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Theorem caucvgb 14410

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 5320	. . . 4 ⊢ (𝐹 ∈ dom ⇝ → (𝐹 ∈ dom ⇝ ↔ ∃𝑚⟨𝐹, 𝑚⟩ ∈ ⇝ ))
2	1	ibi 256	. . 3 ⊢ (𝐹 ∈ dom ⇝ → ∃𝑚⟨𝐹, 𝑚⟩ ∈ ⇝ )
3		df-br 4654	. . . . 5 ⊢ (𝐹 ⇝ 𝑚 ↔ ⟨𝐹, 𝑚⟩ ∈ ⇝ )
4		caucvgb.1	. . . . . . . 8 ⊢ 𝑍 = (ℤ_≥‘𝑀)
5		simpll 790	. . . . . . . 8 ⊢ (((𝑀 ∈ ℤ ∧ 𝐹 ∈ 𝑉) ∧ 𝐹 ⇝ 𝑚) → 𝑀 ∈ ℤ)
6		1rp 11836	. . . . . . . . 9 ⊢ 1 ∈ ℝ⁺
7	6	a1i 11	. . . . . . . 8 ⊢ (((𝑀 ∈ ℤ ∧ 𝐹 ∈ 𝑉) ∧ 𝐹 ⇝ 𝑚) → 1 ∈ ℝ⁺)
8		eqidd 2623	. . . . . . . 8 ⊢ ((((𝑀 ∈ ℤ ∧ 𝐹 ∈ 𝑉) ∧ 𝐹 ⇝ 𝑚) ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = (𝐹‘𝑘))
9		simpr 477	. . . . . . . 8 ⊢ (((𝑀 ∈ ℤ ∧ 𝐹 ∈ 𝑉) ∧ 𝐹 ⇝ 𝑚) → 𝐹 ⇝ 𝑚)
10	4, 5, 7, 8, 9	climi 14241	. . . . . . 7 ⊢ (((𝑀 ∈ ℤ ∧ 𝐹 ∈ 𝑉) ∧ 𝐹 ⇝ 𝑚) → ∃𝑛 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑛)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝑚)) < 1))
11		simpl 473	. . . . . . . . 9 ⊢ (((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝑚)) < 1) → (𝐹‘𝑘) ∈ ℂ)
12	11	ralimi 2952	. . . . . . . 8 ⊢ (∀𝑘 ∈ (ℤ_≥‘𝑛)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝑚)) < 1) → ∀𝑘 ∈ (ℤ_≥‘𝑛)(𝐹‘𝑘) ∈ ℂ)
13	12	reximi 3011	. . . . . . 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 450	. . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ 𝑉) → (𝐹 ⇝ 𝑚 → ∃𝑛 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑛)(𝐹‘𝑘) ∈ ℂ))
16	3, 15	syl5bir 233	. . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ 𝑉) → (⟨𝐹, 𝑚⟩ ∈ ⇝ → ∃𝑛 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑛)(𝐹‘𝑘) ∈ ℂ))
17	16	exlimdv 1861	. . 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		simpl 473	. . . . . . 7 ⊢ (((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥) → (𝐹‘𝑘) ∈ ℂ)
20	19	ralimi 2952	. . . . . 6 ⊢ (∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥) → ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) ∈ ℂ)
21	20	reximi 3011	. . . . 5 ⊢ (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) ∈ ℂ)
22	21	ralimi 2952	. . . 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)
23		fveq2 6191	. . . . . . . 8 ⊢ (𝑗 = 𝑛 → (ℤ_≥‘𝑗) = (ℤ_≥‘𝑛))
24	23	raleqdv 3144	. . . . . . 7 ⊢ (𝑗 = 𝑛 → (∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) ∈ ℂ ↔ ∀𝑘 ∈ (ℤ_≥‘𝑛)(𝐹‘𝑘) ∈ ℂ))
25	24	cbvrexv 3172	. . . . . 6 ⊢ (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) ∈ ℂ ↔ ∃𝑛 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑛)(𝐹‘𝑘) ∈ ℂ)
26	25	a1i 11	. . . . 5 ⊢ (𝑥 = 1 → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) ∈ ℂ ↔ ∃𝑛 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑛)(𝐹‘𝑘) ∈ ℂ))
27	26	rspcv 3305	. . . 4 ⊢ (1 ∈ ℝ⁺ → (∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) ∈ ℂ → ∃𝑛 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑛)(𝐹‘𝑘) ∈ ℂ))
28	6, 22, 27	mpsyl 68	. . 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 11697	. . . . . . . . . 10 ⊢ (𝑛 ∈ (ℤ_≥‘𝑀) → 𝑛 ∈ ℤ)
31	30, 4	eleq2s 2719	. . . . . . . . 9 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ ℤ)
32		eqid 2622	. . . . . . . . . 10 ⊢ (ℤ_≥‘𝑛) = (ℤ_≥‘𝑛)
33	32	climcau 14401	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℤ ∧ 𝐹 ∈ dom ⇝ ) → ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ (ℤ_≥‘𝑛)∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥)
34	31, 33	sylan 488	. . . . . . . 8 ⊢ ((𝑛 ∈ 𝑍 ∧ 𝐹 ∈ dom ⇝ ) → ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ (ℤ_≥‘𝑛)∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥)
35	32	r19.29uz 14090	. . . . . . . . . 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 450	. . . . . . . . 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 2963	. . . . . . . 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 486	. . . . . . 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 846	. . . . . 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 750	. . . . 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 801	. . . . . . . 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 6191	. . . . . . . . . 10 ⊢ (𝑘 = 𝑚 → (𝐹‘𝑘) = (𝐹‘𝑚))
43	42	eleq1d 2686	. . . . . . . . 9 ⊢ (𝑘 = 𝑚 → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘𝑚) ∈ ℂ))
44	43	rspccva 3308	. . . . . . . 8 ⊢ ((∀𝑘 ∈ (ℤ_≥‘𝑛)(𝐹‘𝑘) ∈ ℂ ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → (𝐹‘𝑚) ∈ ℂ)
45	41, 44	sylan 488	. . . . . . 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 477	. . . . . . . . . . . . 13 ⊢ (((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥) → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥)
47	46	ralimi 2952	. . . . . . . . . . . 12 ⊢ (∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥) → ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥)
48	42	oveq1d 6665	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑚 → ((𝐹‘𝑘) − (𝐹‘𝑗)) = ((𝐹‘𝑚) − (𝐹‘𝑗)))
49	48	fveq2d 6195	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑚 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) = (abs‘((𝐹‘𝑚) − (𝐹‘𝑗))))
50	49	breq1d 4663	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑚 → ((abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥 ↔ (abs‘((𝐹‘𝑚) − (𝐹‘𝑗))) < 𝑥))
51	50	cbvralv 3171	. . . . . . . . . . . 12 ⊢ (∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥 ↔ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑚) − (𝐹‘𝑗))) < 𝑥)
52	47, 51	sylib 208	. . . . . . . . . . 11 ⊢ (∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥) → ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑚) − (𝐹‘𝑗))) < 𝑥)
53	52	reximi 3011	. . . . . . . . . 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)
54	53	ralimi 2952	. . . . . . . . 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)
55	54	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)
56		fveq2 6191	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑖 → (ℤ_≥‘𝑗) = (ℤ_≥‘𝑖))
57		fveq2 6191	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑖 → (𝐹‘𝑗) = (𝐹‘𝑖))
58	57	oveq2d 6666	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑖 → ((𝐹‘𝑚) − (𝐹‘𝑗)) = ((𝐹‘𝑚) − (𝐹‘𝑖)))
59	58	fveq2d 6195	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑖 → (abs‘((𝐹‘𝑚) − (𝐹‘𝑗))) = (abs‘((𝐹‘𝑚) − (𝐹‘𝑖))))
60	59	breq1d 4663	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑖 → ((abs‘((𝐹‘𝑚) − (𝐹‘𝑗))) < 𝑥 ↔ (abs‘((𝐹‘𝑚) − (𝐹‘𝑖))) < 𝑥))
61	56, 60	raleqbidv 3152	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑖 → (∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑚) − (𝐹‘𝑗))) < 𝑥 ↔ ∀𝑚 ∈ (ℤ_≥‘𝑖)(abs‘((𝐹‘𝑚) − (𝐹‘𝑖))) < 𝑥))
62	61	cbvrexv 3172	. . . . . . . . . 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)
63		breq2 4657	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → ((abs‘((𝐹‘𝑚) − (𝐹‘𝑖))) < 𝑥 ↔ (abs‘((𝐹‘𝑚) − (𝐹‘𝑖))) < 𝑦))
64	63	rexralbidv 3058	. . . . . . . . . 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))
65	62, 64	syl5bb 272	. . . . . . . . 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))
66	65	cbvralv 3171	. . . . . . . 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)
67	55, 66	sylib 208	. . . . . . 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)
68		simpll 790	. . . . . . 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)
69	32, 45, 67, 68	caucvg 14409	. . . . . 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 ⇝ )
70	69	adantlll 754	. . . . 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 ⇝ )
71	40, 70	impbida 877	. . . 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)))
72	4, 32	cau4 14096	. . . . 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)))
73	72	ad2antrl 764	. . . 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)))
74	71, 73	bitr4d 271	. . 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)))
75	74	rexlimdvaa 3032	. 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))))
76	18, 29, 75	pm5.21ndd 369	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 196 ∧ wa 384 = wceq 1483 ∃wex 1704 ∈ wcel 1990 ∀wral 2912 ∃wrex 2913 ⟨cop 4183 class class class wbr 4653 dom cdm 5114 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 1c1 9937 < clt 10074 − cmin 10266 ℤcz 11377 ℤ_≥cuz 11687 ℝ⁺crp 11832 abscabs 13974 ⇝ cli 14215

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-inf 8349 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-ico 12181 df-fl 12593 df-seq 12802 df-exp 12861 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-limsup 14202 df-clim 14219 df-rlim 14220

This theorem is referenced by: serf0 14411

W3C validator