Definition df-div 7761

Description: Define division. Theorem divmulap 7763 relates it to multiplication, and divclap 7766 and redivclap 7819 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 e. CC , y e. ( CC \ { 0 } ) |-> ( iota_ z e. CC ( y x. z ) = x ) )

Distinct variable group:   x, y, z

Detailed syntax breakdown of Definition df-div

Step	Hyp	Ref	Expression
1		cdiv 7760	. 2 class /
2		vx	. . 3 setvar x
3		vy	. . 3 setvar y
4		cc 6979	. . 3 class CC
5		cc0 6981	. . . . 5 class 0
6	5	csn 3398	. . . 4 class { 0 }
7	4, 6	cdif 2970	. . 3 class ( CC \ { 0 } )
8	3	cv 1283	. . . . . 6 class y
9		vz	. . . . . . 7 setvar z
10	9	cv 1283	. . . . . 6 class z
11		cmul 6986	. . . . . 6 class x.
12	8, 10, 11	co 5532	. . . . 5 class ( y x. z )
13	2	cv 1283	. . . . 5 class x
14	12, 13	wceq 1284	. . . 4 wff ( y x. z ) = x
15	14, 9, 4	crio 5487	. . 3 class ( iota_ z e. CC ( y x. z ) = x )
16	2, 3, 4, 7, 15	cmpt2 5534	. 2 class ( x e. CC , y e. ( CC \ { 0 } ) |-> ( iota_ z e. CC ( y x. z ) = x ) )
17	1, 16	wceq 1284	1 wff / = ( x e. CC , y e. ( CC \ { 0 } ) |-> ( iota_ z e. CC ( y x. z ) = x ) )

This definition is referenced by:  divvalap  7762  divfnzn  8706