	pm5.21ndd 380	1 ⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ 𝑉) → (𝐹 ∈ dom ⇝ ↔ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1539 ∃wex 1783 ∈ wcel 2108 ∀wral 3063 ∃wrex 3064 ⟨cop 4564 class class class wbr 5070 dom cdm 5580 ‘cfv 6418 (class class class)co 7255 ℂcc 10800 1c1 10803 < clt 10940 − cmin 11135 ℤcz 12249 ℤ_≥cuz 12511 ℝ⁺crp 12659 abscabs 14873 ⇝ cli 15121

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-er 8456 df-pm 8576 df-en 8692 df-dom 8693 df-sdom 8694 df-sup 9131 df-inf 9132 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-n0 12164 df-z 12250 df-uz 12512 df-rp 12660 df-ico 13014 df-fl 13440 df-seq 13650 df-exp 13711 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-limsup 15108 df-clim 15125 df-rlim 15126

This theorem is referenced by: serf0 15320

W3C validator