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Theorem mpomulcn 15650

Description: Complex number multiplication is a continuous function. (Contributed by GG, 16-Mar-2025.)

**Hypothesis**
Ref	Expression
mpomulcn.j	⊢ 𝐽 = (TopOpen‘ℂ_fld)

**Assertion**
Ref	Expression
mpomulcn	⊢ (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ∈ ((𝐽 ×_t 𝐽) Cn 𝐽)

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

Proof of Theorem mpomulcn Dummy variables 𝑎 𝑏 𝑐 𝑢 𝑣 𝑤 𝑧 𝑑 𝑒 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mpomulcn.j	. . 3 ⊢ 𝐽 = (TopOpen‘ℂ_fld)
2	1	cnfldtopn 15623	. 2 ⊢ 𝐽 = (MetOpen‘(abs ∘ − ))
3		mpomulf 8310	. 2 ⊢ (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)):(ℂ × ℂ)⟶ℂ
4		mulcn2 12061	. . 3 ⊢ ((𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℂ ∧ 𝑐 ∈ ℂ) → ∃𝑧 ∈ ℝ⁺ ∃𝑤 ∈ ℝ⁺ ∀𝑑 ∈ ℂ ∀𝑒 ∈ ℂ (((abs‘(𝑑 − 𝑏)) < 𝑧 ∧ (abs‘(𝑒 − 𝑐)) < 𝑤) → (abs‘((𝑑 · 𝑒) − (𝑏 · 𝑐))) < 𝑎))
5		simplr 533	. . . . . . . . . . . 12 ⊢ (((𝑣 ∈ ℂ ∧ 𝑢 ∈ ℂ) ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℂ ∧ 𝑐 ∈ ℂ)) → 𝑢 ∈ ℂ)
6		simplll 539	. . . . . . . . . . . . 13 ⊢ ((((𝑣 ∈ ℂ ∧ 𝑢 ∈ ℂ) ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℂ ∧ 𝑐 ∈ ℂ)) ∧ 𝑑 = 𝑢) → 𝑣 ∈ ℂ)
7		simplr 533	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑣 ∈ ℂ ∧ 𝑢 ∈ ℂ) ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℂ ∧ 𝑐 ∈ ℂ)) ∧ 𝑑 = 𝑢) ∧ 𝑒 = 𝑣) → 𝑑 = 𝑢)
8	7	fvoveq1d 6101	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑣 ∈ ℂ ∧ 𝑢 ∈ ℂ) ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℂ ∧ 𝑐 ∈ ℂ)) ∧ 𝑑 = 𝑢) ∧ 𝑒 = 𝑣) → (abs‘(𝑑 − 𝑏)) = (abs‘(𝑢 − 𝑏)))
9	8	breq1d 4138	. . . . . . . . . . . . . . 15 ⊢ (((((𝑣 ∈ ℂ ∧ 𝑢 ∈ ℂ) ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℂ ∧ 𝑐 ∈ ℂ)) ∧ 𝑑 = 𝑢) ∧ 𝑒 = 𝑣) → ((abs‘(𝑑 − 𝑏)) < 𝑧 ↔ (abs‘(𝑢 − 𝑏)) < 𝑧))
10		simpr 110	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑣 ∈ ℂ ∧ 𝑢 ∈ ℂ) ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℂ ∧ 𝑐 ∈ ℂ)) ∧ 𝑑 = 𝑢) ∧ 𝑒 = 𝑣) → 𝑒 = 𝑣)
11	10	fvoveq1d 6101	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑣 ∈ ℂ ∧ 𝑢 ∈ ℂ) ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℂ ∧ 𝑐 ∈ ℂ)) ∧ 𝑑 = 𝑢) ∧ 𝑒 = 𝑣) → (abs‘(𝑒 − 𝑐)) = (abs‘(𝑣 − 𝑐)))
12	11	breq1d 4138	. . . . . . . . . . . . . . 15 ⊢ (((((𝑣 ∈ ℂ ∧ 𝑢 ∈ ℂ) ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℂ ∧ 𝑐 ∈ ℂ)) ∧ 𝑑 = 𝑢) ∧ 𝑒 = 𝑣) → ((abs‘(𝑒 − 𝑐)) < 𝑤 ↔ (abs‘(𝑣 − 𝑐)) < 𝑤))
13	9, 12	anbi12d 477	. . . . . . . . . . . . . 14 ⊢ (((((𝑣 ∈ ℂ ∧ 𝑢 ∈ ℂ) ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℂ ∧ 𝑐 ∈ ℂ)) ∧ 𝑑 = 𝑢) ∧ 𝑒 = 𝑣) → (((abs‘(𝑑 − 𝑏)) < 𝑧 ∧ (abs‘(𝑒 − 𝑐)) < 𝑤) ↔ ((abs‘(𝑢 − 𝑏)) < 𝑧 ∧ (abs‘(𝑣 − 𝑐)) < 𝑤)))
14		simplr 533	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑣 ∈ ℂ ∧ 𝑢 ∈ ℂ) ∧ 𝑑 = 𝑢) ∧ 𝑒 = 𝑣) → 𝑑 = 𝑢)
15	14	eqcomd 2244	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑣 ∈ ℂ ∧ 𝑢 ∈ ℂ) ∧ 𝑑 = 𝑢) ∧ 𝑒 = 𝑣) → 𝑢 = 𝑑)
16		simpr 110	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑣 ∈ ℂ ∧ 𝑢 ∈ ℂ) ∧ 𝑑 = 𝑢) ∧ 𝑒 = 𝑣) → 𝑒 = 𝑣)
17	16	eqcomd 2244	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑣 ∈ ℂ ∧ 𝑢 ∈ ℂ) ∧ 𝑑 = 𝑢) ∧ 𝑒 = 𝑣) → 𝑣 = 𝑒)
18	15, 17	oveq12d 6097	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑣 ∈ ℂ ∧ 𝑢 ∈ ℂ) ∧ 𝑑 = 𝑢) ∧ 𝑒 = 𝑣) → (𝑢 · 𝑣) = (𝑑 · 𝑒))
19		simplr 533	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑣 ∈ ℂ ∧ 𝑢 ∈ ℂ) ∧ 𝑑 = 𝑢) → 𝑢 ∈ ℂ)
20		simplll 539	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑣 ∈ ℂ ∧ 𝑢 ∈ ℂ) ∧ 𝑑 = 𝑢) ∧ 𝑒 = 𝑣) → 𝑣 ∈ ℂ)
21		tru 1406	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ⊤
22		oveq1 6086	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑥 = 𝑢 → (𝑥 · 𝑦) = (𝑢 · 𝑦))
23		oveq2 6087	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑦 = 𝑣 → (𝑢 · 𝑦) = (𝑢 · 𝑣))
24	22, 23	cbvmpov 6162	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) = (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))
25	24	a1i 9	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (⊤ → (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) = (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣)))
26		eqidd 2239	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (⊤ → ⟨𝑢, 𝑣⟩ = ⟨𝑢, 𝑣⟩)
27		mulcl 8300	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ) → (𝑢 · 𝑣) ∈ ℂ)
28	27	3adant1 1046	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((⊤ ∧ 𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ) → (𝑢 · 𝑣) ∈ ℂ)
29	25, 26, 28	fvmpopr2d 6219	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((⊤ ∧ 𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ) → ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘⟨𝑢, 𝑣⟩) = (𝑢 · 𝑣))
30	29	eqcomd 2244	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((⊤ ∧ 𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ) → (𝑢 · 𝑣) = ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘⟨𝑢, 𝑣⟩))
31	21, 30	mp3an1 1365	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ) → (𝑢 · 𝑣) = ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘⟨𝑢, 𝑣⟩))
32		df-ov 6082	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑢(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑣) = ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘⟨𝑢, 𝑣⟩)
33	31, 32	eqtr4di 2289	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ) → (𝑢 · 𝑣) = (𝑢(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑣))
34	19, 20, 33	syl2an2r 603	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑣 ∈ ℂ ∧ 𝑢 ∈ ℂ) ∧ 𝑑 = 𝑢) ∧ 𝑒 = 𝑣) → (𝑢 · 𝑣) = (𝑢(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑣))
35	18, 34	eqtr3d 2273	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑣 ∈ ℂ ∧ 𝑢 ∈ ℂ) ∧ 𝑑 = 𝑢) ∧ 𝑒 = 𝑣) → (𝑑 · 𝑒) = (𝑢(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑣))
36	35	adantllr 485	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑣 ∈ ℂ ∧ 𝑢 ∈ ℂ) ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℂ ∧ 𝑐 ∈ ℂ)) ∧ 𝑑 = 𝑢) ∧ 𝑒 = 𝑣) → (𝑑 · 𝑒) = (𝑢(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑣))
37		df-ov 6082	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑐) = ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘⟨𝑏, 𝑐⟩)
38		oveq1 6086	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑏 → (𝑥 · 𝑦) = (𝑏 · 𝑦))
39		oveq2 6087	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 = 𝑐 → (𝑏 · 𝑦) = (𝑏 · 𝑐))
40	38, 39	cbvmpov 6162	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) = (𝑏 ∈ ℂ, 𝑐 ∈ ℂ ↦ (𝑏 · 𝑐))
41	40	a1i 9	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 ∈ ℝ⁺ → (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) = (𝑏 ∈ ℂ, 𝑐 ∈ ℂ ↦ (𝑏 · 𝑐)))
42		eqidd 2239	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 ∈ ℝ⁺ → ⟨𝑏, 𝑐⟩ = ⟨𝑏, 𝑐⟩)
43		mulcl 8300	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑏 ∈ ℂ ∧ 𝑐 ∈ ℂ) → (𝑏 · 𝑐) ∈ ℂ)
44	43	3adant1 1046	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℂ ∧ 𝑐 ∈ ℂ) → (𝑏 · 𝑐) ∈ ℂ)
45	41, 42, 44	fvmpopr2d 6219	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℂ ∧ 𝑐 ∈ ℂ) → ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘⟨𝑏, 𝑐⟩) = (𝑏 · 𝑐))
46	37, 45	eqtr2id 2284	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℂ ∧ 𝑐 ∈ ℂ) → (𝑏 · 𝑐) = (𝑏(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑐))
47	46	ad3antlr 497	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑣 ∈ ℂ ∧ 𝑢 ∈ ℂ) ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℂ ∧ 𝑐 ∈ ℂ)) ∧ 𝑑 = 𝑢) ∧ 𝑒 = 𝑣) → (𝑏 · 𝑐) = (𝑏(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑐))
48	36, 47	oveq12d 6097	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑣 ∈ ℂ ∧ 𝑢 ∈ ℂ) ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℂ ∧ 𝑐 ∈ ℂ)) ∧ 𝑑 = 𝑢) ∧ 𝑒 = 𝑣) → ((𝑑 · 𝑒) − (𝑏 · 𝑐)) = ((𝑢(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑣) − (𝑏(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑐)))
49	48	fveq2d 5697	. . . . . . . . . . . . . . 15 ⊢ (((((𝑣 ∈ ℂ ∧ 𝑢 ∈ ℂ) ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℂ ∧ 𝑐 ∈ ℂ)) ∧ 𝑑 = 𝑢) ∧ 𝑒 = 𝑣) → (abs‘((𝑑 · 𝑒) − (𝑏 · 𝑐))) = (abs‘((𝑢(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑣) − (𝑏(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑐))))
50	49	breq1d 4138	. . . . . . . . . . . . . 14 ⊢ (((((𝑣 ∈ ℂ ∧ 𝑢 ∈ ℂ) ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℂ ∧ 𝑐 ∈ ℂ)) ∧ 𝑑 = 𝑢) ∧ 𝑒 = 𝑣) → ((abs‘((𝑑 · 𝑒) − (𝑏 · 𝑐))) < 𝑎 ↔ (abs‘((𝑢(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑣) − (𝑏(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑐))) < 𝑎))
51	13, 50	imbi12d 234	. . . . . . . . . . . . 13 ⊢ (((((𝑣 ∈ ℂ ∧ 𝑢 ∈ ℂ) ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℂ ∧ 𝑐 ∈ ℂ)) ∧ 𝑑 = 𝑢) ∧ 𝑒 = 𝑣) → ((((abs‘(𝑑 − 𝑏)) < 𝑧 ∧ (abs‘(𝑒 − 𝑐)) < 𝑤) → (abs‘((𝑑 · 𝑒) − (𝑏 · 𝑐))) < 𝑎) ↔ (((abs‘(𝑢 − 𝑏)) < 𝑧 ∧ (abs‘(𝑣 − 𝑐)) < 𝑤) → (abs‘((𝑢(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑣) − (𝑏(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑐))) < 𝑎)))
52	6, 51	rspcdv 2932	. . . . . . . . . . . 12 ⊢ ((((𝑣 ∈ ℂ ∧ 𝑢 ∈ ℂ) ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℂ ∧ 𝑐 ∈ ℂ)) ∧ 𝑑 = 𝑢) → (∀𝑒 ∈ ℂ (((abs‘(𝑑 − 𝑏)) < 𝑧 ∧ (abs‘(𝑒 − 𝑐)) < 𝑤) → (abs‘((𝑑 · 𝑒) − (𝑏 · 𝑐))) < 𝑎) → (((abs‘(𝑢 − 𝑏)) < 𝑧 ∧ (abs‘(𝑣 − 𝑐)) < 𝑤) → (abs‘((𝑢(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑣) − (𝑏(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑐))) < 𝑎)))
53	5, 52	rspcimdv 2930	. . . . . . . . . . 11 ⊢ (((𝑣 ∈ ℂ ∧ 𝑢 ∈ ℂ) ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℂ ∧ 𝑐 ∈ ℂ)) → (∀𝑑 ∈ ℂ ∀𝑒 ∈ ℂ (((abs‘(𝑑 − 𝑏)) < 𝑧 ∧ (abs‘(𝑒 − 𝑐)) < 𝑤) → (abs‘((𝑑 · 𝑒) − (𝑏 · 𝑐))) < 𝑎) → (((abs‘(𝑢 − 𝑏)) < 𝑧 ∧ (abs‘(𝑣 − 𝑐)) < 𝑤) → (abs‘((𝑢(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑣) − (𝑏(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑐))) < 𝑎)))
54	53	expimpd 363	. . . . . . . . . 10 ⊢ ((𝑣 ∈ ℂ ∧ 𝑢 ∈ ℂ) → (((𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℂ ∧ 𝑐 ∈ ℂ) ∧ ∀𝑑 ∈ ℂ ∀𝑒 ∈ ℂ (((abs‘(𝑑 − 𝑏)) < 𝑧 ∧ (abs‘(𝑒 − 𝑐)) < 𝑤) → (abs‘((𝑑 · 𝑒) − (𝑏 · 𝑐))) < 𝑎)) → (((abs‘(𝑢 − 𝑏)) < 𝑧 ∧ (abs‘(𝑣 − 𝑐)) < 𝑤) → (abs‘((𝑢(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑣) − (𝑏(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑐))) < 𝑎)))
55	54	ex 115	. . . . . . . . 9 ⊢ (𝑣 ∈ ℂ → (𝑢 ∈ ℂ → (((𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℂ ∧ 𝑐 ∈ ℂ) ∧ ∀𝑑 ∈ ℂ ∀𝑒 ∈ ℂ (((abs‘(𝑑 − 𝑏)) < 𝑧 ∧ (abs‘(𝑒 − 𝑐)) < 𝑤) → (abs‘((𝑑 · 𝑒) − (𝑏 · 𝑐))) < 𝑎)) → (((abs‘(𝑢 − 𝑏)) < 𝑧 ∧ (abs‘(𝑣 − 𝑐)) < 𝑤) → (abs‘((𝑢(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑣) − (𝑏(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑐))) < 𝑎))))
56	55	com13 80	. . . . . . . 8 ⊢ (((𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℂ ∧ 𝑐 ∈ ℂ) ∧ ∀𝑑 ∈ ℂ ∀𝑒 ∈ ℂ (((abs‘(𝑑 − 𝑏)) < 𝑧 ∧ (abs‘(𝑒 − 𝑐)) < 𝑤) → (abs‘((𝑑 · 𝑒) − (𝑏 · 𝑐))) < 𝑎)) → (𝑢 ∈ ℂ → (𝑣 ∈ ℂ → (((abs‘(𝑢 − 𝑏)) < 𝑧 ∧ (abs‘(𝑣 − 𝑐)) < 𝑤) → (abs‘((𝑢(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑣) − (𝑏(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑐))) < 𝑎))))
57	56	ralrimdv 2629	. . . . . . 7 ⊢ (((𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℂ ∧ 𝑐 ∈ ℂ) ∧ ∀𝑑 ∈ ℂ ∀𝑒 ∈ ℂ (((abs‘(𝑑 − 𝑏)) < 𝑧 ∧ (abs‘(𝑒 − 𝑐)) < 𝑤) → (abs‘((𝑑 · 𝑒) − (𝑏 · 𝑐))) < 𝑎)) → (𝑢 ∈ ℂ → ∀𝑣 ∈ ℂ (((abs‘(𝑢 − 𝑏)) < 𝑧 ∧ (abs‘(𝑣 − 𝑐)) < 𝑤) → (abs‘((𝑢(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑣) − (𝑏(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑐))) < 𝑎)))
58	57	ex 115	. . . . . 6 ⊢ ((𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℂ ∧ 𝑐 ∈ ℂ) → (∀𝑑 ∈ ℂ ∀𝑒 ∈ ℂ (((abs‘(𝑑 − 𝑏)) < 𝑧 ∧ (abs‘(𝑒 − 𝑐)) < 𝑤) → (abs‘((𝑑 · 𝑒) − (𝑏 · 𝑐))) < 𝑎) → (𝑢 ∈ ℂ → ∀𝑣 ∈ ℂ (((abs‘(𝑢 − 𝑏)) < 𝑧 ∧ (abs‘(𝑣 − 𝑐)) < 𝑤) → (abs‘((𝑢(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑣) − (𝑏(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑐))) < 𝑎))))
59	58	ralrimdv 2629	. . . . 5 ⊢ ((𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℂ ∧ 𝑐 ∈ ℂ) → (∀𝑑 ∈ ℂ ∀𝑒 ∈ ℂ (((abs‘(𝑑 − 𝑏)) < 𝑧 ∧ (abs‘(𝑒 − 𝑐)) < 𝑤) → (abs‘((𝑑 · 𝑒) − (𝑏 · 𝑐))) < 𝑎) → ∀𝑢 ∈ ℂ ∀𝑣 ∈ ℂ (((abs‘(𝑢 − 𝑏)) < 𝑧 ∧ (abs‘(𝑣 − 𝑐)) < 𝑤) → (abs‘((𝑢(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑣) − (𝑏(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑐))) < 𝑎)))
60	59	reximdv 2651	. . . 4 ⊢ ((𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℂ ∧ 𝑐 ∈ ℂ) → (∃𝑤 ∈ ℝ⁺ ∀𝑑 ∈ ℂ ∀𝑒 ∈ ℂ (((abs‘(𝑑 − 𝑏)) < 𝑧 ∧ (abs‘(𝑒 − 𝑐)) < 𝑤) → (abs‘((𝑑 · 𝑒) − (𝑏 · 𝑐))) < 𝑎) → ∃𝑤 ∈ ℝ⁺ ∀𝑢 ∈ ℂ ∀𝑣 ∈ ℂ (((abs‘(𝑢 − 𝑏)) < 𝑧 ∧ (abs‘(𝑣 − 𝑐)) < 𝑤) → (abs‘((𝑢(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑣) − (𝑏(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑐))) < 𝑎)))
61	60	reximdv 2651	. . 3 ⊢ ((𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℂ ∧ 𝑐 ∈ ℂ) → (∃𝑧 ∈ ℝ⁺ ∃𝑤 ∈ ℝ⁺ ∀𝑑 ∈ ℂ ∀𝑒 ∈ ℂ (((abs‘(𝑑 − 𝑏)) < 𝑧 ∧ (abs‘(𝑒 − 𝑐)) < 𝑤) → (abs‘((𝑑 · 𝑒) − (𝑏 · 𝑐))) < 𝑎) → ∃𝑧 ∈ ℝ⁺ ∃𝑤 ∈ ℝ⁺ ∀𝑢 ∈ ℂ ∀𝑣 ∈ ℂ (((abs‘(𝑢 − 𝑏)) < 𝑧 ∧ (abs‘(𝑣 − 𝑐)) < 𝑤) → (abs‘((𝑢(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑣) − (𝑏(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑐))) < 𝑎)))
62	4, 61	mpd 13	. 2 ⊢ ((𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℂ ∧ 𝑐 ∈ ℂ) → ∃𝑧 ∈ ℝ⁺ ∃𝑤 ∈ ℝ⁺ ∀𝑢 ∈ ℂ ∀𝑣 ∈ ℂ (((abs‘(𝑢 − 𝑏)) < 𝑧 ∧ (abs‘(𝑣 − 𝑐)) < 𝑤) → (abs‘((𝑢(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑣) − (𝑏(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑐))) < 𝑎))
63	2, 3, 62	addcncntoplem 15645	1 ⊢ (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ∈ ((𝐽 ×_t 𝐽) Cn 𝐽)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ∧ w3a 1009 = wceq 1402 ⊤wtru 1403 ∈ wcel 2209 ∀wral 2528 ∃wrex 2529 ⟨cop 3711 class class class wbr 4128 ‘cfv 5375 (class class class)co 6079 ∈ cmpo 6081 ℂcc 8171 · cmul 8178 < clt 8354 − cmin 8491 ℝ⁺crp 10037 abscabs 11746 TopOpenctopn 13577 ℂ_fldccnfld 14876 Cn ccn 15269 ×_t ctx 15336

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 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4244 ax-sep 4247 ax-nul 4257 ax-pow 4309 ax-pr 4344 ax-un 4576 ax-setind 4682 ax-iinf 4733 ax-cnex 8264 ax-resscn 8265 ax-1cn 8266 ax-1re 8267 ax-icn 8268 ax-addcl 8269 ax-addrcl 8270 ax-mulcl 8271 ax-mulrcl 8272 ax-addcom 8273 ax-mulcom 8274 ax-addass 8275 ax-mulass 8276 ax-distr 8277 ax-i2m1 8278 ax-0lt1 8279 ax-1rid 8280 ax-0id 8281 ax-rnegex 8282 ax-precex 8283 ax-cnre 8284 ax-pre-ltirr 8285 ax-pre-ltwlin 8286 ax-pre-lttrn 8287 ax-pre-apti 8288 ax-pre-ltadd 8289 ax-pre-mulgt0 8290 ax-pre-mulext 8291 ax-arch 8292 ax-caucvg 8293

This theorem depends on definitions: df-bi 117 df-stab 843 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rmo 2536 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3639 df-pw 3690 df-sn 3714 df-pr 3715 df-tp 3716 df-op 3717 df-uni 3934 df-int 3969 df-iun 4012 df-br 4129 df-opab 4191 df-mpt 4192 df-tr 4228 df-id 4436 df-po 4439 df-iso 4440 df-iord 4509 df-on 4511 df-ilim 4512 df-suc 4514 df-iom 4736 df-xp 4778 df-rel 4779 df-cnv 4780 df-co 4781 df-dm 4782 df-rn 4783 df-res 4784 df-ima 4785 df-iota 5335 df-fun 5377 df-fn 5378 df-f 5379 df-f1 5380 df-fo 5381 df-f1o 5382 df-fv 5383 df-isom 5384 df-riota 6032 df-ov 6082 df-oprab 6083 df-mpo 6084 df-1st 6368 df-2nd 6369 df-recs 6570 df-frec 6656 df-map 6918 df-sup 7318 df-inf 7319 df-pnf 8356 df-mnf 8357 df-xr 8358 df-ltxr 8359 df-le 8360 df-sub 8493 df-neg 8494 df-reap 8897 df-ap 8904 df-div 8997 df-inn 9288 df-2 9346 df-3 9347 df-4 9348 df-5 9349 df-6 9350 df-7 9351 df-8 9352 df-9 9353 df-n0 9547 df-z 9628 df-dec 9761 df-uz 9905 df-q 10003 df-rp 10038 df-xneg 10157 df-xadd 10158 df-fz 10395 df-seqfrec 10868 df-exp 10959 df-cj 11590 df-re 11591 df-im 11592 df-rsqrt 11747 df-abs 11748 df-struct 13337 df-ndx 13338 df-slot 13339 df-base 13341 df-plusg 13427 df-mulr 13428 df-starv 13429 df-tset 13433 df-ple 13434 df-ds 13436 df-unif 13437 df-rest 13578 df-topn 13579 df-topgen 13597 df-psmet 14863 df-xmet 14864 df-met 14865 df-bl 14866 df-mopn 14867 df-fg 14869 df-metu 14870 df-cnfld 14877 df-top 15082 df-topon 15095 df-bases 15127 df-cn 15272 df-cnp 15273 df-tx 15337

This theorem is referenced by: expcn 15653 plycn 15846

W3C validator