Definition df-sub 7928

Description: Define subtraction. Theorem subval 7947 shows its value (and describes how this definition works), theorem subaddi 8042 relates it to addition, and theorems subcli 8031 and resubcli 8018 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 7926	. 2 class -
2		vx	. . 3 setvar x
3		vy	. . 3 setvar y
4		cc 7611	. . 3 class CC
5	3	cv 1330	. . . . . 6 class y
6		vz	. . . . . . 7 setvar z
7	6	cv 1330	. . . . . 6 class z
8		caddc 7616	. . . . . 6 class +
9	5, 7, 8	co 5767	. . . . 5 class ( y + z )
10	2	cv 1330	. . . . 5 class x
11	9, 10	wceq 1331	. . . 4 wff ( y + z ) = x
12	11, 6, 4	crio 5722	. . 3 class ( iota_ z e. CC ( y + z ) = x )
13	2, 3, 4, 4, 12	cmpo 5769	. 2 class ( x e. CC , y e. CC |-> ( iota_ z e. CC ( y + z ) = x ) )
14	1, 13	wceq 1331	1 wff - = ( x e. CC , y e. CC |-> ( iota_ z e. CC ( y + z ) = x ) )

This definition is referenced by:  subval  7947  subf  7957