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Theorem caucvgb 15391

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 5808	. . . 4 ⊢ (𝐹 ∈ dom ⇝ → (𝐹 ∈ dom ⇝ ↔ ∃𝑚⟨𝐹, 𝑚⟩ ∈ ⇝ ))
2	1	ibi 266	. . 3 ⊢ (𝐹 ∈ dom ⇝ → ∃𝑚⟨𝐹, 𝑚⟩ ∈ ⇝ )
3		df-br 5075	. . . . 5 ⊢ (𝐹 ⇝ 𝑚 ↔ ⟨𝐹, 𝑚⟩ ∈ ⇝ )
4		caucvgb.1	. . . . . . . 8 ⊢ 𝑍 = (ℤ_≥‘𝑀)
5		simpll 764	. . . . . . . 8 ⊢ (((𝑀 ∈ ℤ ∧ 𝐹 ∈ 𝑉) ∧ 𝐹 ⇝ 𝑚) → 𝑀 ∈ ℤ)
6		1rp 12734	. . . . . . . . 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 485	. . . . . . . 8 ⊢ (((𝑀 ∈ ℤ ∧ 𝐹 ∈ 𝑉) ∧ 𝐹 ⇝ 𝑚) → 𝐹 ⇝ 𝑚)
10	4, 5, 7, 8, 9	climi 15219	. . . . . . 7 ⊢ (((𝑀 ∈ ℤ ∧ 𝐹 ∈ 𝑉) ∧ 𝐹 ⇝ 𝑚) → ∃𝑛 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑛)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝑚)) < 1))
11		simpl 483	. . . . . . . . 9 ⊢ (((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝑚)) < 1) → (𝐹‘𝑘) ∈ ℂ)
12	11	ralimi 3087	. . . . . . . 8 ⊢ (∀𝑘 ∈ (ℤ_≥‘𝑛)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝑚)) < 1) → ∀𝑘 ∈ (ℤ_≥‘𝑛)(𝐹‘𝑘) ∈ ℂ)
13	12	reximi 3178	. . . . . . 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 413	. . . . 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 1936	. . 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 6774	. . . . . . 7 ⊢ (𝑗 = 𝑛 → (ℤ_≥‘𝑗) = (ℤ_≥‘𝑛))
20	19	raleqdv 3348	. . . . . 6 ⊢ (𝑗 = 𝑛 → (∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) ∈ ℂ ↔ ∀𝑘 ∈ (ℤ_≥‘𝑛)(𝐹‘𝑘) ∈ ℂ))
21	20	cbvrexvw 3384	. . . . 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 483	. . . . . . 7 ⊢ (((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥) → (𝐹‘𝑘) ∈ ℂ)
24	23	ralimi 3087	. . . . . 6 ⊢ (∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥) → ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) ∈ ℂ)
25	24	reximi 3178	. . . . 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 3087	. . . 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 3562	. . 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 12592	. . . . . . . . . 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 15382	. . . . . . . . 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 15062	. . . . . . . . . 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 413	. . . . . . . . 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 3109	. . . . . . . 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 507	. . . . . . 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 649	. . . . . 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 711	. . . . 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 775	. . . . . . . 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 6774	. . . . . . . . . 10 ⊢ (𝑘 = 𝑚 → (𝐹‘𝑘) = (𝐹‘𝑚))
43	42	eleq1d 2823	. . . . . . . . 9 ⊢ (𝑘 = 𝑚 → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘𝑚) ∈ ℂ))
44	43	rspccva 3560	. . . . . . . 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 485	. . . . . . . . . . . . 13 ⊢ (((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥) → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥)
47	46	ralimi 3087	. . . . . . . . . . . 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 7297	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑚 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) = (abs‘((𝐹‘𝑚) − (𝐹‘𝑗))))
49	48	breq1d 5084	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑚 → ((abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥 ↔ (abs‘((𝐹‘𝑚) − (𝐹‘𝑗))) < 𝑥))
50	49	cbvralvw 3383	. . . . . . . . . . . 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 3178	. . . . . . . . . 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 3087	. . . . . . . . 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 482	. . . . . . . 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 6774	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑖 → (ℤ_≥‘𝑗) = (ℤ_≥‘𝑖))
56		fveq2 6774	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑖 → (𝐹‘𝑗) = (𝐹‘𝑖))
57	56	oveq2d 7291	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑖 → ((𝐹‘𝑚) − (𝐹‘𝑗)) = ((𝐹‘𝑚) − (𝐹‘𝑖)))
58	57	fveq2d 6778	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑖 → (abs‘((𝐹‘𝑚) − (𝐹‘𝑗))) = (abs‘((𝐹‘𝑚) − (𝐹‘𝑖))))
59	58	breq1d 5084	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑖 → ((abs‘((𝐹‘𝑚) − (𝐹‘𝑗))) < 𝑥 ↔ (abs‘((𝐹‘𝑚) − (𝐹‘𝑖))) < 𝑥))
60	55, 59	raleqbidv 3336	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑖 → (∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑚) − (𝐹‘𝑗))) < 𝑥 ↔ ∀𝑚 ∈ (ℤ_≥‘𝑖)(abs‘((𝐹‘𝑚) − (𝐹‘𝑖))) < 𝑥))
61	60	cbvrexvw 3384	. . . . . . . . . 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 5078	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → ((abs‘((𝐹‘𝑚) − (𝐹‘𝑖))) < 𝑥 ↔ (abs‘((𝐹‘𝑚) − (𝐹‘𝑖))) < 𝑦))
63	62	rexralbidv 3230	. . . . . . . . . 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 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 3383	. . . . . . . 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 764	. . . . . . 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 15390	. . . . . 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 715	. . . . 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 798	. . . 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 15068	. . . . 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 725	. . . 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 3214	. 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 381	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 396 = wceq 1539 ∃wex 1782 ∈ wcel 2106 ∀wral 3064 ∃wrex 3065 ⟨cop 4567 class class class wbr 5074 dom cdm 5589 ‘cfv 6433 (class class class)co 7275 ℂcc 10869 1c1 10872 < clt 11009 − cmin 11205 ℤcz 12319 ℤ_≥cuz 12582 ℝ⁺crp 12730 abscabs 14945 ⇝ cli 15193

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-er 8498 df-pm 8618 df-en 8734 df-dom 8735 df-sdom 8736 df-sup 9201 df-inf 9202 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-n0 12234 df-z 12320 df-uz 12583 df-rp 12731 df-ico 13085 df-fl 13512 df-seq 13722 df-exp 13783 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-limsup 15180 df-clim 15197 df-rlim 15198

This theorem is referenced by: serf0 15392

W3C validator