Definition df-div 7414

Description: Define division. Theorem divmulap 7416 relates it to multiplication, and divclap 7419 and redivclap 7469 prove its closure laws. (Contributed by NM, 2-Feb-1995.) (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 7413	. 2 class /
2		vx	. . 3 setvar x
3		vy	. . 3 setvar y
4		cc 6689	. . 3 class ℂ
5		cc0 6691	. . . . 5 class 0
6	5	csn 3367	. . . 4 class {0}
7	4, 6	cdif 2908	. . 3 class (ℂ ∖ {0})
8	3	cv 1241	. . . . . 6 class y
9		vz	. . . . . . 7 setvar z
10	9	cv 1241	. . . . . 6 class z
11		cmul 6696	. . . . . 6 class ·
12	8, 10, 11	co 5455	. . . . 5 class (y · z)
13	2	cv 1241	. . . . 5 class x
14	12, 13	wceq 1242	. . . 4 wff (y · z) = x
15	14, 9, 4	crio 5410	. . 3 class (℩z ∈ ℂ (y · z) = x)
16	2, 3, 4, 7, 15	cmpt2 5457	. 2 class (x ∈ ℂ, y ∈ (ℂ ∖ {0}) ↦ (℩z ∈ ℂ (y · z) = x))
17	1, 16	wceq 1242	1 wff / = (x ∈ ℂ, y ∈ (ℂ ∖ {0}) ↦ (℩z ∈ ℂ (y · z) = x))

This definition is referenced by:  divvalap  7415  divfnzn  8312