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Theorem cdivcncfap 14758

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 4032	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → (𝑦 # 0 ↔ 𝑥 # 0))
4	3	elrab 2916	. . . . . . . 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 8812	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0}) → (𝐴 / 𝑥) = (𝐴 · (1 / 𝑥)))
10	9	mpteq2dva 4119	. . 3 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↦ (𝐴 / 𝑥)) = (𝑥 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↦ (𝐴 · (1 / 𝑥))))
11		recclap 8698	. . . . . . 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 5926	. . . . . . 7 ⊢ (𝑤 = 𝑥 → (1 / 𝑤) = (1 / 𝑥))
15	14	cbvmptv 4125	. . . . . 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 2194	. . . . 5 ⊢ (𝐴 ∈ ℂ → (𝑧 ∈ ℂ ↦ (𝐴 · 𝑧)) = (𝑧 ∈ ℂ ↦ (𝐴 · 𝑧)))
18		oveq2 5926	. . . . 5 ⊢ (𝑧 = (1 / 𝑥) → (𝐴 · 𝑧) = (𝐴 · (1 / 𝑥)))
19	13, 16, 17, 18	fmptco 5724	. . . 4 ⊢ (𝐴 ∈ ℂ → ((𝑧 ∈ ℂ ↦ (𝐴 · 𝑧)) ∘ (𝑤 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↦ (1 / 𝑤))) = (𝑥 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↦ (𝐴 · (1 / 𝑥))))
20		breq1 4032	. . . . . . . . . 10 ⊢ (𝑦 = 𝑤 → (𝑦 # 0 ↔ 𝑤 # 0))
21	20	elrab 2916	. . . . . . . . 9 ⊢ (𝑤 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↔ (𝑤 ∈ ℂ ∧ 𝑤 # 0))
22		recclap 8698	. . . . . . . . 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 5713	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (𝑤 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↦ (1 / 𝑤)):{𝑦 ∈ ℂ ∣ 𝑦 # 0}⟶ℂ)
26		breq1 4032	. . . . . . . . 9 ⊢ (𝑦 = 𝑏 → (𝑦 # 0 ↔ 𝑏 # 0))
27	26	elrab 2916	. . . . . . . 8 ⊢ (𝑏 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↔ (𝑏 ∈ ℂ ∧ 𝑏 # 0))
28		eqid 2193	. . . . . . . . . . . 12 ⊢ (inf({1, ((abs‘𝑏) · 𝑒)}, ℝ, < ) · ((abs‘𝑏) / 2)) = (inf({1, ((abs‘𝑏) · 𝑒)}, ℝ, < ) · ((abs‘𝑏) / 2))
29	28	reccn2ap 11456	. . . . . . . . . . 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 2194	. . . . . . . . . . . . . . . . . 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 5926	. . . . . . . . . . . . . . . . . . 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 4032	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = 𝑎 → (𝑦 # 0 ↔ 𝑎 # 0))
35	34	elrab 2916	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↔ (𝑎 ∈ ℂ ∧ 𝑎 # 0))
36		recclap 8698	. . . . . . . . . . . . . . . . . . . 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 5638	. . . . . . . . . . . . . . . . 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 5926	. . . . . . . . . . . . . . . . . . 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 2918	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑏 ∈ ℂ ∧ 𝑏 # 0 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑎 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0}) → 𝑏 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0})
45	42, 43	recclapd 8800	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑏 ∈ ℂ ∧ 𝑏 # 0 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑎 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0}) → (1 / 𝑏) ∈ ℂ)
46	30, 41, 44, 45	fvmptd 5638	. . . . . . . . . . . . . . . . 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 5936	. . . . . . . . . . . . . . . 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 5558	. . . . . . . . . . . . . . 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 4039	. . . . . . . . . . . . . 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 2490	. . . . . . . . . . . 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 2491	. . . . . . . . . . 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 2567	. . . . . . . 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 2547	. . . . . 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 3264	. . . . . . 7 ⊢ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ⊆ ℂ
59		ssid 3199	. . . . . . 7 ⊢ ℂ ⊆ ℂ
60		elcncf2 14729	. . . . . . 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 2193	. . . . . 6 ⊢ (𝑧 ∈ ℂ ↦ (𝐴 · 𝑧)) = (𝑧 ∈ ℂ ↦ (𝐴 · 𝑧))
64	63	mulc1cncf 14744	. . . . 5 ⊢ (𝐴 ∈ ℂ → (𝑧 ∈ ℂ ↦ (𝐴 · 𝑧)) ∈ (ℂ–cn→ℂ))
65	62, 64	cncfco 14746	. . . 4 ⊢ (𝐴 ∈ ℂ → ((𝑧 ∈ ℂ ↦ (𝐴 · 𝑧)) ∘ (𝑤 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↦ (1 / 𝑤))) ∈ ({𝑦 ∈ ℂ ∣ 𝑦 # 0}–cn→ℂ))
66	19, 65	eqeltrrd 2271	. . 3 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↦ (𝐴 · (1 / 𝑥))) ∈ ({𝑦 ∈ ℂ ∣ 𝑦 # 0}–cn→ℂ))
67	10, 66	eqeltrd 2270	. 2 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↦ (𝐴 / 𝑥)) ∈ ({𝑦 ∈ ℂ ∣ 𝑦 # 0}–cn→ℂ))
68	1, 67	eqeltrid 2280	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 2164 ∀wral 2472 ∃wrex 2473 {crab 2476 ⊆ wss 3153 {cpr 3619 class class class wbr 4029 ↦ cmpt 4090 ∘ ccom 4663 ⟶wf 5250 ‘cfv 5254 (class class class)co 5918 infcinf 7042 ℂcc 7870 ℝcr 7871 0cc0 7872 1c1 7873 · cmul 7877 < clt 8054 − cmin 8190 # cap 8600 / cdiv 8691 2c2 9033 ℝ⁺crp 9719 abscabs 11141 –cn→ccncf 14725

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 2166 ax-14 2167 ax-ext 2175 ax-coll 4144 ax-sep 4147 ax-nul 4155 ax-pow 4203 ax-pr 4238 ax-un 4464 ax-setind 4569 ax-iinf 4620 ax-cnex 7963 ax-resscn 7964 ax-1cn 7965 ax-1re 7966 ax-icn 7967 ax-addcl 7968 ax-addrcl 7969 ax-mulcl 7970 ax-mulrcl 7971 ax-addcom 7972 ax-mulcom 7973 ax-addass 7974 ax-mulass 7975 ax-distr 7976 ax-i2m1 7977 ax-0lt1 7978 ax-1rid 7979 ax-0id 7980 ax-rnegex 7981 ax-precex 7982 ax-cnre 7983 ax-pre-ltirr 7984 ax-pre-ltwlin 7985 ax-pre-lttrn 7986 ax-pre-apti 7987 ax-pre-ltadd 7988 ax-pre-mulgt0 7989 ax-pre-mulext 7990 ax-arch 7991 ax-caucvg 7992

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 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rmo 2480 df-rab 2481 df-v 2762 df-sbc 2986 df-csb 3081 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-nul 3447 df-if 3558 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-int 3871 df-iun 3914 df-br 4030 df-opab 4091 df-mpt 4092 df-tr 4128 df-id 4324 df-po 4327 df-iso 4328 df-iord 4397 df-on 4399 df-ilim 4400 df-suc 4402 df-iom 4623 df-xp 4665 df-rel 4666 df-cnv 4667 df-co 4668 df-dm 4669 df-rn 4670 df-res 4671 df-ima 4672 df-iota 5215 df-fun 5256 df-fn 5257 df-f 5258 df-f1 5259 df-fo 5260 df-f1o 5261 df-fv 5262 df-isom 5263 df-riota 5873 df-ov 5921 df-oprab 5922 df-mpo 5923 df-1st 6193 df-2nd 6194 df-recs 6358 df-frec 6444 df-map 6704 df-sup 7043 df-inf 7044 df-pnf 8056 df-mnf 8057 df-xr 8058 df-ltxr 8059 df-le 8060 df-sub 8192 df-neg 8193 df-reap 8594 df-ap 8601 df-div 8692 df-inn 8983 df-2 9041 df-3 9042 df-4 9043 df-n0 9241 df-z 9318 df-uz 9593 df-rp 9720 df-seqfrec 10519 df-exp 10610 df-cj 10986 df-re 10987 df-im 10988 df-rsqrt 11142 df-abs 11143 df-cncf 14726

This theorem is referenced by: divcncfap 14768 dvrecap 14862

W3C validator