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Theorem cdivcncfap 14564

Description: Division with a constant numerator is continuous. (Contributed by Mario Carneiro, 28-Dec-2016.) (Revised by Jim Kingdon, 26-May-2023.)

**Hypothesis**
Ref	Expression
cdivcncf.1	⊢ 𝐹 = (𝑥 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↦ (𝐴 / 𝑥))

**Assertion**
Ref	Expression
cdivcncfap	⊢ (𝐴 ∈ ℂ → 𝐹 ∈ ({𝑦 ∈ ℂ ∣ 𝑦 # 0}–cn→ℂ))

Distinct variable group: 𝑥,𝐴,𝑦 Allowed substitution hints: 𝐹(𝑥,𝑦)

Proof of Theorem cdivcncfap Dummy variables 𝑤 𝑧 𝑎 𝑏 𝑑 𝑒 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cdivcncf.1	. 2 ⊢ 𝐹 = (𝑥 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↦ (𝐴 / 𝑥))
2		simpl 109	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0}) → 𝐴 ∈ ℂ)
3		breq1 4021	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → (𝑦 # 0 ↔ 𝑥 # 0))
4	3	elrab 2908	. . . . . . . 8 ⊢ (𝑥 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↔ (𝑥 ∈ ℂ ∧ 𝑥 # 0))
5	4	biimpi 120	. . . . . . 7 ⊢ (𝑥 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} → (𝑥 ∈ ℂ ∧ 𝑥 # 0))
6	5	adantl 277	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0}) → (𝑥 ∈ ℂ ∧ 𝑥 # 0))
7	6	simpld 112	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0}) → 𝑥 ∈ ℂ)
8	6	simprd 114	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0}) → 𝑥 # 0)
9	2, 7, 8	divrecapd 8781	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0}) → (𝐴 / 𝑥) = (𝐴 · (1 / 𝑥)))
10	9	mpteq2dva 4108	. . 3 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↦ (𝐴 / 𝑥)) = (𝑥 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↦ (𝐴 · (1 / 𝑥))))
11		recclap 8667	. . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑥 # 0) → (1 / 𝑥) ∈ ℂ)
12	4, 11	sylbi 121	. . . . . 6 ⊢ (𝑥 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} → (1 / 𝑥) ∈ ℂ)
13	12	adantl 277	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0}) → (1 / 𝑥) ∈ ℂ)
14		oveq2 5905	. . . . . . 7 ⊢ (𝑤 = 𝑥 → (1 / 𝑤) = (1 / 𝑥))
15	14	cbvmptv 4114	. . . . . 6 ⊢ (𝑤 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↦ (1 / 𝑤)) = (𝑥 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↦ (1 / 𝑥))
16	15	a1i 9	. . . . 5 ⊢ (𝐴 ∈ ℂ → (𝑤 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↦ (1 / 𝑤)) = (𝑥 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↦ (1 / 𝑥)))
17		eqidd 2190	. . . . 5 ⊢ (𝐴 ∈ ℂ → (𝑧 ∈ ℂ ↦ (𝐴 · 𝑧)) = (𝑧 ∈ ℂ ↦ (𝐴 · 𝑧)))
18		oveq2 5905	. . . . 5 ⊢ (𝑧 = (1 / 𝑥) → (𝐴 · 𝑧) = (𝐴 · (1 / 𝑥)))
19	13, 16, 17, 18	fmptco 5703	. . . 4 ⊢ (𝐴 ∈ ℂ → ((𝑧 ∈ ℂ ↦ (𝐴 · 𝑧)) ∘ (𝑤 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↦ (1 / 𝑤))) = (𝑥 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↦ (𝐴 · (1 / 𝑥))))
20		breq1 4021	. . . . . . . . . 10 ⊢ (𝑦 = 𝑤 → (𝑦 # 0 ↔ 𝑤 # 0))
21	20	elrab 2908	. . . . . . . . 9 ⊢ (𝑤 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↔ (𝑤 ∈ ℂ ∧ 𝑤 # 0))
22		recclap 8667	. . . . . . . . 9 ⊢ ((𝑤 ∈ ℂ ∧ 𝑤 # 0) → (1 / 𝑤) ∈ ℂ)
23	21, 22	sylbi 121	. . . . . . . 8 ⊢ (𝑤 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} → (1 / 𝑤) ∈ ℂ)
24	23	adantl 277	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑤 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0}) → (1 / 𝑤) ∈ ℂ)
25	24	fmpttd 5692	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (𝑤 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↦ (1 / 𝑤)):{𝑦 ∈ ℂ ∣ 𝑦 # 0}⟶ℂ)
26		breq1 4021	. . . . . . . . 9 ⊢ (𝑦 = 𝑏 → (𝑦 # 0 ↔ 𝑏 # 0))
27	26	elrab 2908	. . . . . . . 8 ⊢ (𝑏 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↔ (𝑏 ∈ ℂ ∧ 𝑏 # 0))
28		eqid 2189	. . . . . . . . . . . 12 ⊢ (inf({1, ((abs‘𝑏) · 𝑒)}, ℝ, < ) · ((abs‘𝑏) / 2)) = (inf({1, ((abs‘𝑏) · 𝑒)}, ℝ, < ) · ((abs‘𝑏) / 2))
29	28	reccn2ap 11356	. . . . . . . . . . 11 ⊢ ((𝑏 ∈ ℂ ∧ 𝑏 # 0 ∧ 𝑒 ∈ ℝ⁺) → ∃𝑑 ∈ ℝ⁺ ∀𝑎 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ((abs‘(𝑎 − 𝑏)) < 𝑑 → (abs‘((1 / 𝑎) − (1 / 𝑏))) < 𝑒))
30		eqidd 2190	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑏 ∈ ℂ ∧ 𝑏 # 0 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑎 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0}) → (𝑤 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↦ (1 / 𝑤)) = (𝑤 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↦ (1 / 𝑤)))
31		oveq2 5905	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 = 𝑎 → (1 / 𝑤) = (1 / 𝑎))
32	31	adantl 277	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑏 ∈ ℂ ∧ 𝑏 # 0 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑎 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0}) ∧ 𝑤 = 𝑎) → (1 / 𝑤) = (1 / 𝑎))
33		simpr 110	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑏 ∈ ℂ ∧ 𝑏 # 0 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑎 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0}) → 𝑎 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0})
34		breq1 4021	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = 𝑎 → (𝑦 # 0 ↔ 𝑎 # 0))
35	34	elrab 2908	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↔ (𝑎 ∈ ℂ ∧ 𝑎 # 0))
36		recclap 8667	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑎 ∈ ℂ ∧ 𝑎 # 0) → (1 / 𝑎) ∈ ℂ)
37	35, 36	sylbi 121	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} → (1 / 𝑎) ∈ ℂ)
38	37	adantl 277	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑏 ∈ ℂ ∧ 𝑏 # 0 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑎 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0}) → (1 / 𝑎) ∈ ℂ)
39	30, 32, 33, 38	fvmptd 5618	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑏 ∈ ℂ ∧ 𝑏 # 0 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑎 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0}) → ((𝑤 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↦ (1 / 𝑤))‘𝑎) = (1 / 𝑎))
40		oveq2 5905	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 = 𝑏 → (1 / 𝑤) = (1 / 𝑏))
41	40	adantl 277	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑏 ∈ ℂ ∧ 𝑏 # 0 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑎 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0}) ∧ 𝑤 = 𝑏) → (1 / 𝑤) = (1 / 𝑏))
42		simpll1 1038	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑏 ∈ ℂ ∧ 𝑏 # 0 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑎 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0}) → 𝑏 ∈ ℂ)
43		simpll2 1039	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑏 ∈ ℂ ∧ 𝑏 # 0 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑎 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0}) → 𝑏 # 0)
44	26, 42, 43	elrabd 2910	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑏 ∈ ℂ ∧ 𝑏 # 0 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑎 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0}) → 𝑏 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0})
45	42, 43	recclapd 8769	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑏 ∈ ℂ ∧ 𝑏 # 0 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑎 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0}) → (1 / 𝑏) ∈ ℂ)
46	30, 41, 44, 45	fvmptd 5618	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑏 ∈ ℂ ∧ 𝑏 # 0 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑎 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0}) → ((𝑤 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↦ (1 / 𝑤))‘𝑏) = (1 / 𝑏))
47	39, 46	oveq12d 5915	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑏 ∈ ℂ ∧ 𝑏 # 0 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑎 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0}) → (((𝑤 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↦ (1 / 𝑤))‘𝑎) − ((𝑤 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↦ (1 / 𝑤))‘𝑏)) = ((1 / 𝑎) − (1 / 𝑏)))
48	47	fveq2d 5538	. . . . . . . . . . . . . . 15 ⊢ ((((𝑏 ∈ ℂ ∧ 𝑏 # 0 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑎 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0}) → (abs‘(((𝑤 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↦ (1 / 𝑤))‘𝑎) − ((𝑤 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↦ (1 / 𝑤))‘𝑏))) = (abs‘((1 / 𝑎) − (1 / 𝑏))))
49	48	breq1d 4028	. . . . . . . . . . . . . 14 ⊢ ((((𝑏 ∈ ℂ ∧ 𝑏 # 0 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑎 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0}) → ((abs‘(((𝑤 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↦ (1 / 𝑤))‘𝑎) − ((𝑤 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↦ (1 / 𝑤))‘𝑏))) < 𝑒 ↔ (abs‘((1 / 𝑎) − (1 / 𝑏))) < 𝑒))
50	49	imbi2d 230	. . . . . . . . . . . . 13 ⊢ ((((𝑏 ∈ ℂ ∧ 𝑏 # 0 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑎 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0}) → (((abs‘(𝑎 − 𝑏)) < 𝑑 → (abs‘(((𝑤 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↦ (1 / 𝑤))‘𝑎) − ((𝑤 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↦ (1 / 𝑤))‘𝑏))) < 𝑒) ↔ ((abs‘(𝑎 − 𝑏)) < 𝑑 → (abs‘((1 / 𝑎) − (1 / 𝑏))) < 𝑒)))
51	50	ralbidva 2486	. . . . . . . . . . . 12 ⊢ (((𝑏 ∈ ℂ ∧ 𝑏 # 0 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) → (∀𝑎 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ((abs‘(𝑎 − 𝑏)) < 𝑑 → (abs‘(((𝑤 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↦ (1 / 𝑤))‘𝑎) − ((𝑤 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↦ (1 / 𝑤))‘𝑏))) < 𝑒) ↔ ∀𝑎 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ((abs‘(𝑎 − 𝑏)) < 𝑑 → (abs‘((1 / 𝑎) − (1 / 𝑏))) < 𝑒)))
52	51	rexbidva 2487	. . . . . . . . . . 11 ⊢ ((𝑏 ∈ ℂ ∧ 𝑏 # 0 ∧ 𝑒 ∈ ℝ⁺) → (∃𝑑 ∈ ℝ⁺ ∀𝑎 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ((abs‘(𝑎 − 𝑏)) < 𝑑 → (abs‘(((𝑤 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↦ (1 / 𝑤))‘𝑎) − ((𝑤 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↦ (1 / 𝑤))‘𝑏))) < 𝑒) ↔ ∃𝑑 ∈ ℝ⁺ ∀𝑎 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ((abs‘(𝑎 − 𝑏)) < 𝑑 → (abs‘((1 / 𝑎) − (1 / 𝑏))) < 𝑒)))
53	29, 52	mpbird 167	. . . . . . . . . 10 ⊢ ((𝑏 ∈ ℂ ∧ 𝑏 # 0 ∧ 𝑒 ∈ ℝ⁺) → ∃𝑑 ∈ ℝ⁺ ∀𝑎 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ((abs‘(𝑎 − 𝑏)) < 𝑑 → (abs‘(((𝑤 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↦ (1 / 𝑤))‘𝑎) − ((𝑤 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↦ (1 / 𝑤))‘𝑏))) < 𝑒))
54	53	3expa 1205	. . . . . . . . 9 ⊢ (((𝑏 ∈ ℂ ∧ 𝑏 # 0) ∧ 𝑒 ∈ ℝ⁺) → ∃𝑑 ∈ ℝ⁺ ∀𝑎 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ((abs‘(𝑎 − 𝑏)) < 𝑑 → (abs‘(((𝑤 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↦ (1 / 𝑤))‘𝑎) − ((𝑤 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↦ (1 / 𝑤))‘𝑏))) < 𝑒))
55	54	ralrimiva 2563	. . . . . . . 8 ⊢ ((𝑏 ∈ ℂ ∧ 𝑏 # 0) → ∀𝑒 ∈ ℝ⁺ ∃𝑑 ∈ ℝ⁺ ∀𝑎 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ((abs‘(𝑎 − 𝑏)) < 𝑑 → (abs‘(((𝑤 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↦ (1 / 𝑤))‘𝑎) − ((𝑤 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↦ (1 / 𝑤))‘𝑏))) < 𝑒))
56	27, 55	sylbi 121	. . . . . . 7 ⊢ (𝑏 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} → ∀𝑒 ∈ ℝ⁺ ∃𝑑 ∈ ℝ⁺ ∀𝑎 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ((abs‘(𝑎 − 𝑏)) < 𝑑 → (abs‘(((𝑤 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↦ (1 / 𝑤))‘𝑎) − ((𝑤 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↦ (1 / 𝑤))‘𝑏))) < 𝑒))
57	56	rgen 2543	. . . . . 6 ⊢ ∀𝑏 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0}∀𝑒 ∈ ℝ⁺ ∃𝑑 ∈ ℝ⁺ ∀𝑎 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ((abs‘(𝑎 − 𝑏)) < 𝑑 → (abs‘(((𝑤 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↦ (1 / 𝑤))‘𝑎) − ((𝑤 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↦ (1 / 𝑤))‘𝑏))) < 𝑒)
58		ssrab2 3255	. . . . . . 7 ⊢ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ⊆ ℂ
59		ssid 3190	. . . . . . 7 ⊢ ℂ ⊆ ℂ
60		elcncf2 14538	. . . . . . 7 ⊢ (({𝑦 ∈ ℂ ∣ 𝑦 # 0} ⊆ ℂ ∧ ℂ ⊆ ℂ) → ((𝑤 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↦ (1 / 𝑤)) ∈ ({𝑦 ∈ ℂ ∣ 𝑦 # 0}–cn→ℂ) ↔ ((𝑤 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↦ (1 / 𝑤)):{𝑦 ∈ ℂ ∣ 𝑦 # 0}⟶ℂ ∧ ∀𝑏 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0}∀𝑒 ∈ ℝ⁺ ∃𝑑 ∈ ℝ⁺ ∀𝑎 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ((abs‘(𝑎 − 𝑏)) < 𝑑 → (abs‘(((𝑤 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↦ (1 / 𝑤))‘𝑎) − ((𝑤 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↦ (1 / 𝑤))‘𝑏))) < 𝑒))))
61	58, 59, 60	mp2an 426	. . . . . 6 ⊢ ((𝑤 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↦ (1 / 𝑤)) ∈ ({𝑦 ∈ ℂ ∣ 𝑦 # 0}–cn→ℂ) ↔ ((𝑤 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↦ (1 / 𝑤)):{𝑦 ∈ ℂ ∣ 𝑦 # 0}⟶ℂ ∧ ∀𝑏 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0}∀𝑒 ∈ ℝ⁺ ∃𝑑 ∈ ℝ⁺ ∀𝑎 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ((abs‘(𝑎 − 𝑏)) < 𝑑 → (abs‘(((𝑤 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↦ (1 / 𝑤))‘𝑎) − ((𝑤 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↦ (1 / 𝑤))‘𝑏))) < 𝑒)))
62	25, 57, 61	sylanblrc 416	. . . . 5 ⊢ (𝐴 ∈ ℂ → (𝑤 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↦ (1 / 𝑤)) ∈ ({𝑦 ∈ ℂ ∣ 𝑦 # 0}–cn→ℂ))
63		eqid 2189	. . . . . 6 ⊢ (𝑧 ∈ ℂ ↦ (𝐴 · 𝑧)) = (𝑧 ∈ ℂ ↦ (𝐴 · 𝑧))
64	63	mulc1cncf 14553	. . . . 5 ⊢ (𝐴 ∈ ℂ → (𝑧 ∈ ℂ ↦ (𝐴 · 𝑧)) ∈ (ℂ–cn→ℂ))
65	62, 64	cncfco 14555	. . . 4 ⊢ (𝐴 ∈ ℂ → ((𝑧 ∈ ℂ ↦ (𝐴 · 𝑧)) ∘ (𝑤 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↦ (1 / 𝑤))) ∈ ({𝑦 ∈ ℂ ∣ 𝑦 # 0}–cn→ℂ))
66	19, 65	eqeltrrd 2267	. . 3 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↦ (𝐴 · (1 / 𝑥))) ∈ ({𝑦 ∈ ℂ ∣ 𝑦 # 0}–cn→ℂ))
67	10, 66	eqeltrd 2266	. 2 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↦ (𝐴 / 𝑥)) ∈ ({𝑦 ∈ ℂ ∣ 𝑦 # 0}–cn→ℂ))
68	1, 67	eqeltrid 2276	1 ⊢ (𝐴 ∈ ℂ → 𝐹 ∈ ({𝑦 ∈ ℂ ∣ 𝑦 # 0}–cn→ℂ))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 980 = wceq 1364 ∈ wcel 2160 ∀wral 2468 ∃wrex 2469 {crab 2472 ⊆ wss 3144 {cpr 3608 class class class wbr 4018 ↦ cmpt 4079 ∘ ccom 4648 ⟶wf 5231 ‘cfv 5235 (class class class)co 5897 infcinf 7013 ℂcc 7840 ℝcr 7841 0cc0 7842 1c1 7843 · cmul 7847 < clt 8023 − cmin 8159 # cap 8569 / cdiv 8660 2c2 9001 ℝ⁺crp 9685 abscabs 11041 –cn→ccncf 14534

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-nul 4144 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-iinf 4605 ax-cnex 7933 ax-resscn 7934 ax-1cn 7935 ax-1re 7936 ax-icn 7937 ax-addcl 7938 ax-addrcl 7939 ax-mulcl 7940 ax-mulrcl 7941 ax-addcom 7942 ax-mulcom 7943 ax-addass 7944 ax-mulass 7945 ax-distr 7946 ax-i2m1 7947 ax-0lt1 7948 ax-1rid 7949 ax-0id 7950 ax-rnegex 7951 ax-precex 7952 ax-cnre 7953 ax-pre-ltirr 7954 ax-pre-ltwlin 7955 ax-pre-lttrn 7956 ax-pre-apti 7957 ax-pre-ltadd 7958 ax-pre-mulgt0 7959 ax-pre-mulext 7960 ax-arch 7961 ax-caucvg 7962

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-if 3550 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-tr 4117 df-id 4311 df-po 4314 df-iso 4315 df-iord 4384 df-on 4386 df-ilim 4387 df-suc 4389 df-iom 4608 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-f1 5240 df-fo 5241 df-f1o 5242 df-fv 5243 df-isom 5244 df-riota 5852 df-ov 5900 df-oprab 5901 df-mpo 5902 df-1st 6166 df-2nd 6167 df-recs 6331 df-frec 6417 df-map 6677 df-sup 7014 df-inf 7015 df-pnf 8025 df-mnf 8026 df-xr 8027 df-ltxr 8028 df-le 8029 df-sub 8161 df-neg 8162 df-reap 8563 df-ap 8570 df-div 8661 df-inn 8951 df-2 9009 df-3 9010 df-4 9011 df-n0 9208 df-z 9285 df-uz 9560 df-rp 9686 df-seqfrec 10479 df-exp 10554 df-cj 10886 df-re 10887 df-im 10888 df-rsqrt 11042 df-abs 11043 df-cncf 14535

This theorem is referenced by: dvrecap 14654

W3C validator