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Theorem divcn 22479

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 10564	. . 3 ⊢ / = (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥))
2		eldifsn 4260	. . . . 5 ⊢ (𝑦 ∈ (ℂ ∖ {0}) ↔ (𝑦 ∈ ℂ ∧ 𝑦 ≠ 0))
3		divval 10566	. . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑦 ≠ 0) → (𝑥 / 𝑦) = (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥))
4		divrec 10580	. . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑦 ≠ 0) → (𝑥 / 𝑦) = (𝑥 · (1 / 𝑦)))
5	3, 4	eqtr3d 2646	. . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑦 ≠ 0) → (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥) = (𝑥 · (1 / 𝑦)))
6	5	3expb 1258	. . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ (𝑦 ∈ ℂ ∧ 𝑦 ≠ 0)) → (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥) = (𝑥 · (1 / 𝑦)))
7	2, 6	sylan2b 491	. . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0})) → (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥) = (𝑥 · (1 / 𝑦)))
8	7	mpt2eq3ia 6618	. . 3 ⊢ (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥)) = (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (𝑥 · (1 / 𝑦)))
9	1, 8	eqtri 2632	. 2 ⊢ / = (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (𝑥 · (1 / 𝑦)))
10		addcn.j	. . . . . 6 ⊢ 𝐽 = (TopOpen‘ℂ_fld)
11	10	cnfldtopon 22396	. . . . 5 ⊢ 𝐽 ∈ (TopOn‘ℂ)
12	11	a1i 11	. . . 4 ⊢ (⊤ → 𝐽 ∈ (TopOn‘ℂ))
13		divcn.k	. . . . 5 ⊢ 𝐾 = (𝐽 ↾_t (ℂ ∖ {0}))
14		difss 3699	. . . . . 6 ⊢ (ℂ ∖ {0}) ⊆ ℂ
15		resttopon 20775	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘ℂ) ∧ (ℂ ∖ {0}) ⊆ ℂ) → (𝐽 ↾_t (ℂ ∖ {0})) ∈ (TopOn‘(ℂ ∖ {0})))
16	12, 14, 15	sylancl 693	. . . . 5 ⊢ (⊤ → (𝐽 ↾_t (ℂ ∖ {0})) ∈ (TopOn‘(ℂ ∖ {0})))
17	13, 16	syl5eqel 2692	. . . 4 ⊢ (⊤ → 𝐾 ∈ (TopOn‘(ℂ ∖ {0})))
18	12, 17	cnmpt1st 21281	. . . 4 ⊢ (⊤ → (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ 𝑥) ∈ ((𝐽 ×_t 𝐾) Cn 𝐽))
19	12, 17	cnmpt2nd 21282	. . . . 5 ⊢ (⊤ → (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ 𝑦) ∈ ((𝐽 ×_t 𝐾) Cn 𝐾))
20		eqid 2610	. . . . . . . 8 ⊢ (𝑧 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑧)) = (𝑧 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑧))
21		eldifsn 4260	. . . . . . . . 9 ⊢ (𝑧 ∈ (ℂ ∖ {0}) ↔ (𝑧 ∈ ℂ ∧ 𝑧 ≠ 0))
22		reccl 10571	. . . . . . . . 9 ⊢ ((𝑧 ∈ ℂ ∧ 𝑧 ≠ 0) → (1 / 𝑧) ∈ ℂ)
23	21, 22	sylbi 206	. . . . . . . 8 ⊢ (𝑧 ∈ (ℂ ∖ {0}) → (1 / 𝑧) ∈ ℂ)
24	20, 23	fmpti 6291	. . . . . . 7 ⊢ (𝑧 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑧)):(ℂ ∖ {0})⟶ℂ
25		eqid 2610	. . . . . . . . . 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 14175	. . . . . . . . 9 ⊢ ((𝑥 ∈ (ℂ ∖ {0}) ∧ 𝑦 ∈ ℝ⁺) → ∃𝑢 ∈ ℝ⁺ ∀𝑤 ∈ (ℂ ∖ {0})((abs‘(𝑤 − 𝑥)) < 𝑢 → (abs‘((1 / 𝑤) − (1 / 𝑥))) < 𝑦))
27		ovres 6698	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (ℂ ∖ {0}) ∧ 𝑤 ∈ (ℂ ∖ {0})) → (𝑥((abs ∘ − ) ↾ ((ℂ ∖ {0}) × (ℂ ∖ {0})))𝑤) = (𝑥(abs ∘ − )𝑤))
28		eldifi 3694	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (ℂ ∖ {0}) → 𝑥 ∈ ℂ)
29		eldifi 3694	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ (ℂ ∖ {0}) → 𝑤 ∈ ℂ)
30		eqid 2610	. . . . . . . . . . . . . . . . . 18 ⊢ (abs ∘ − ) = (abs ∘ − )
31	30	cnmetdval 22384	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℂ ∧ 𝑤 ∈ ℂ) → (𝑥(abs ∘ − )𝑤) = (abs‘(𝑥 − 𝑤)))
32		abssub 13914	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℂ ∧ 𝑤 ∈ ℂ) → (abs‘(𝑥 − 𝑤)) = (abs‘(𝑤 − 𝑥)))
33	31, 32	eqtrd 2644	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℂ ∧ 𝑤 ∈ ℂ) → (𝑥(abs ∘ − )𝑤) = (abs‘(𝑤 − 𝑥)))
34	28, 29, 33	syl2an 493	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (ℂ ∖ {0}) ∧ 𝑤 ∈ (ℂ ∖ {0})) → (𝑥(abs ∘ − )𝑤) = (abs‘(𝑤 − 𝑥)))
35	27, 34	eqtrd 2644	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (ℂ ∖ {0}) ∧ 𝑤 ∈ (ℂ ∖ {0})) → (𝑥((abs ∘ − ) ↾ ((ℂ ∖ {0}) × (ℂ ∖ {0})))𝑤) = (abs‘(𝑤 − 𝑥)))
36	35	breq1d 4593	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (ℂ ∖ {0}) ∧ 𝑤 ∈ (ℂ ∖ {0})) → ((𝑥((abs ∘ − ) ↾ ((ℂ ∖ {0}) × (ℂ ∖ {0})))𝑤) < 𝑢 ↔ (abs‘(𝑤 − 𝑥)) < 𝑢))
37		oveq2 6557	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = 𝑥 → (1 / 𝑧) = (1 / 𝑥))
38		ovex 6577	. . . . . . . . . . . . . . . . 17 ⊢ (1 / 𝑥) ∈ V
39	37, 20, 38	fvmpt 6191	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (ℂ ∖ {0}) → ((𝑧 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑧))‘𝑥) = (1 / 𝑥))
40		oveq2 6557	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = 𝑤 → (1 / 𝑧) = (1 / 𝑤))
41		ovex 6577	. . . . . . . . . . . . . . . . 17 ⊢ (1 / 𝑤) ∈ V
42	40, 20, 41	fvmpt 6191	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ (ℂ ∖ {0}) → ((𝑧 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑧))‘𝑤) = (1 / 𝑤))
43	39, 42	oveqan12d 6568	. . . . . . . . . . . . . . 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 4260	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (ℂ ∖ {0}) ↔ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
45		reccl 10571	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) → (1 / 𝑥) ∈ ℂ)
46	44, 45	sylbi 206	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (ℂ ∖ {0}) → (1 / 𝑥) ∈ ℂ)
47		eldifsn 4260	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 ∈ (ℂ ∖ {0}) ↔ (𝑤 ∈ ℂ ∧ 𝑤 ≠ 0))
48		reccl 10571	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑤 ∈ ℂ ∧ 𝑤 ≠ 0) → (1 / 𝑤) ∈ ℂ)
49	47, 48	sylbi 206	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ (ℂ ∖ {0}) → (1 / 𝑤) ∈ ℂ)
50	30	cnmetdval 22384	. . . . . . . . . . . . . . . . 17 ⊢ (((1 / 𝑥) ∈ ℂ ∧ (1 / 𝑤) ∈ ℂ) → ((1 / 𝑥)(abs ∘ − )(1 / 𝑤)) = (abs‘((1 / 𝑥) − (1 / 𝑤))))
51		abssub 13914	. . . . . . . . . . . . . . . . 17 ⊢ (((1 / 𝑥) ∈ ℂ ∧ (1 / 𝑤) ∈ ℂ) → (abs‘((1 / 𝑥) − (1 / 𝑤))) = (abs‘((1 / 𝑤) − (1 / 𝑥))))
52	50, 51	eqtrd 2644	. . . . . . . . . . . . . . . 16 ⊢ (((1 / 𝑥) ∈ ℂ ∧ (1 / 𝑤) ∈ ℂ) → ((1 / 𝑥)(abs ∘ − )(1 / 𝑤)) = (abs‘((1 / 𝑤) − (1 / 𝑥))))
53	46, 49, 52	syl2an 493	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (ℂ ∖ {0}) ∧ 𝑤 ∈ (ℂ ∖ {0})) → ((1 / 𝑥)(abs ∘ − )(1 / 𝑤)) = (abs‘((1 / 𝑤) − (1 / 𝑥))))
54	43, 53	eqtrd 2644	. . . . . . . . . . . . . 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 4593	. . . . . . . . . . . . 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 333	. . . . . . . . . . . 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 2968	. . . . . . . . . . 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 3034	. . . . . . . . . 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 480	. . . . . . . . 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 246	. . . . . . . 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 2958	. . . . . . 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 22386	. . . . . . . . 9 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
63		xmetres2 21976	. . . . . . . . 9 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ (ℂ ∖ {0}) ⊆ ℂ) → ((abs ∘ − ) ↾ ((ℂ ∖ {0}) × (ℂ ∖ {0}))) ∈ (∞Met‘(ℂ ∖ {0})))
64	62, 14, 63	mp2an 704	. . . . . . . 8 ⊢ ((abs ∘ − ) ↾ ((ℂ ∖ {0}) × (ℂ ∖ {0}))) ∈ (∞Met‘(ℂ ∖ {0}))
65		eqid 2610	. . . . . . . . . . . 12 ⊢ ((abs ∘ − ) ↾ ((ℂ ∖ {0}) × (ℂ ∖ {0}))) = ((abs ∘ − ) ↾ ((ℂ ∖ {0}) × (ℂ ∖ {0})))
66	10	cnfldtopn 22395	. . . . . . . . . . . 12 ⊢ 𝐽 = (MetOpen‘(abs ∘ − ))
67		eqid 2610	. . . . . . . . . . . 12 ⊢ (MetOpen‘((abs ∘ − ) ↾ ((ℂ ∖ {0}) × (ℂ ∖ {0})))) = (MetOpen‘((abs ∘ − ) ↾ ((ℂ ∖ {0}) × (ℂ ∖ {0}))))
68	65, 66, 67	metrest 22139	. . . . . . . . . . 11 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ (ℂ ∖ {0}) ⊆ ℂ) → (𝐽 ↾_t (ℂ ∖ {0})) = (MetOpen‘((abs ∘ − ) ↾ ((ℂ ∖ {0}) × (ℂ ∖ {0})))))
69	62, 14, 68	mp2an 704	. . . . . . . . . 10 ⊢ (𝐽 ↾_t (ℂ ∖ {0})) = (MetOpen‘((abs ∘ − ) ↾ ((ℂ ∖ {0}) × (ℂ ∖ {0}))))
70	13, 69	eqtri 2632	. . . . . . . . 9 ⊢ 𝐾 = (MetOpen‘((abs ∘ − ) ↾ ((ℂ ∖ {0}) × (ℂ ∖ {0}))))
71	70, 66	metcn 22158	. . . . . . . 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 704	. . . . . . 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 957	. . . . . 6 ⊢ (𝑧 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑧)) ∈ (𝐾 Cn 𝐽)
74	73	a1i 11	. . . . 5 ⊢ (⊤ → (𝑧 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑧)) ∈ (𝐾 Cn 𝐽))
75		oveq2 6557	. . . . 5 ⊢ (𝑧 = 𝑦 → (1 / 𝑧) = (1 / 𝑦))
76	12, 17, 19, 17, 74, 75	cnmpt21 21284	. . . 4 ⊢ (⊤ → (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑦)) ∈ ((𝐽 ×_t 𝐾) Cn 𝐽))
77	10	mulcn 22478	. . . . 5 ⊢ · ∈ ((𝐽 ×_t 𝐽) Cn 𝐽)
78	77	a1i 11	. . . 4 ⊢ (⊤ → · ∈ ((𝐽 ×_t 𝐽) Cn 𝐽))
79	12, 17, 18, 76, 78	cnmpt22f 21288	. . 3 ⊢ (⊤ → (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (𝑥 · (1 / 𝑦))) ∈ ((𝐽 ×_t 𝐾) Cn 𝐽))
80	79	trud 1484	. 2 ⊢ (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (𝑥 · (1 / 𝑦))) ∈ ((𝐽 ×_t 𝐾) Cn 𝐽)
81	9, 80	eqeltri 2684	1 ⊢ / ∈ ((𝐽 ×_t 𝐾) Cn 𝐽)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ⊤wtru 1476 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 ∃wrex 2897 ∖ cdif 3537 ⊆ wss 3540 ifcif 4036 {csn 4125 class class class wbr 4583 ↦ cmpt 4643 × cxp 5036 ↾ cres 5040 ∘ ccom 5042 ⟶wf 5800 ‘cfv 5804 ℩crio 6510 (class class class)co 6549 ↦ cmpt2 6551 ℂcc 9813 0cc0 9815 1c1 9816 · cmul 9820 < clt 9953 ≤ cle 9954 − cmin 10145 / cdiv 10563 2c2 10947 ℝ⁺crp 11708 abscabs 13822 ↾_t crest 15904 TopOpenctopn 15905 ∞Metcxmt 19552 MetOpencmopn 19557 ℂ_fldccnfld 19567 TopOnctopon 20518 Cn ccn 20838 ×_t ctx 21173

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-mulf 9895

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-fi 8200 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-icc 12053 df-fz 12198 df-fzo 12335 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-hom 15793 df-cco 15794 df-rest 15906 df-topn 15907 df-0g 15925 df-gsum 15926 df-topgen 15927 df-pt 15928 df-prds 15931 df-xrs 15985 df-qtop 15990 df-imas 15991 df-xps 15993 df-mre 16069 df-mrc 16070 df-acs 16072 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-submnd 17159 df-mulg 17364 df-cntz 17573 df-cmn 18018 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-cnfld 19568 df-top 20521 df-bases 20522 df-topon 20523 df-topsp 20524 df-cn 20841 df-cnp 20842 df-tx 21175 df-hmeo 21368 df-xms 21935 df-ms 21936 df-tms 21937

This theorem is referenced by: cdivcncf 22528 evth 22566 dvcnvlem 23543 lhop1lem 23580

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