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Theorem cdivcncfap 13381

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 108	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0}) → 𝐴 ∈ ℂ)
3		breq1 3992	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → (𝑦 # 0 ↔ 𝑥 # 0))
4	3	elrab 2886	. . . . . . . 8 ⊢ (𝑥 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↔ (𝑥 ∈ ℂ ∧ 𝑥 # 0))
5	4	biimpi 119	. . . . . . 7 ⊢ (𝑥 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} → (𝑥 ∈ ℂ ∧ 𝑥 # 0))
6	5	adantl 275	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0}) → (𝑥 ∈ ℂ ∧ 𝑥 # 0))
7	6	simpld 111	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0}) → 𝑥 ∈ ℂ)
8	6	simprd 113	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0}) → 𝑥 # 0)
9	2, 7, 8	divrecapd 8710	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0}) → (𝐴 / 𝑥) = (𝐴 · (1 / 𝑥)))
10	9	mpteq2dva 4079	. . 3 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↦ (𝐴 / 𝑥)) = (𝑥 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↦ (𝐴 · (1 / 𝑥))))
11		recclap 8596	. . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑥 # 0) → (1 / 𝑥) ∈ ℂ)
12	4, 11	sylbi 120	. . . . . 6 ⊢ (𝑥 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} → (1 / 𝑥) ∈ ℂ)
13	12	adantl 275	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0}) → (1 / 𝑥) ∈ ℂ)
14		oveq2 5861	. . . . . . 7 ⊢ (𝑤 = 𝑥 → (1 / 𝑤) = (1 / 𝑥))
15	14	cbvmptv 4085	. . . . . 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 2171	. . . . 5 ⊢ (𝐴 ∈ ℂ → (𝑧 ∈ ℂ ↦ (𝐴 · 𝑧)) = (𝑧 ∈ ℂ ↦ (𝐴 · 𝑧)))
18		oveq2 5861	. . . . 5 ⊢ (𝑧 = (1 / 𝑥) → (𝐴 · 𝑧) = (𝐴 · (1 / 𝑥)))
19	13, 16, 17, 18	fmptco 5662	. . . 4 ⊢ (𝐴 ∈ ℂ → ((𝑧 ∈ ℂ ↦ (𝐴 · 𝑧)) ∘ (𝑤 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↦ (1 / 𝑤))) = (𝑥 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↦ (𝐴 · (1 / 𝑥))))
20		breq1 3992	. . . . . . . . . 10 ⊢ (𝑦 = 𝑤 → (𝑦 # 0 ↔ 𝑤 # 0))
21	20	elrab 2886	. . . . . . . . 9 ⊢ (𝑤 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↔ (𝑤 ∈ ℂ ∧ 𝑤 # 0))
22		recclap 8596	. . . . . . . . 9 ⊢ ((𝑤 ∈ ℂ ∧ 𝑤 # 0) → (1 / 𝑤) ∈ ℂ)
23	21, 22	sylbi 120	. . . . . . . 8 ⊢ (𝑤 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} → (1 / 𝑤) ∈ ℂ)
24	23	adantl 275	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑤 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0}) → (1 / 𝑤) ∈ ℂ)
25	24	fmpttd 5651	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (𝑤 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↦ (1 / 𝑤)):{𝑦 ∈ ℂ ∣ 𝑦 # 0}⟶ℂ)
26		breq1 3992	. . . . . . . . 9 ⊢ (𝑦 = 𝑏 → (𝑦 # 0 ↔ 𝑏 # 0))
27	26	elrab 2886	. . . . . . . 8 ⊢ (𝑏 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↔ (𝑏 ∈ ℂ ∧ 𝑏 # 0))
28		eqid 2170	. . . . . . . . . . . 12 ⊢ (inf({1, ((abs‘𝑏) · 𝑒)}, ℝ, < ) · ((abs‘𝑏) / 2)) = (inf({1, ((abs‘𝑏) · 𝑒)}, ℝ, < ) · ((abs‘𝑏) / 2))
29	28	reccn2ap 11276	. . . . . . . . . . 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 2171	. . . . . . . . . . . . . . . . . 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 5861	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 = 𝑎 → (1 / 𝑤) = (1 / 𝑎))
32	31	adantl 275	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑏 ∈ ℂ ∧ 𝑏 # 0 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑎 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0}) ∧ 𝑤 = 𝑎) → (1 / 𝑤) = (1 / 𝑎))
33		simpr 109	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑏 ∈ ℂ ∧ 𝑏 # 0 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑎 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0}) → 𝑎 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0})
34		breq1 3992	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = 𝑎 → (𝑦 # 0 ↔ 𝑎 # 0))
35	34	elrab 2886	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↔ (𝑎 ∈ ℂ ∧ 𝑎 # 0))
36		recclap 8596	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑎 ∈ ℂ ∧ 𝑎 # 0) → (1 / 𝑎) ∈ ℂ)
37	35, 36	sylbi 120	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} → (1 / 𝑎) ∈ ℂ)
38	37	adantl 275	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑏 ∈ ℂ ∧ 𝑏 # 0 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑎 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0}) → (1 / 𝑎) ∈ ℂ)
39	30, 32, 33, 38	fvmptd 5577	. . . . . . . . . . . . . . . . 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 5861	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 = 𝑏 → (1 / 𝑤) = (1 / 𝑏))
41	40	adantl 275	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑏 ∈ ℂ ∧ 𝑏 # 0 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑎 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0}) ∧ 𝑤 = 𝑏) → (1 / 𝑤) = (1 / 𝑏))
42		simpll1 1031	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑏 ∈ ℂ ∧ 𝑏 # 0 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑎 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0}) → 𝑏 ∈ ℂ)
43		simpll2 1032	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑏 ∈ ℂ ∧ 𝑏 # 0 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑎 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0}) → 𝑏 # 0)
44	26, 42, 43	elrabd 2888	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑏 ∈ ℂ ∧ 𝑏 # 0 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑎 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0}) → 𝑏 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0})
45	42, 43	recclapd 8698	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑏 ∈ ℂ ∧ 𝑏 # 0 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑎 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0}) → (1 / 𝑏) ∈ ℂ)
46	30, 41, 44, 45	fvmptd 5577	. . . . . . . . . . . . . . . . 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 5871	. . . . . . . . . . . . . . . 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 5500	. . . . . . . . . . . . . . 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 3999	. . . . . . . . . . . . . 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 229	. . . . . . . . . . . . 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 2466	. . . . . . . . . . . 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 2467	. . . . . . . . . . 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 166	. . . . . . . . . 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 1198	. . . . . . . . 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 2543	. . . . . . . 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 120	. . . . . . 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 2523	. . . . . 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 3232	. . . . . . 7 ⊢ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ⊆ ℂ
59		ssid 3167	. . . . . . 7 ⊢ ℂ ⊆ ℂ
60		elcncf2 13355	. . . . . . 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 424	. . . . . 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 414	. . . . 5 ⊢ (𝐴 ∈ ℂ → (𝑤 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↦ (1 / 𝑤)) ∈ ({𝑦 ∈ ℂ ∣ 𝑦 # 0}–cn→ℂ))
63		eqid 2170	. . . . . 6 ⊢ (𝑧 ∈ ℂ ↦ (𝐴 · 𝑧)) = (𝑧 ∈ ℂ ↦ (𝐴 · 𝑧))
64	63	mulc1cncf 13370	. . . . 5 ⊢ (𝐴 ∈ ℂ → (𝑧 ∈ ℂ ↦ (𝐴 · 𝑧)) ∈ (ℂ–cn→ℂ))
65	62, 64	cncfco 13372	. . . 4 ⊢ (𝐴 ∈ ℂ → ((𝑧 ∈ ℂ ↦ (𝐴 · 𝑧)) ∘ (𝑤 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↦ (1 / 𝑤))) ∈ ({𝑦 ∈ ℂ ∣ 𝑦 # 0}–cn→ℂ))
66	19, 65	eqeltrrd 2248	. . 3 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↦ (𝐴 · (1 / 𝑥))) ∈ ({𝑦 ∈ ℂ ∣ 𝑦 # 0}–cn→ℂ))
67	10, 66	eqeltrd 2247	. 2 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↦ (𝐴 / 𝑥)) ∈ ({𝑦 ∈ ℂ ∣ 𝑦 # 0}–cn→ℂ))
68	1, 67	eqeltrid 2257	1 ⊢ (𝐴 ∈ ℂ → 𝐹 ∈ ({𝑦 ∈ ℂ ∣ 𝑦 # 0}–cn→ℂ))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∧ w3a 973 = wceq 1348 ∈ wcel 2141 ∀wral 2448 ∃wrex 2449 {crab 2452 ⊆ wss 3121 {cpr 3584 class class class wbr 3989 ↦ cmpt 4050 ∘ ccom 4615 ⟶wf 5194 ‘cfv 5198 (class class class)co 5853 infcinf 6960 ℂcc 7772 ℝcr 7773 0cc0 7774 1c1 7775 · cmul 7779 < clt 7954 − cmin 8090 # cap 8500 / cdiv 8589 2c2 8929 ℝ⁺crp 9610 abscabs 10961 –cn→ccncf 13351

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-mulrcl 7873 ax-addcom 7874 ax-mulcom 7875 ax-addass 7876 ax-mulass 7877 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-1rid 7881 ax-0id 7882 ax-rnegex 7883 ax-precex 7884 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-apti 7889 ax-pre-ltadd 7890 ax-pre-mulgt0 7891 ax-pre-mulext 7892 ax-arch 7893 ax-caucvg 7894

This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3527 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-id 4278 df-po 4281 df-iso 4282 df-iord 4351 df-on 4353 df-ilim 4354 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-isom 5207 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-recs 6284 df-frec 6370 df-map 6628 df-sup 6961 df-inf 6962 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-reap 8494 df-ap 8501 df-div 8590 df-inn 8879 df-2 8937 df-3 8938 df-4 8939 df-n0 9136 df-z 9213 df-uz 9488 df-rp 9611 df-seqfrec 10402 df-exp 10476 df-cj 10806 df-re 10807 df-im 10808 df-rsqrt 10962 df-abs 10963 df-cncf 13352

This theorem is referenced by: dvrecap 13471

W3C validator