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Theorem cdivcncfap 12756

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 3932	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → (𝑦 # 0 ↔ 𝑥 # 0))
4	3	elrab 2840	. . . . . . . 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 8553	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0}) → (𝐴 / 𝑥) = (𝐴 · (1 / 𝑥)))
10	9	mpteq2dva 4018	. . 3 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↦ (𝐴 / 𝑥)) = (𝑥 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↦ (𝐴 · (1 / 𝑥))))
11		recclap 8439	. . . . . . 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 5782	. . . . . . 7 ⊢ (𝑤 = 𝑥 → (1 / 𝑤) = (1 / 𝑥))
15	14	cbvmptv 4024	. . . . . 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 2140	. . . . 5 ⊢ (𝐴 ∈ ℂ → (𝑧 ∈ ℂ ↦ (𝐴 · 𝑧)) = (𝑧 ∈ ℂ ↦ (𝐴 · 𝑧)))
18		oveq2 5782	. . . . 5 ⊢ (𝑧 = (1 / 𝑥) → (𝐴 · 𝑧) = (𝐴 · (1 / 𝑥)))
19	13, 16, 17, 18	fmptco 5586	. . . 4 ⊢ (𝐴 ∈ ℂ → ((𝑧 ∈ ℂ ↦ (𝐴 · 𝑧)) ∘ (𝑤 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↦ (1 / 𝑤))) = (𝑥 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↦ (𝐴 · (1 / 𝑥))))
20		breq1 3932	. . . . . . . . . 10 ⊢ (𝑦 = 𝑤 → (𝑦 # 0 ↔ 𝑤 # 0))
21	20	elrab 2840	. . . . . . . . 9 ⊢ (𝑤 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↔ (𝑤 ∈ ℂ ∧ 𝑤 # 0))
22		recclap 8439	. . . . . . . . 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 5575	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (𝑤 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↦ (1 / 𝑤)):{𝑦 ∈ ℂ ∣ 𝑦 # 0}⟶ℂ)
26		breq1 3932	. . . . . . . . 9 ⊢ (𝑦 = 𝑏 → (𝑦 # 0 ↔ 𝑏 # 0))
27	26	elrab 2840	. . . . . . . 8 ⊢ (𝑏 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↔ (𝑏 ∈ ℂ ∧ 𝑏 # 0))
28		eqid 2139	. . . . . . . . . . . 12 ⊢ (inf({1, ((abs‘𝑏) · 𝑒)}, ℝ, < ) · ((abs‘𝑏) / 2)) = (inf({1, ((abs‘𝑏) · 𝑒)}, ℝ, < ) · ((abs‘𝑏) / 2))
29	28	reccn2ap 11082	. . . . . . . . . . 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 2140	. . . . . . . . . . . . . . . . . 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 5782	. . . . . . . . . . . . . . . . . . 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 3932	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = 𝑎 → (𝑦 # 0 ↔ 𝑎 # 0))
35	34	elrab 2840	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↔ (𝑎 ∈ ℂ ∧ 𝑎 # 0))
36		recclap 8439	. . . . . . . . . . . . . . . . . . . 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 5502	. . . . . . . . . . . . . . . . 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 5782	. . . . . . . . . . . . . . . . . . 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 1020	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑏 ∈ ℂ ∧ 𝑏 # 0 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑎 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0}) → 𝑏 ∈ ℂ)
43		simpll2 1021	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑏 ∈ ℂ ∧ 𝑏 # 0 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑎 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0}) → 𝑏 # 0)
44	26, 42, 43	elrabd 2842	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑏 ∈ ℂ ∧ 𝑏 # 0 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑎 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0}) → 𝑏 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0})
45	42, 43	recclapd 8541	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑏 ∈ ℂ ∧ 𝑏 # 0 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑎 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0}) → (1 / 𝑏) ∈ ℂ)
46	30, 41, 44, 45	fvmptd 5502	. . . . . . . . . . . . . . . . 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 5792	. . . . . . . . . . . . . . . 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 5425	. . . . . . . . . . . . . . 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 3939	. . . . . . . . . . . . . 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 2433	. . . . . . . . . . . 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 2434	. . . . . . . . . . 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 1181	. . . . . . . . 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 2505	. . . . . . . 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 2485	. . . . . 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 3182	. . . . . . 7 ⊢ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ⊆ ℂ
59		ssid 3117	. . . . . . 7 ⊢ ℂ ⊆ ℂ
60		elcncf2 12730	. . . . . . 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 422	. . . . . 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 412	. . . . 5 ⊢ (𝐴 ∈ ℂ → (𝑤 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↦ (1 / 𝑤)) ∈ ({𝑦 ∈ ℂ ∣ 𝑦 # 0}–cn→ℂ))
63		eqid 2139	. . . . . 6 ⊢ (𝑧 ∈ ℂ ↦ (𝐴 · 𝑧)) = (𝑧 ∈ ℂ ↦ (𝐴 · 𝑧))
64	63	mulc1cncf 12745	. . . . 5 ⊢ (𝐴 ∈ ℂ → (𝑧 ∈ ℂ ↦ (𝐴 · 𝑧)) ∈ (ℂ–cn→ℂ))
65	62, 64	cncfco 12747	. . . 4 ⊢ (𝐴 ∈ ℂ → ((𝑧 ∈ ℂ ↦ (𝐴 · 𝑧)) ∘ (𝑤 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↦ (1 / 𝑤))) ∈ ({𝑦 ∈ ℂ ∣ 𝑦 # 0}–cn→ℂ))
66	19, 65	eqeltrrd 2217	. . 3 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↦ (𝐴 · (1 / 𝑥))) ∈ ({𝑦 ∈ ℂ ∣ 𝑦 # 0}–cn→ℂ))
67	10, 66	eqeltrd 2216	. 2 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↦ (𝐴 / 𝑥)) ∈ ({𝑦 ∈ ℂ ∣ 𝑦 # 0}–cn→ℂ))
68	1, 67	eqeltrid 2226	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 962 = wceq 1331 ∈ wcel 1480 ∀wral 2416 ∃wrex 2417 {crab 2420 ⊆ wss 3071 {cpr 3528 class class class wbr 3929 ↦ cmpt 3989 ∘ ccom 4543 ⟶wf 5119 ‘cfv 5123 (class class class)co 5774 infcinf 6870 ℂcc 7618 ℝcr 7619 0cc0 7620 1c1 7621 · cmul 7625 < clt 7800 − cmin 7933 # cap 8343 / cdiv 8432 2c2 8771 ℝ⁺crp 9441 abscabs 10769 –cn→ccncf 12726

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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-precex 7730 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 ax-pre-mulgt0 7737 ax-pre-mulext 7738 ax-arch 7739 ax-caucvg 7740

This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-po 4218 df-iso 4219 df-iord 4288 df-on 4290 df-ilim 4291 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-isom 5132 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-frec 6288 df-map 6544 df-sup 6871 df-inf 6872 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-reap 8337 df-ap 8344 df-div 8433 df-inn 8721 df-2 8779 df-3 8780 df-4 8781 df-n0 8978 df-z 9055 df-uz 9327 df-rp 9442 df-seqfrec 10219 df-exp 10293 df-cj 10614 df-re 10615 df-im 10616 df-rsqrt 10770 df-abs 10771 df-cncf 12727

This theorem is referenced by: dvrecap 12846

W3C validator