Definition df-div 10685

Description: Define division. Theorem divmuli 10779 relates it to multiplication, and divcli 10767 and redivcli 10792 prove its closure laws. (Contributed by NM, 2-Feb-1995.) Use divval 10687 instead. (Revised by Mario Carneiro, 1-Apr-2014.) (New usage is discouraged.)

Assertion

Ref	Expression
df-div	⊢ / = (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥))

Distinct variable group:   𝑥,𝑦,𝑧

Detailed syntax breakdown of Definition df-div

Step	Hyp	Ref	Expression
1		cdiv 10684	. 2 class /
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cc 9934	. . 3 class ℂ
5		cc0 9936	. . . . 5 class 0
6	5	csn 4177	. . . 4 class {0}
7	4, 6	cdif 3571	. . 3 class (ℂ ∖ {0})
8	3	cv 1482	. . . . . 6 class 𝑦
9		vz	. . . . . . 7 setvar 𝑧
10	9	cv 1482	. . . . . 6 class 𝑧
11		cmul 9941	. . . . . 6 class ·
12	8, 10, 11	co 6650	. . . . 5 class (𝑦 · 𝑧)
13	2	cv 1482	. . . . 5 class 𝑥
14	12, 13	wceq 1483	. . . 4 wff (𝑦 · 𝑧) = 𝑥
15	14, 9, 4	crio 6610	. . 3 class (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥)
16	2, 3, 4, 7, 15	cmpt2 6652	. 2 class (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥))
17	1, 16	wceq 1483	1 wff / = (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥))

This definition is referenced by:  1div0  10686  divval  10687  elq  11790  cnflddiv  19776  divcn  22671  1div0apr  27324