Definition df-sub 8130

Description: Define subtraction. Theorem subval 8149 shows its value (and describes how this definition works), Theorem subaddi 8244 relates it to addition, and Theorems subcli 8233 and resubcli 8220 prove its closure laws. (Contributed by NM, 26-Nov-1994.)

Assertion

Ref	Expression
df-sub	|- - = ( x e. CC , y e. CC |-> ( iota_ z e. CC ( y + z ) = x ) )

Distinct variable group:   x, y, z

Detailed syntax breakdown of Definition df-sub

Step	Hyp	Ref	Expression
1		cmin 8128	. 2 class -
2		vx	. . 3 setvar x
3		vy	. . 3 setvar y
4		cc 7809	. . 3 class CC
5	3	cv 1352	. . . . . 6 class y
6		vz	. . . . . . 7 setvar z
7	6	cv 1352	. . . . . 6 class z
8		caddc 7814	. . . . . 6 class +
9	5, 7, 8	co 5875	. . . . 5 class ( y + z )
10	2	cv 1352	. . . . 5 class x
11	9, 10	wceq 1353	. . . 4 wff ( y + z ) = x
12	11, 6, 4	crio 5830	. . 3 class ( iota_ z e. CC ( y + z ) = x )
13	2, 3, 4, 4, 12	cmpo 5877	. 2 class ( x e. CC , y e. CC |-> ( iota_ z e. CC ( y + z ) = x ) )
14	1, 13	wceq 1353	1 wff - = ( x e. CC , y e. CC |-> ( iota_ z e. CC ( y + z ) = x ) )

This definition is referenced by:  subval  8149  subf  8159