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Theorem divcn 22671

Description: Complex number division is a continuous function, when the second argument is nonzero. (Contributed by Mario Carneiro, 12-Aug-2014.)

**Hypotheses**
Ref	Expression
addcn.j	⊢ 𝐽 = (TopOpen‘ℂ_fld)
divcn.k	⊢ 𝐾 = (𝐽 ↾_t (ℂ ∖ {0}))

**Assertion**
Ref	Expression
divcn	⊢ / ∈ ((𝐽 ×_t 𝐾) Cn 𝐽)

Proof of Theorem divcn Dummy variables 𝑢 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-div 10685	. . 3 ⊢ / = (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥))
2		eldifsn 4317	. . . . 5 ⊢ (𝑦 ∈ (ℂ ∖ {0}) ↔ (𝑦 ∈ ℂ ∧ 𝑦 ≠ 0))
3		divval 10687	. . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑦 ≠ 0) → (𝑥 / 𝑦) = (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥))
4		divrec 10701	. . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑦 ≠ 0) → (𝑥 / 𝑦) = (𝑥 · (1 / 𝑦)))
5	3, 4	eqtr3d 2658	. . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑦 ≠ 0) → (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥) = (𝑥 · (1 / 𝑦)))
6	5	3expb 1266	. . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ (𝑦 ∈ ℂ ∧ 𝑦 ≠ 0)) → (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥) = (𝑥 · (1 / 𝑦)))
7	2, 6	sylan2b 492	. . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0})) → (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥) = (𝑥 · (1 / 𝑦)))
8	7	mpt2eq3ia 6720	. . 3 ⊢ (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥)) = (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (𝑥 · (1 / 𝑦)))
9	1, 8	eqtri 2644	. 2 ⊢ / = (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (𝑥 · (1 / 𝑦)))
10		addcn.j	. . . . . 6 ⊢ 𝐽 = (TopOpen‘ℂ_fld)
11	10	cnfldtopon 22586	. . . . 5 ⊢ 𝐽 ∈ (TopOn‘ℂ)
12	11	a1i 11	. . . 4 ⊢ (⊤ → 𝐽 ∈ (TopOn‘ℂ))
13		divcn.k	. . . . 5 ⊢ 𝐾 = (𝐽 ↾_t (ℂ ∖ {0}))
14		difss 3737	. . . . . 6 ⊢ (ℂ ∖ {0}) ⊆ ℂ
15		resttopon 20965	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘ℂ) ∧ (ℂ ∖ {0}) ⊆ ℂ) → (𝐽 ↾_t (ℂ ∖ {0})) ∈ (TopOn‘(ℂ ∖ {0})))
16	12, 14, 15	sylancl 694	. . . . 5 ⊢ (⊤ → (𝐽 ↾_t (ℂ ∖ {0})) ∈ (TopOn‘(ℂ ∖ {0})))
17	13, 16	syl5eqel 2705	. . . 4 ⊢ (⊤ → 𝐾 ∈ (TopOn‘(ℂ ∖ {0})))
18	12, 17	cnmpt1st 21471	. . . 4 ⊢ (⊤ → (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ 𝑥) ∈ ((𝐽 ×_t 𝐾) Cn 𝐽))
19	12, 17	cnmpt2nd 21472	. . . . 5 ⊢ (⊤ → (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ 𝑦) ∈ ((𝐽 ×_t 𝐾) Cn 𝐾))
20		eqid 2622	. . . . . . . 8 ⊢ (𝑧 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑧)) = (𝑧 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑧))
21		eldifsn 4317	. . . . . . . . 9 ⊢ (𝑧 ∈ (ℂ ∖ {0}) ↔ (𝑧 ∈ ℂ ∧ 𝑧 ≠ 0))
22		reccl 10692	. . . . . . . . 9 ⊢ ((𝑧 ∈ ℂ ∧ 𝑧 ≠ 0) → (1 / 𝑧) ∈ ℂ)
23	21, 22	sylbi 207	. . . . . . . 8 ⊢ (𝑧 ∈ (ℂ ∖ {0}) → (1 / 𝑧) ∈ ℂ)
24	20, 23	fmpti 6383	. . . . . . 7 ⊢ (𝑧 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑧)):(ℂ ∖ {0})⟶ℂ
25		eqid 2622	. . . . . . . . . 10 ⊢ (if(1 ≤ ((abs‘𝑥) · 𝑦), 1, ((abs‘𝑥) · 𝑦)) · ((abs‘𝑥) / 2)) = (if(1 ≤ ((abs‘𝑥) · 𝑦), 1, ((abs‘𝑥) · 𝑦)) · ((abs‘𝑥) / 2))
26	25	reccn2 14327	. . . . . . . . 9 ⊢ ((𝑥 ∈ (ℂ ∖ {0}) ∧ 𝑦 ∈ ℝ⁺) → ∃𝑢 ∈ ℝ⁺ ∀𝑤 ∈ (ℂ ∖ {0})((abs‘(𝑤 − 𝑥)) < 𝑢 → (abs‘((1 / 𝑤) − (1 / 𝑥))) < 𝑦))
27		ovres 6800	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (ℂ ∖ {0}) ∧ 𝑤 ∈ (ℂ ∖ {0})) → (𝑥((abs ∘ − ) ↾ ((ℂ ∖ {0}) × (ℂ ∖ {0})))𝑤) = (𝑥(abs ∘ − )𝑤))
28		eldifi 3732	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (ℂ ∖ {0}) → 𝑥 ∈ ℂ)
29		eldifi 3732	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ (ℂ ∖ {0}) → 𝑤 ∈ ℂ)
30		eqid 2622	. . . . . . . . . . . . . . . . . 18 ⊢ (abs ∘ − ) = (abs ∘ − )
31	30	cnmetdval 22574	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℂ ∧ 𝑤 ∈ ℂ) → (𝑥(abs ∘ − )𝑤) = (abs‘(𝑥 − 𝑤)))
32		abssub 14066	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℂ ∧ 𝑤 ∈ ℂ) → (abs‘(𝑥 − 𝑤)) = (abs‘(𝑤 − 𝑥)))
33	31, 32	eqtrd 2656	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℂ ∧ 𝑤 ∈ ℂ) → (𝑥(abs ∘ − )𝑤) = (abs‘(𝑤 − 𝑥)))
34	28, 29, 33	syl2an 494	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (ℂ ∖ {0}) ∧ 𝑤 ∈ (ℂ ∖ {0})) → (𝑥(abs ∘ − )𝑤) = (abs‘(𝑤 − 𝑥)))
35	27, 34	eqtrd 2656	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (ℂ ∖ {0}) ∧ 𝑤 ∈ (ℂ ∖ {0})) → (𝑥((abs ∘ − ) ↾ ((ℂ ∖ {0}) × (ℂ ∖ {0})))𝑤) = (abs‘(𝑤 − 𝑥)))
36	35	breq1d 4663	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (ℂ ∖ {0}) ∧ 𝑤 ∈ (ℂ ∖ {0})) → ((𝑥((abs ∘ − ) ↾ ((ℂ ∖ {0}) × (ℂ ∖ {0})))𝑤) < 𝑢 ↔ (abs‘(𝑤 − 𝑥)) < 𝑢))
37		oveq2 6658	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = 𝑥 → (1 / 𝑧) = (1 / 𝑥))
38		ovex 6678	. . . . . . . . . . . . . . . . 17 ⊢ (1 / 𝑥) ∈ V
39	37, 20, 38	fvmpt 6282	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (ℂ ∖ {0}) → ((𝑧 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑧))‘𝑥) = (1 / 𝑥))
40		oveq2 6658	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = 𝑤 → (1 / 𝑧) = (1 / 𝑤))
41		ovex 6678	. . . . . . . . . . . . . . . . 17 ⊢ (1 / 𝑤) ∈ V
42	40, 20, 41	fvmpt 6282	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ (ℂ ∖ {0}) → ((𝑧 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑧))‘𝑤) = (1 / 𝑤))
43	39, 42	oveqan12d 6669	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (ℂ ∖ {0}) ∧ 𝑤 ∈ (ℂ ∖ {0})) → (((𝑧 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑧))‘𝑥)(abs ∘ − )((𝑧 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑧))‘𝑤)) = ((1 / 𝑥)(abs ∘ − )(1 / 𝑤)))
44		eldifsn 4317	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (ℂ ∖ {0}) ↔ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
45		reccl 10692	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) → (1 / 𝑥) ∈ ℂ)
46	44, 45	sylbi 207	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (ℂ ∖ {0}) → (1 / 𝑥) ∈ ℂ)
47		eldifsn 4317	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 ∈ (ℂ ∖ {0}) ↔ (𝑤 ∈ ℂ ∧ 𝑤 ≠ 0))
48		reccl 10692	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑤 ∈ ℂ ∧ 𝑤 ≠ 0) → (1 / 𝑤) ∈ ℂ)
49	47, 48	sylbi 207	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ (ℂ ∖ {0}) → (1 / 𝑤) ∈ ℂ)
50	30	cnmetdval 22574	. . . . . . . . . . . . . . . . 17 ⊢ (((1 / 𝑥) ∈ ℂ ∧ (1 / 𝑤) ∈ ℂ) → ((1 / 𝑥)(abs ∘ − )(1 / 𝑤)) = (abs‘((1 / 𝑥) − (1 / 𝑤))))
51		abssub 14066	. . . . . . . . . . . . . . . . 17 ⊢ (((1 / 𝑥) ∈ ℂ ∧ (1 / 𝑤) ∈ ℂ) → (abs‘((1 / 𝑥) − (1 / 𝑤))) = (abs‘((1 / 𝑤) − (1 / 𝑥))))
52	50, 51	eqtrd 2656	. . . . . . . . . . . . . . . 16 ⊢ (((1 / 𝑥) ∈ ℂ ∧ (1 / 𝑤) ∈ ℂ) → ((1 / 𝑥)(abs ∘ − )(1 / 𝑤)) = (abs‘((1 / 𝑤) − (1 / 𝑥))))
53	46, 49, 52	syl2an 494	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (ℂ ∖ {0}) ∧ 𝑤 ∈ (ℂ ∖ {0})) → ((1 / 𝑥)(abs ∘ − )(1 / 𝑤)) = (abs‘((1 / 𝑤) − (1 / 𝑥))))
54	43, 53	eqtrd 2656	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (ℂ ∖ {0}) ∧ 𝑤 ∈ (ℂ ∖ {0})) → (((𝑧 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑧))‘𝑥)(abs ∘ − )((𝑧 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑧))‘𝑤)) = (abs‘((1 / 𝑤) − (1 / 𝑥))))
55	54	breq1d 4663	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (ℂ ∖ {0}) ∧ 𝑤 ∈ (ℂ ∖ {0})) → ((((𝑧 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑧))‘𝑥)(abs ∘ − )((𝑧 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑧))‘𝑤)) < 𝑦 ↔ (abs‘((1 / 𝑤) − (1 / 𝑥))) < 𝑦))
56	36, 55	imbi12d 334	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (ℂ ∖ {0}) ∧ 𝑤 ∈ (ℂ ∖ {0})) → (((𝑥((abs ∘ − ) ↾ ((ℂ ∖ {0}) × (ℂ ∖ {0})))𝑤) < 𝑢 → (((𝑧 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑧))‘𝑥)(abs ∘ − )((𝑧 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑧))‘𝑤)) < 𝑦) ↔ ((abs‘(𝑤 − 𝑥)) < 𝑢 → (abs‘((1 / 𝑤) − (1 / 𝑥))) < 𝑦)))
57	56	ralbidva 2985	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (ℂ ∖ {0}) → (∀𝑤 ∈ (ℂ ∖ {0})((𝑥((abs ∘ − ) ↾ ((ℂ ∖ {0}) × (ℂ ∖ {0})))𝑤) < 𝑢 → (((𝑧 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑧))‘𝑥)(abs ∘ − )((𝑧 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑧))‘𝑤)) < 𝑦) ↔ ∀𝑤 ∈ (ℂ ∖ {0})((abs‘(𝑤 − 𝑥)) < 𝑢 → (abs‘((1 / 𝑤) − (1 / 𝑥))) < 𝑦)))
58	57	rexbidv 3052	. . . . . . . . . 10 ⊢ (𝑥 ∈ (ℂ ∖ {0}) → (∃𝑢 ∈ ℝ⁺ ∀𝑤 ∈ (ℂ ∖ {0})((𝑥((abs ∘ − ) ↾ ((ℂ ∖ {0}) × (ℂ ∖ {0})))𝑤) < 𝑢 → (((𝑧 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑧))‘𝑥)(abs ∘ − )((𝑧 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑧))‘𝑤)) < 𝑦) ↔ ∃𝑢 ∈ ℝ⁺ ∀𝑤 ∈ (ℂ ∖ {0})((abs‘(𝑤 − 𝑥)) < 𝑢 → (abs‘((1 / 𝑤) − (1 / 𝑥))) < 𝑦)))
59	58	adantr 481	. . . . . . . . 9 ⊢ ((𝑥 ∈ (ℂ ∖ {0}) ∧ 𝑦 ∈ ℝ⁺) → (∃𝑢 ∈ ℝ⁺ ∀𝑤 ∈ (ℂ ∖ {0})((𝑥((abs ∘ − ) ↾ ((ℂ ∖ {0}) × (ℂ ∖ {0})))𝑤) < 𝑢 → (((𝑧 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑧))‘𝑥)(abs ∘ − )((𝑧 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑧))‘𝑤)) < 𝑦) ↔ ∃𝑢 ∈ ℝ⁺ ∀𝑤 ∈ (ℂ ∖ {0})((abs‘(𝑤 − 𝑥)) < 𝑢 → (abs‘((1 / 𝑤) − (1 / 𝑥))) < 𝑦)))
60	26, 59	mpbird 247	. . . . . . . 8 ⊢ ((𝑥 ∈ (ℂ ∖ {0}) ∧ 𝑦 ∈ ℝ⁺) → ∃𝑢 ∈ ℝ⁺ ∀𝑤 ∈ (ℂ ∖ {0})((𝑥((abs ∘ − ) ↾ ((ℂ ∖ {0}) × (ℂ ∖ {0})))𝑤) < 𝑢 → (((𝑧 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑧))‘𝑥)(abs ∘ − )((𝑧 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑧))‘𝑤)) < 𝑦))
61	60	rgen2 2975	. . . . . . 7 ⊢ ∀𝑥 ∈ (ℂ ∖ {0})∀𝑦 ∈ ℝ⁺ ∃𝑢 ∈ ℝ⁺ ∀𝑤 ∈ (ℂ ∖ {0})((𝑥((abs ∘ − ) ↾ ((ℂ ∖ {0}) × (ℂ ∖ {0})))𝑤) < 𝑢 → (((𝑧 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑧))‘𝑥)(abs ∘ − )((𝑧 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑧))‘𝑤)) < 𝑦)
62		cnxmet 22576	. . . . . . . . 9 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
63		xmetres2 22166	. . . . . . . . 9 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ (ℂ ∖ {0}) ⊆ ℂ) → ((abs ∘ − ) ↾ ((ℂ ∖ {0}) × (ℂ ∖ {0}))) ∈ (∞Met‘(ℂ ∖ {0})))
64	62, 14, 63	mp2an 708	. . . . . . . 8 ⊢ ((abs ∘ − ) ↾ ((ℂ ∖ {0}) × (ℂ ∖ {0}))) ∈ (∞Met‘(ℂ ∖ {0}))
65		eqid 2622	. . . . . . . . . . . 12 ⊢ ((abs ∘ − ) ↾ ((ℂ ∖ {0}) × (ℂ ∖ {0}))) = ((abs ∘ − ) ↾ ((ℂ ∖ {0}) × (ℂ ∖ {0})))
66	10	cnfldtopn 22585	. . . . . . . . . . . 12 ⊢ 𝐽 = (MetOpen‘(abs ∘ − ))
67		eqid 2622	. . . . . . . . . . . 12 ⊢ (MetOpen‘((abs ∘ − ) ↾ ((ℂ ∖ {0}) × (ℂ ∖ {0})))) = (MetOpen‘((abs ∘ − ) ↾ ((ℂ ∖ {0}) × (ℂ ∖ {0}))))
68	65, 66, 67	metrest 22329	. . . . . . . . . . 11 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ (ℂ ∖ {0}) ⊆ ℂ) → (𝐽 ↾_t (ℂ ∖ {0})) = (MetOpen‘((abs ∘ − ) ↾ ((ℂ ∖ {0}) × (ℂ ∖ {0})))))
69	62, 14, 68	mp2an 708	. . . . . . . . . 10 ⊢ (𝐽 ↾_t (ℂ ∖ {0})) = (MetOpen‘((abs ∘ − ) ↾ ((ℂ ∖ {0}) × (ℂ ∖ {0}))))
70	13, 69	eqtri 2644	. . . . . . . . 9 ⊢ 𝐾 = (MetOpen‘((abs ∘ − ) ↾ ((ℂ ∖ {0}) × (ℂ ∖ {0}))))
71	70, 66	metcn 22348	. . . . . . . 8 ⊢ ((((abs ∘ − ) ↾ ((ℂ ∖ {0}) × (ℂ ∖ {0}))) ∈ (∞Met‘(ℂ ∖ {0})) ∧ (abs ∘ − ) ∈ (∞Met‘ℂ)) → ((𝑧 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑧)) ∈ (𝐾 Cn 𝐽) ↔ ((𝑧 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑧)):(ℂ ∖ {0})⟶ℂ ∧ ∀𝑥 ∈ (ℂ ∖ {0})∀𝑦 ∈ ℝ⁺ ∃𝑢 ∈ ℝ⁺ ∀𝑤 ∈ (ℂ ∖ {0})((𝑥((abs ∘ − ) ↾ ((ℂ ∖ {0}) × (ℂ ∖ {0})))𝑤) < 𝑢 → (((𝑧 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑧))‘𝑥)(abs ∘ − )((𝑧 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑧))‘𝑤)) < 𝑦))))
72	64, 62, 71	mp2an 708	. . . . . . 7 ⊢ ((𝑧 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑧)) ∈ (𝐾 Cn 𝐽) ↔ ((𝑧 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑧)):(ℂ ∖ {0})⟶ℂ ∧ ∀𝑥 ∈ (ℂ ∖ {0})∀𝑦 ∈ ℝ⁺ ∃𝑢 ∈ ℝ⁺ ∀𝑤 ∈ (ℂ ∖ {0})((𝑥((abs ∘ − ) ↾ ((ℂ ∖ {0}) × (ℂ ∖ {0})))𝑤) < 𝑢 → (((𝑧 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑧))‘𝑥)(abs ∘ − )((𝑧 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑧))‘𝑤)) < 𝑦)))
73	24, 61, 72	mpbir2an 955	. . . . . 6 ⊢ (𝑧 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑧)) ∈ (𝐾 Cn 𝐽)
74	73	a1i 11	. . . . 5 ⊢ (⊤ → (𝑧 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑧)) ∈ (𝐾 Cn 𝐽))
75		oveq2 6658	. . . . 5 ⊢ (𝑧 = 𝑦 → (1 / 𝑧) = (1 / 𝑦))
76	12, 17, 19, 17, 74, 75	cnmpt21 21474	. . . 4 ⊢ (⊤ → (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑦)) ∈ ((𝐽 ×_t 𝐾) Cn 𝐽))
77	10	mulcn 22670	. . . . 5 ⊢ · ∈ ((𝐽 ×_t 𝐽) Cn 𝐽)
78	77	a1i 11	. . . 4 ⊢ (⊤ → · ∈ ((𝐽 ×_t 𝐽) Cn 𝐽))
79	12, 17, 18, 76, 78	cnmpt22f 21478	. . 3 ⊢ (⊤ → (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (𝑥 · (1 / 𝑦))) ∈ ((𝐽 ×_t 𝐾) Cn 𝐽))
80	79	trud 1493	. 2 ⊢ (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (𝑥 · (1 / 𝑦))) ∈ ((𝐽 ×_t 𝐾) Cn 𝐽)
81	9, 80	eqeltri 2697	1 ⊢ / ∈ ((𝐽 ×_t 𝐾) Cn 𝐽)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ⊤wtru 1484 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 ∃wrex 2913 ∖ cdif 3571 ⊆ wss 3574 ifcif 4086 {csn 4177 class class class wbr 4653 ↦ cmpt 4729 × cxp 5112 ↾ cres 5116 ∘ ccom 5118 ⟶wf 5884 ‘cfv 5888 ℩crio 6610 (class class class)co 6650 ↦ cmpt2 6652 ℂcc 9934 0cc0 9936 1c1 9937 · cmul 9941 < clt 10074 ≤ cle 10075 − cmin 10266 / cdiv 10684 2c2 11070 ℝ⁺crp 11832 abscabs 13974 ↾_t crest 16081 TopOpenctopn 16082 ∞Metcxmt 19731 MetOpencmopn 19736 ℂ_fldccnfld 19746 TopOnctopon 20715 Cn ccn 21028 ×_t ctx 21363

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-mulf 10016

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-icc 12182 df-fz 12327 df-fzo 12466 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-rest 16083 df-topn 16084 df-0g 16102 df-gsum 16103 df-topgen 16104 df-pt 16105 df-prds 16108 df-xrs 16162 df-qtop 16167 df-imas 16168 df-xps 16170 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-mulg 17541 df-cntz 17750 df-cmn 18195 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-cnfld 19747 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cn 21031 df-cnp 21032 df-tx 21365 df-hmeo 21558 df-xms 22125 df-ms 22126 df-tms 22127

This theorem is referenced by: cdivcncf 22720 evth 22758 dvcnvlem 23739 lhop1lem 23776

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