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Theorem cdivcncfap 15272

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 4085	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → (𝑦 # 0 ↔ 𝑥 # 0))
4	3	elrab 2959	. . . . . . . 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 8936	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0}) → (𝐴 / 𝑥) = (𝐴 · (1 / 𝑥)))
10	9	mpteq2dva 4173	. . 3 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↦ (𝐴 / 𝑥)) = (𝑥 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↦ (𝐴 · (1 / 𝑥))))
11		recclap 8822	. . . . . . 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 6008	. . . . . . 7 ⊢ (𝑤 = 𝑥 → (1 / 𝑤) = (1 / 𝑥))
15	14	cbvmptv 4179	. . . . . 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 2230	. . . . 5 ⊢ (𝐴 ∈ ℂ → (𝑧 ∈ ℂ ↦ (𝐴 · 𝑧)) = (𝑧 ∈ ℂ ↦ (𝐴 · 𝑧)))
18		oveq2 6008	. . . . 5 ⊢ (𝑧 = (1 / 𝑥) → (𝐴 · 𝑧) = (𝐴 · (1 / 𝑥)))
19	13, 16, 17, 18	fmptco 5800	. . . 4 ⊢ (𝐴 ∈ ℂ → ((𝑧 ∈ ℂ ↦ (𝐴 · 𝑧)) ∘ (𝑤 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↦ (1 / 𝑤))) = (𝑥 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↦ (𝐴 · (1 / 𝑥))))
20		breq1 4085	. . . . . . . . . 10 ⊢ (𝑦 = 𝑤 → (𝑦 # 0 ↔ 𝑤 # 0))
21	20	elrab 2959	. . . . . . . . 9 ⊢ (𝑤 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↔ (𝑤 ∈ ℂ ∧ 𝑤 # 0))
22		recclap 8822	. . . . . . . . 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 5789	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (𝑤 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↦ (1 / 𝑤)):{𝑦 ∈ ℂ ∣ 𝑦 # 0}⟶ℂ)
26		breq1 4085	. . . . . . . . 9 ⊢ (𝑦 = 𝑏 → (𝑦 # 0 ↔ 𝑏 # 0))
27	26	elrab 2959	. . . . . . . 8 ⊢ (𝑏 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↔ (𝑏 ∈ ℂ ∧ 𝑏 # 0))
28		eqid 2229	. . . . . . . . . . . 12 ⊢ (inf({1, ((abs‘𝑏) · 𝑒)}, ℝ, < ) · ((abs‘𝑏) / 2)) = (inf({1, ((abs‘𝑏) · 𝑒)}, ℝ, < ) · ((abs‘𝑏) / 2))
29	28	reccn2ap 11819	. . . . . . . . . . 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 2230	. . . . . . . . . . . . . . . . . 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 6008	. . . . . . . . . . . . . . . . . . 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 4085	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = 𝑎 → (𝑦 # 0 ↔ 𝑎 # 0))
35	34	elrab 2959	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↔ (𝑎 ∈ ℂ ∧ 𝑎 # 0))
36		recclap 8822	. . . . . . . . . . . . . . . . . . . 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 5714	. . . . . . . . . . . . . . . . 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 6008	. . . . . . . . . . . . . . . . . . 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 1060	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑏 ∈ ℂ ∧ 𝑏 # 0 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑎 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0}) → 𝑏 ∈ ℂ)
43		simpll2 1061	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑏 ∈ ℂ ∧ 𝑏 # 0 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑎 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0}) → 𝑏 # 0)
44	26, 42, 43	elrabd 2961	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑏 ∈ ℂ ∧ 𝑏 # 0 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑎 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0}) → 𝑏 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0})
45	42, 43	recclapd 8924	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑏 ∈ ℂ ∧ 𝑏 # 0 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑎 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0}) → (1 / 𝑏) ∈ ℂ)
46	30, 41, 44, 45	fvmptd 5714	. . . . . . . . . . . . . . . . 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 6018	. . . . . . . . . . . . . . . 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 5630	. . . . . . . . . . . . . . 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 4092	. . . . . . . . . . . . . 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 2526	. . . . . . . . . . . 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 2527	. . . . . . . . . . 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 1227	. . . . . . . . 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 2603	. . . . . . . 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 2583	. . . . . 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 3309	. . . . . . 7 ⊢ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ⊆ ℂ
59		ssid 3244	. . . . . . 7 ⊢ ℂ ⊆ ℂ
60		elcncf2 15242	. . . . . . 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 2229	. . . . . 6 ⊢ (𝑧 ∈ ℂ ↦ (𝐴 · 𝑧)) = (𝑧 ∈ ℂ ↦ (𝐴 · 𝑧))
64	63	mulc1cncf 15257	. . . . 5 ⊢ (𝐴 ∈ ℂ → (𝑧 ∈ ℂ ↦ (𝐴 · 𝑧)) ∈ (ℂ–cn→ℂ))
65	62, 64	cncfco 15259	. . . 4 ⊢ (𝐴 ∈ ℂ → ((𝑧 ∈ ℂ ↦ (𝐴 · 𝑧)) ∘ (𝑤 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↦ (1 / 𝑤))) ∈ ({𝑦 ∈ ℂ ∣ 𝑦 # 0}–cn→ℂ))
66	19, 65	eqeltrrd 2307	. . 3 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↦ (𝐴 · (1 / 𝑥))) ∈ ({𝑦 ∈ ℂ ∣ 𝑦 # 0}–cn→ℂ))
67	10, 66	eqeltrd 2306	. 2 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↦ (𝐴 / 𝑥)) ∈ ({𝑦 ∈ ℂ ∣ 𝑦 # 0}–cn→ℂ))
68	1, 67	eqeltrid 2316	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 1002 = wceq 1395 ∈ wcel 2200 ∀wral 2508 ∃wrex 2509 {crab 2512 ⊆ wss 3197 {cpr 3667 class class class wbr 4082 ↦ cmpt 4144 ∘ ccom 4722 ⟶wf 5313 ‘cfv 5317 (class class class)co 6000 infcinf 7146 ℂcc 7993 ℝcr 7994 0cc0 7995 1c1 7996 · cmul 8000 < clt 8177 − cmin 8313 # cap 8724 / cdiv 8815 2c2 9157 ℝ⁺crp 9845 abscabs 11503 –cn→ccncf 15238

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4198 ax-sep 4201 ax-nul 4209 ax-pow 4257 ax-pr 4292 ax-un 4523 ax-setind 4628 ax-iinf 4679 ax-cnex 8086 ax-resscn 8087 ax-1cn 8088 ax-1re 8089 ax-icn 8090 ax-addcl 8091 ax-addrcl 8092 ax-mulcl 8093 ax-mulrcl 8094 ax-addcom 8095 ax-mulcom 8096 ax-addass 8097 ax-mulass 8098 ax-distr 8099 ax-i2m1 8100 ax-0lt1 8101 ax-1rid 8102 ax-0id 8103 ax-rnegex 8104 ax-precex 8105 ax-cnre 8106 ax-pre-ltirr 8107 ax-pre-ltwlin 8108 ax-pre-lttrn 8109 ax-pre-apti 8110 ax-pre-ltadd 8111 ax-pre-mulgt0 8112 ax-pre-mulext 8113 ax-arch 8114 ax-caucvg 8115

This theorem depends on definitions: df-bi 117 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rmo 2516 df-rab 2517 df-v 2801 df-sbc 3029 df-csb 3125 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-nul 3492 df-if 3603 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3888 df-int 3923 df-iun 3966 df-br 4083 df-opab 4145 df-mpt 4146 df-tr 4182 df-id 4383 df-po 4386 df-iso 4387 df-iord 4456 df-on 4458 df-ilim 4459 df-suc 4461 df-iom 4682 df-xp 4724 df-rel 4725 df-cnv 4726 df-co 4727 df-dm 4728 df-rn 4729 df-res 4730 df-ima 4731 df-iota 5277 df-fun 5319 df-fn 5320 df-f 5321 df-f1 5322 df-fo 5323 df-f1o 5324 df-fv 5325 df-isom 5326 df-riota 5953 df-ov 6003 df-oprab 6004 df-mpo 6005 df-1st 6284 df-2nd 6285 df-recs 6449 df-frec 6535 df-map 6795 df-sup 7147 df-inf 7148 df-pnf 8179 df-mnf 8180 df-xr 8181 df-ltxr 8182 df-le 8183 df-sub 8315 df-neg 8316 df-reap 8718 df-ap 8725 df-div 8816 df-inn 9107 df-2 9165 df-3 9166 df-4 9167 df-n0 9366 df-z 9443 df-uz 9719 df-rp 9846 df-seqfrec 10665 df-exp 10756 df-cj 11348 df-re 11349 df-im 11350 df-rsqrt 11504 df-abs 11505 df-cncf 15239

This theorem is referenced by: divcncfap 15282 dvrecap 15381

W3C validator