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Theorem caucvgb 15601

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 5846	. . . 4 ⊢ (𝐹 ∈ dom ⇝ → (𝐹 ∈ dom ⇝ ↔ ∃𝑚⟨𝐹, 𝑚⟩ ∈ ⇝ ))
2	1	ibi 267	. . 3 ⊢ (𝐹 ∈ dom ⇝ → ∃𝑚⟨𝐹, 𝑚⟩ ∈ ⇝ )
3		df-br 5097	. . . . 5 ⊢ (𝐹 ⇝ 𝑚 ↔ ⟨𝐹, 𝑚⟩ ∈ ⇝ )
4		caucvgb.1	. . . . . . . 8 ⊢ 𝑍 = (ℤ_≥‘𝑀)
5		simpll 766	. . . . . . . 8 ⊢ (((𝑀 ∈ ℤ ∧ 𝐹 ∈ 𝑉) ∧ 𝐹 ⇝ 𝑚) → 𝑀 ∈ ℤ)
6		1rp 12907	. . . . . . . . 9 ⊢ 1 ∈ ℝ⁺
7	6	a1i 11	. . . . . . . 8 ⊢ (((𝑀 ∈ ℤ ∧ 𝐹 ∈ 𝑉) ∧ 𝐹 ⇝ 𝑚) → 1 ∈ ℝ⁺)
8		eqidd 2735	. . . . . . . 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 15431	. . . . . . 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 3071	. . . . . . . 8 ⊢ (∀𝑘 ∈ (ℤ_≥‘𝑛)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝑚)) < 1) → ∀𝑘 ∈ (ℤ_≥‘𝑛)(𝐹‘𝑘) ∈ ℂ)
13	12	reximi 3072	. . . . . . 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 1934	. . 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 6832	. . . . . . 7 ⊢ (𝑗 = 𝑛 → (ℤ_≥‘𝑗) = (ℤ_≥‘𝑛))
20	19	raleqdv 3294	. . . . . 6 ⊢ (𝑗 = 𝑛 → (∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) ∈ ℂ ↔ ∀𝑘 ∈ (ℤ_≥‘𝑛)(𝐹‘𝑘) ∈ ℂ))
21	20	cbvrexvw 3213	. . . . 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 3071	. . . . . 6 ⊢ (∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥) → ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) ∈ ℂ)
25	24	reximi 3072	. . . . 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 3071	. . . 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 3575	. . 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 12759	. . . . . . . . . 10 ⊢ (𝑛 ∈ (ℤ_≥‘𝑀) → 𝑛 ∈ ℤ)
31	30, 4	eleq2s 2852	. . . . . . . . 9 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ ℤ)
32		eqid 2734	. . . . . . . . . 10 ⊢ (ℤ_≥‘𝑛) = (ℤ_≥‘𝑛)
33	32	climcau 15592	. . . . . . . . 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 15272	. . . . . . . . . 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 3148	. . . . . . . 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 777	. . . . . . . 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 6832	. . . . . . . . . 10 ⊢ (𝑘 = 𝑚 → (𝐹‘𝑘) = (𝐹‘𝑚))
43	42	eleq1d 2819	. . . . . . . . 9 ⊢ (𝑘 = 𝑚 → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘𝑚) ∈ ℂ))
44	43	rspccva 3573	. . . . . . . 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 3071	. . . . . . . . . . . 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 7378	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑚 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) = (abs‘((𝐹‘𝑚) − (𝐹‘𝑗))))
49	48	breq1d 5106	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑚 → ((abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥 ↔ (abs‘((𝐹‘𝑚) − (𝐹‘𝑗))) < 𝑥))
50	49	cbvralvw 3212	. . . . . . . . . . . 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 3072	. . . . . . . . . 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 3071	. . . . . . . . 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 6832	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑖 → (ℤ_≥‘𝑗) = (ℤ_≥‘𝑖))
56		fveq2 6832	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑖 → (𝐹‘𝑗) = (𝐹‘𝑖))
57	56	oveq2d 7372	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑖 → ((𝐹‘𝑚) − (𝐹‘𝑗)) = ((𝐹‘𝑚) − (𝐹‘𝑖)))
58	57	fveq2d 6836	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑖 → (abs‘((𝐹‘𝑚) − (𝐹‘𝑗))) = (abs‘((𝐹‘𝑚) − (𝐹‘𝑖))))
59	58	breq1d 5106	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑖 → ((abs‘((𝐹‘𝑚) − (𝐹‘𝑗))) < 𝑥 ↔ (abs‘((𝐹‘𝑚) − (𝐹‘𝑖))) < 𝑥))
60	55, 59	raleqbidv 3314	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑖 → (∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑚) − (𝐹‘𝑗))) < 𝑥 ↔ ∀𝑚 ∈ (ℤ_≥‘𝑖)(abs‘((𝐹‘𝑚) − (𝐹‘𝑖))) < 𝑥))
61	60	cbvrexvw 3213	. . . . . . . . . 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 5100	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → ((abs‘((𝐹‘𝑚) − (𝐹‘𝑖))) < 𝑥 ↔ (abs‘((𝐹‘𝑚) − (𝐹‘𝑖))) < 𝑦))
63	62	rexralbidv 3200	. . . . . . . . . 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 3212	. . . . . . . 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 766	. . . . . . 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 15600	. . . . . 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 800	. . . 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 15278	. . . . 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 3136	. 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 1541 ∃wex 1780 ∈ wcel 2113 ∀wral 3049 ∃wrex 3058 ⟨cop 4584 class class class wbr 5096 dom cdm 5622 ‘cfv 6490 (class class class)co 7356 ℂcc 11022 1c1 11025 < clt 11164 − cmin 11362 ℤcz 12486 ℤ_≥cuz 12749 ℝ⁺crp 12903 abscabs 15155 ⇝ cli 15405

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2706 ax-rep 5222 ax-sep 5239 ax-nul 5249 ax-pow 5308 ax-pr 5375 ax-un 7678 ax-cnex 11080 ax-resscn 11081 ax-1cn 11082 ax-icn 11083 ax-addcl 11084 ax-addrcl 11085 ax-mulcl 11086 ax-mulrcl 11087 ax-mulcom 11088 ax-addass 11089 ax-mulass 11090 ax-distr 11091 ax-i2m1 11092 ax-1ne0 11093 ax-1rid 11094 ax-rnegex 11095 ax-rrecex 11096 ax-cnre 11097 ax-pre-lttri 11098 ax-pre-lttrn 11099 ax-pre-ltadd 11100 ax-pre-mulgt0 11101 ax-pre-sup 11102

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3059 df-rmo 3348 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-op 4585 df-uni 4862 df-iun 4946 df-br 5097 df-opab 5159 df-mpt 5178 df-tr 5204 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-er 8633 df-pm 8764 df-en 8882 df-dom 8883 df-sdom 8884 df-sup 9343 df-inf 9344 df-pnf 11166 df-mnf 11167 df-xr 11168 df-ltxr 11169 df-le 11170 df-sub 11364 df-neg 11365 df-div 11793 df-nn 12144 df-2 12206 df-3 12207 df-n0 12400 df-z 12487 df-uz 12750 df-rp 12904 df-ico 13265 df-fl 13710 df-seq 13923 df-exp 13983 df-cj 15020 df-re 15021 df-im 15022 df-sqrt 15156 df-abs 15157 df-limsup 15392 df-clim 15409 df-rlim 15410

This theorem is referenced by: serf0 15602 caucvgbf 45675

W3C validator