Definition df-xdiv 32900

Description: Define division over extended real numbers. (Contributed by Thierry Arnoux, 17-Dec-2016.)

Assertion

Ref	Expression
df-xdiv	⊢ /𝑒 = (𝑥 ∈ ℝ*, 𝑦 ∈ (ℝ ∖ {0}) ↦ (℩𝑧 ∈ ℝ* (𝑦 ·e 𝑧) = 𝑥))

Distinct variable group:   𝑥,𝑦,𝑧

Detailed syntax breakdown of Definition df-xdiv

Step	Hyp	Ref	Expression
1		cxdiv 32899	. 2 class /𝑒
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cxr 11294	. . 3 class ℝ*
5		cr 11154	. . . 4 class ℝ
6		cc0 11155	. . . . 5 class 0
7	6	csn 4626	. . . 4 class {0}
8	5, 7	cdif 3948	. . 3 class (ℝ ∖ {0})
9	3	cv 1539	. . . . . 6 class 𝑦
10		vz	. . . . . . 7 setvar 𝑧
11	10	cv 1539	. . . . . 6 class 𝑧
12		cxmu 13153	. . . . . 6 class ·e
13	9, 11, 12	co 7431	. . . . 5 class (𝑦 ·e 𝑧)
14	2	cv 1539	. . . . 5 class 𝑥
15	13, 14	wceq 1540	. . . 4 wff (𝑦 ·e 𝑧) = 𝑥
16	15, 10, 4	crio 7387	. . 3 class (℩𝑧 ∈ ℝ* (𝑦 ·e 𝑧) = 𝑥)
17	2, 3, 4, 8, 16	cmpo 7433	. 2 class (𝑥 ∈ ℝ*, 𝑦 ∈ (ℝ ∖ {0}) ↦ (℩𝑧 ∈ ℝ* (𝑦 ·e 𝑧) = 𝑥))
18	1, 17	wceq 1540	1 wff /𝑒 = (𝑥 ∈ ℝ*, 𝑦 ∈ (ℝ ∖ {0}) ↦ (℩𝑧 ∈ ℝ* (𝑦 ·e 𝑧) = 𝑥))

This definition is referenced by:  xdivval  32901