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Theorem divcn 24031

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 11633	. . 3 ⊢ / = (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥))
2		eldifsn 4720	. . . . 5 ⊢ (𝑦 ∈ (ℂ ∖ {0}) ↔ (𝑦 ∈ ℂ ∧ 𝑦 ≠ 0))
3		divval 11635	. . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑦 ≠ 0) → (𝑥 / 𝑦) = (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥))
4		divrec 11649	. . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑦 ≠ 0) → (𝑥 / 𝑦) = (𝑥 · (1 / 𝑦)))
5	3, 4	eqtr3d 2780	. . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑦 ≠ 0) → (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥) = (𝑥 · (1 / 𝑦)))
6	5	3expb 1119	. . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ (𝑦 ∈ ℂ ∧ 𝑦 ≠ 0)) → (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥) = (𝑥 · (1 / 𝑦)))
7	2, 6	sylan2b 594	. . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0})) → (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥) = (𝑥 · (1 / 𝑦)))
8	7	mpoeq3ia 7353	. . 3 ⊢ (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥)) = (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (𝑥 · (1 / 𝑦)))
9	1, 8	eqtri 2766	. 2 ⊢ / = (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (𝑥 · (1 / 𝑦)))
10		addcn.j	. . . . . 6 ⊢ 𝐽 = (TopOpen‘ℂ_fld)
11	10	cnfldtopon 23946	. . . . 5 ⊢ 𝐽 ∈ (TopOn‘ℂ)
12	11	a1i 11	. . . 4 ⊢ (⊤ → 𝐽 ∈ (TopOn‘ℂ))
13		divcn.k	. . . . 5 ⊢ 𝐾 = (𝐽 ↾_t (ℂ ∖ {0}))
14		difss 4066	. . . . . 6 ⊢ (ℂ ∖ {0}) ⊆ ℂ
15		resttopon 22312	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘ℂ) ∧ (ℂ ∖ {0}) ⊆ ℂ) → (𝐽 ↾_t (ℂ ∖ {0})) ∈ (TopOn‘(ℂ ∖ {0})))
16	12, 14, 15	sylancl 586	. . . . 5 ⊢ (⊤ → (𝐽 ↾_t (ℂ ∖ {0})) ∈ (TopOn‘(ℂ ∖ {0})))
17	13, 16	eqeltrid 2843	. . . 4 ⊢ (⊤ → 𝐾 ∈ (TopOn‘(ℂ ∖ {0})))
18	12, 17	cnmpt1st 22819	. . . 4 ⊢ (⊤ → (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ 𝑥) ∈ ((𝐽 ×_t 𝐾) Cn 𝐽))
19	12, 17	cnmpt2nd 22820	. . . . 5 ⊢ (⊤ → (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ 𝑦) ∈ ((𝐽 ×_t 𝐾) Cn 𝐾))
20		eqid 2738	. . . . . . . 8 ⊢ (𝑧 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑧)) = (𝑧 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑧))
21		eldifsn 4720	. . . . . . . . 9 ⊢ (𝑧 ∈ (ℂ ∖ {0}) ↔ (𝑧 ∈ ℂ ∧ 𝑧 ≠ 0))
22		reccl 11640	. . . . . . . . 9 ⊢ ((𝑧 ∈ ℂ ∧ 𝑧 ≠ 0) → (1 / 𝑧) ∈ ℂ)
23	21, 22	sylbi 216	. . . . . . . 8 ⊢ (𝑧 ∈ (ℂ ∖ {0}) → (1 / 𝑧) ∈ ℂ)
24	20, 23	fmpti 6986	. . . . . . 7 ⊢ (𝑧 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑧)):(ℂ ∖ {0})⟶ℂ
25		eqid 2738	. . . . . . . . . 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 15306	. . . . . . . . 9 ⊢ ((𝑥 ∈ (ℂ ∖ {0}) ∧ 𝑦 ∈ ℝ⁺) → ∃𝑢 ∈ ℝ⁺ ∀𝑤 ∈ (ℂ ∖ {0})((abs‘(𝑤 − 𝑥)) < 𝑢 → (abs‘((1 / 𝑤) − (1 / 𝑥))) < 𝑦))
27		ovres 7438	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (ℂ ∖ {0}) ∧ 𝑤 ∈ (ℂ ∖ {0})) → (𝑥((abs ∘ − ) ↾ ((ℂ ∖ {0}) × (ℂ ∖ {0})))𝑤) = (𝑥(abs ∘ − )𝑤))
28		eldifi 4061	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (ℂ ∖ {0}) → 𝑥 ∈ ℂ)
29		eldifi 4061	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ (ℂ ∖ {0}) → 𝑤 ∈ ℂ)
30		eqid 2738	. . . . . . . . . . . . . . . . . 18 ⊢ (abs ∘ − ) = (abs ∘ − )
31	30	cnmetdval 23934	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℂ ∧ 𝑤 ∈ ℂ) → (𝑥(abs ∘ − )𝑤) = (abs‘(𝑥 − 𝑤)))
32		abssub 15038	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℂ ∧ 𝑤 ∈ ℂ) → (abs‘(𝑥 − 𝑤)) = (abs‘(𝑤 − 𝑥)))
33	31, 32	eqtrd 2778	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℂ ∧ 𝑤 ∈ ℂ) → (𝑥(abs ∘ − )𝑤) = (abs‘(𝑤 − 𝑥)))
34	28, 29, 33	syl2an 596	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (ℂ ∖ {0}) ∧ 𝑤 ∈ (ℂ ∖ {0})) → (𝑥(abs ∘ − )𝑤) = (abs‘(𝑤 − 𝑥)))
35	27, 34	eqtrd 2778	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (ℂ ∖ {0}) ∧ 𝑤 ∈ (ℂ ∖ {0})) → (𝑥((abs ∘ − ) ↾ ((ℂ ∖ {0}) × (ℂ ∖ {0})))𝑤) = (abs‘(𝑤 − 𝑥)))
36	35	breq1d 5084	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (ℂ ∖ {0}) ∧ 𝑤 ∈ (ℂ ∖ {0})) → ((𝑥((abs ∘ − ) ↾ ((ℂ ∖ {0}) × (ℂ ∖ {0})))𝑤) < 𝑢 ↔ (abs‘(𝑤 − 𝑥)) < 𝑢))
37		oveq2 7283	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = 𝑥 → (1 / 𝑧) = (1 / 𝑥))
38		ovex 7308	. . . . . . . . . . . . . . . . 17 ⊢ (1 / 𝑥) ∈ V
39	37, 20, 38	fvmpt 6875	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (ℂ ∖ {0}) → ((𝑧 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑧))‘𝑥) = (1 / 𝑥))
40		oveq2 7283	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = 𝑤 → (1 / 𝑧) = (1 / 𝑤))
41		ovex 7308	. . . . . . . . . . . . . . . . 17 ⊢ (1 / 𝑤) ∈ V
42	40, 20, 41	fvmpt 6875	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ (ℂ ∖ {0}) → ((𝑧 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑧))‘𝑤) = (1 / 𝑤))
43	39, 42	oveqan12d 7294	. . . . . . . . . . . . . . 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 4720	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (ℂ ∖ {0}) ↔ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
45		reccl 11640	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) → (1 / 𝑥) ∈ ℂ)
46	44, 45	sylbi 216	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (ℂ ∖ {0}) → (1 / 𝑥) ∈ ℂ)
47		eldifsn 4720	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 ∈ (ℂ ∖ {0}) ↔ (𝑤 ∈ ℂ ∧ 𝑤 ≠ 0))
48		reccl 11640	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑤 ∈ ℂ ∧ 𝑤 ≠ 0) → (1 / 𝑤) ∈ ℂ)
49	47, 48	sylbi 216	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ (ℂ ∖ {0}) → (1 / 𝑤) ∈ ℂ)
50	30	cnmetdval 23934	. . . . . . . . . . . . . . . . 17 ⊢ (((1 / 𝑥) ∈ ℂ ∧ (1 / 𝑤) ∈ ℂ) → ((1 / 𝑥)(abs ∘ − )(1 / 𝑤)) = (abs‘((1 / 𝑥) − (1 / 𝑤))))
51		abssub 15038	. . . . . . . . . . . . . . . . 17 ⊢ (((1 / 𝑥) ∈ ℂ ∧ (1 / 𝑤) ∈ ℂ) → (abs‘((1 / 𝑥) − (1 / 𝑤))) = (abs‘((1 / 𝑤) − (1 / 𝑥))))
52	50, 51	eqtrd 2778	. . . . . . . . . . . . . . . 16 ⊢ (((1 / 𝑥) ∈ ℂ ∧ (1 / 𝑤) ∈ ℂ) → ((1 / 𝑥)(abs ∘ − )(1 / 𝑤)) = (abs‘((1 / 𝑤) − (1 / 𝑥))))
53	46, 49, 52	syl2an 596	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (ℂ ∖ {0}) ∧ 𝑤 ∈ (ℂ ∖ {0})) → ((1 / 𝑥)(abs ∘ − )(1 / 𝑤)) = (abs‘((1 / 𝑤) − (1 / 𝑥))))
54	43, 53	eqtrd 2778	. . . . . . . . . . . . . 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 5084	. . . . . . . . . . . . 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 345	. . . . . . . . . . . 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 3111	. . . . . . . . . . 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 3226	. . . . . . . . . 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 256	. . . . . . . 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 3120	. . . . . . 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 23936	. . . . . . . . 9 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
63		xmetres2 23514	. . . . . . . . 9 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ (ℂ ∖ {0}) ⊆ ℂ) → ((abs ∘ − ) ↾ ((ℂ ∖ {0}) × (ℂ ∖ {0}))) ∈ (∞Met‘(ℂ ∖ {0})))
64	62, 14, 63	mp2an 689	. . . . . . . 8 ⊢ ((abs ∘ − ) ↾ ((ℂ ∖ {0}) × (ℂ ∖ {0}))) ∈ (∞Met‘(ℂ ∖ {0}))
65		eqid 2738	. . . . . . . . . . . 12 ⊢ ((abs ∘ − ) ↾ ((ℂ ∖ {0}) × (ℂ ∖ {0}))) = ((abs ∘ − ) ↾ ((ℂ ∖ {0}) × (ℂ ∖ {0})))
66	10	cnfldtopn 23945	. . . . . . . . . . . 12 ⊢ 𝐽 = (MetOpen‘(abs ∘ − ))
67		eqid 2738	. . . . . . . . . . . 12 ⊢ (MetOpen‘((abs ∘ − ) ↾ ((ℂ ∖ {0}) × (ℂ ∖ {0})))) = (MetOpen‘((abs ∘ − ) ↾ ((ℂ ∖ {0}) × (ℂ ∖ {0}))))
68	65, 66, 67	metrest 23680	. . . . . . . . . . 11 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ (ℂ ∖ {0}) ⊆ ℂ) → (𝐽 ↾_t (ℂ ∖ {0})) = (MetOpen‘((abs ∘ − ) ↾ ((ℂ ∖ {0}) × (ℂ ∖ {0})))))
69	62, 14, 68	mp2an 689	. . . . . . . . . 10 ⊢ (𝐽 ↾_t (ℂ ∖ {0})) = (MetOpen‘((abs ∘ − ) ↾ ((ℂ ∖ {0}) × (ℂ ∖ {0}))))
70	13, 69	eqtri 2766	. . . . . . . . 9 ⊢ 𝐾 = (MetOpen‘((abs ∘ − ) ↾ ((ℂ ∖ {0}) × (ℂ ∖ {0}))))
71	70, 66	metcn 23699	. . . . . . . 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 689	. . . . . . 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 708	. . . . . 6 ⊢ (𝑧 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑧)) ∈ (𝐾 Cn 𝐽)
74	73	a1i 11	. . . . 5 ⊢ (⊤ → (𝑧 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑧)) ∈ (𝐾 Cn 𝐽))
75		oveq2 7283	. . . . 5 ⊢ (𝑧 = 𝑦 → (1 / 𝑧) = (1 / 𝑦))
76	12, 17, 19, 17, 74, 75	cnmpt21 22822	. . . 4 ⊢ (⊤ → (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑦)) ∈ ((𝐽 ×_t 𝐾) Cn 𝐽))
77	10	mulcn 24030	. . . . 5 ⊢ · ∈ ((𝐽 ×_t 𝐽) Cn 𝐽)
78	77	a1i 11	. . . 4 ⊢ (⊤ → · ∈ ((𝐽 ×_t 𝐽) Cn 𝐽))
79	12, 17, 18, 76, 78	cnmpt22f 22826	. . 3 ⊢ (⊤ → (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (𝑥 · (1 / 𝑦))) ∈ ((𝐽 ×_t 𝐾) Cn 𝐽))
80	79	mptru 1546	. 2 ⊢ (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (𝑥 · (1 / 𝑦))) ∈ ((𝐽 ×_t 𝐾) Cn 𝐽)
81	9, 80	eqeltri 2835	1 ⊢ / ∈ ((𝐽 ×_t 𝐾) Cn 𝐽)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ⊤wtru 1540 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 ∃wrex 3065 ∖ cdif 3884 ⊆ wss 3887 ifcif 4459 {csn 4561 class class class wbr 5074 ↦ cmpt 5157 × cxp 5587 ↾ cres 5591 ∘ ccom 5593 ⟶wf 6429 ‘cfv 6433 ℩crio 7231 (class class class)co 7275 ∈ cmpo 7277 ℂcc 10869 0cc0 10871 1c1 10872 · cmul 10876 < clt 11009 ≤ cle 11010 − cmin 11205 / cdiv 11632 2c2 12028 ℝ⁺crp 12730 abscabs 14945 ↾_t crest 17131 TopOpenctopn 17132 ∞Metcxmet 20582 MetOpencmopn 20587 ℂ_fldccnfld 20597 TopOnctopon 22059 Cn ccn 22375 ×_t ctx 22711

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 ax-mulf 10951

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7978 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-2o 8298 df-er 8498 df-map 8617 df-ixp 8686 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fsupp 9129 df-fi 9170 df-sup 9201 df-inf 9202 df-oi 9269 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-q 12689 df-rp 12731 df-xneg 12848 df-xadd 12849 df-xmul 12850 df-icc 13086 df-fz 13240 df-fzo 13383 df-seq 13722 df-exp 13783 df-hash 14045 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-starv 16977 df-sca 16978 df-vsca 16979 df-ip 16980 df-tset 16981 df-ple 16982 df-ds 16984 df-unif 16985 df-hom 16986 df-cco 16987 df-rest 17133 df-topn 17134 df-0g 17152 df-gsum 17153 df-topgen 17154 df-pt 17155 df-prds 17158 df-xrs 17213 df-qtop 17218 df-imas 17219 df-xps 17221 df-mre 17295 df-mrc 17296 df-acs 17298 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-submnd 18431 df-mulg 18701 df-cntz 18923 df-cmn 19388 df-psmet 20589 df-xmet 20590 df-met 20591 df-bl 20592 df-mopn 20593 df-cnfld 20598 df-top 22043 df-topon 22060 df-topsp 22082 df-bases 22096 df-cn 22378 df-cnp 22379 df-tx 22713 df-hmeo 22906 df-xms 23473 df-ms 23474 df-tms 23475

This theorem is referenced by: cdivcncf 24084 evth 24122 dvcnvlem 25140 lhop1lem 25177

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