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Theorem caucvgb 15713

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 5913	. . . 4 ⊢ (𝐹 ∈ dom ⇝ → (𝐹 ∈ dom ⇝ ↔ ∃𝑚⟨𝐹, 𝑚⟩ ∈ ⇝ ))
2	1	ibi 267	. . 3 ⊢ (𝐹 ∈ dom ⇝ → ∃𝑚⟨𝐹, 𝑚⟩ ∈ ⇝ )
3		df-br 5149	. . . . 5 ⊢ (𝐹 ⇝ 𝑚 ↔ ⟨𝐹, 𝑚⟩ ∈ ⇝ )
4		caucvgb.1	. . . . . . . 8 ⊢ 𝑍 = (ℤ_≥‘𝑀)
5		simpll 767	. . . . . . . 8 ⊢ (((𝑀 ∈ ℤ ∧ 𝐹 ∈ 𝑉) ∧ 𝐹 ⇝ 𝑚) → 𝑀 ∈ ℤ)
6		1rp 13036	. . . . . . . . 9 ⊢ 1 ∈ ℝ⁺
7	6	a1i 11	. . . . . . . 8 ⊢ (((𝑀 ∈ ℤ ∧ 𝐹 ∈ 𝑉) ∧ 𝐹 ⇝ 𝑚) → 1 ∈ ℝ⁺)
8		eqidd 2736	. . . . . . . 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 15543	. . . . . . 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 3081	. . . . . . . 8 ⊢ (∀𝑘 ∈ (ℤ_≥‘𝑛)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝑚)) < 1) → ∀𝑘 ∈ (ℤ_≥‘𝑛)(𝐹‘𝑘) ∈ ℂ)
13	12	reximi 3082	. . . . . . 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	biimtrrid 243	. . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ 𝑉) → (⟨𝐹, 𝑚⟩ ∈ ⇝ → ∃𝑛 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑛)(𝐹‘𝑘) ∈ ℂ))
17	16	exlimdv 1931	. . 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 6907	. . . . . . 7 ⊢ (𝑗 = 𝑛 → (ℤ_≥‘𝑗) = (ℤ_≥‘𝑛))
20	19	raleqdv 3324	. . . . . 6 ⊢ (𝑗 = 𝑛 → (∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) ∈ ℂ ↔ ∀𝑘 ∈ (ℤ_≥‘𝑛)(𝐹‘𝑘) ∈ ℂ))
21	20	cbvrexvw 3236	. . . . 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 3081	. . . . . 6 ⊢ (∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥) → ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) ∈ ℂ)
25	24	reximi 3082	. . . . 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 3081	. . . 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 3623	. . 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 12886	. . . . . . . . . 10 ⊢ (𝑛 ∈ (ℤ_≥‘𝑀) → 𝑛 ∈ ℤ)
31	30, 4	eleq2s 2857	. . . . . . . . 9 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ ℤ)
32		eqid 2735	. . . . . . . . . 10 ⊢ (ℤ_≥‘𝑛) = (ℤ_≥‘𝑛)
33	32	climcau 15704	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℤ ∧ 𝐹 ∈ dom ⇝ ) → ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ (ℤ_≥‘𝑛)∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥)
34	31, 33	sylan 580	. . . . . . . 8 ⊢ ((𝑛 ∈ 𝑍 ∧ 𝐹 ∈ dom ⇝ ) → ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ (ℤ_≥‘𝑛)∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥)
35	32	r19.29uz 15386	. . . . . . . . . 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 3167	. . . . . . . 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 652	. . . . . 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 714	. . . . 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 778	. . . . . . . 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 6907	. . . . . . . . . 10 ⊢ (𝑘 = 𝑚 → (𝐹‘𝑘) = (𝐹‘𝑚))
43	42	eleq1d 2824	. . . . . . . . 9 ⊢ (𝑘 = 𝑚 → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘𝑚) ∈ ℂ))
44	43	rspccva 3621	. . . . . . . 8 ⊢ ((∀𝑘 ∈ (ℤ_≥‘𝑛)(𝐹‘𝑘) ∈ ℂ ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → (𝐹‘𝑚) ∈ ℂ)
45	41, 44	sylan 580	. . . . . . 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 3081	. . . . . . . . . . . 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 7453	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑚 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) = (abs‘((𝐹‘𝑚) − (𝐹‘𝑗))))
49	48	breq1d 5158	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑚 → ((abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥 ↔ (abs‘((𝐹‘𝑚) − (𝐹‘𝑗))) < 𝑥))
50	49	cbvralvw 3235	. . . . . . . . . . . 12 ⊢ (∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥 ↔ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑚) − (𝐹‘𝑗))) < 𝑥)
51	47, 50	sylib 218	. . . . . . . . . . 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 3082	. . . . . . . . . 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 3081	. . . . . . . . 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 6907	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑖 → (ℤ_≥‘𝑗) = (ℤ_≥‘𝑖))
56		fveq2 6907	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑖 → (𝐹‘𝑗) = (𝐹‘𝑖))
57	56	oveq2d 7447	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑖 → ((𝐹‘𝑚) − (𝐹‘𝑗)) = ((𝐹‘𝑚) − (𝐹‘𝑖)))
58	57	fveq2d 6911	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑖 → (abs‘((𝐹‘𝑚) − (𝐹‘𝑗))) = (abs‘((𝐹‘𝑚) − (𝐹‘𝑖))))
59	58	breq1d 5158	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑖 → ((abs‘((𝐹‘𝑚) − (𝐹‘𝑗))) < 𝑥 ↔ (abs‘((𝐹‘𝑚) − (𝐹‘𝑖))) < 𝑥))
60	55, 59	raleqbidv 3344	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑖 → (∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑚) − (𝐹‘𝑗))) < 𝑥 ↔ ∀𝑚 ∈ (ℤ_≥‘𝑖)(abs‘((𝐹‘𝑚) − (𝐹‘𝑖))) < 𝑥))
61	60	cbvrexvw 3236	. . . . . . . . . 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 5152	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → ((abs‘((𝐹‘𝑚) − (𝐹‘𝑖))) < 𝑥 ↔ (abs‘((𝐹‘𝑚) − (𝐹‘𝑖))) < 𝑦))
63	62	rexralbidv 3221	. . . . . . . . . 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	bitrid 283	. . . . . . . . 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 3235	. . . . . . . 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 218	. . . . . . 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 767	. . . . . . 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 15712	. . . . . 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 718	. . . . 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 801	. . . 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 15392	. . . . 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 728	. . . 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 282	. . 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 3154	. 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 379	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 206 ∧ wa 395 = wceq 1537 ∃wex 1776 ∈ wcel 2106 ∀wral 3059 ∃wrex 3068 ⟨cop 4637 class class class wbr 5148 dom cdm 5689 ‘cfv 6563 (class class class)co 7431 ℂcc 11151 1c1 11154 < clt 11293 − cmin 11490 ℤcz 12611 ℤ_≥cuz 12876 ℝ⁺crp 13032 abscabs 15270 ⇝ cli 15517

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-pm 8868 df-en 8985 df-dom 8986 df-sdom 8987 df-sup 9480 df-inf 9481 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12612 df-uz 12877 df-rp 13033 df-ico 13390 df-fl 13829 df-seq 14040 df-exp 14100 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-limsup 15504 df-clim 15521 df-rlim 15522

This theorem is referenced by: serf0 15714 caucvgbf 45440

W3C validator