Definition df-sub 7418

Description: Define subtraction. Theorem subval 7437 shows its value (and describes how this definition works), theorem subaddi 7532 relates it to addition, and theorems subcli 7521 and resubcli 7508 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 7416	. 2 class -
2		vx	. . 3 setvar x
3		vy	. . 3 setvar y
4		cc 7111	. . 3 class CC
5	3	cv 1284	. . . . . 6 class y
6		vz	. . . . . . 7 setvar z
7	6	cv 1284	. . . . . 6 class z
8		caddc 7116	. . . . . 6 class +
9	5, 7, 8	co 5564	. . . . 5 class ( y + z )
10	2	cv 1284	. . . . 5 class x
11	9, 10	wceq 1285	. . . 4 wff ( y + z ) = x
12	11, 6, 4	crio 5519	. . 3 class ( iota_ z e. CC ( y + z ) = x )
13	2, 3, 4, 4, 12	cmpt2 5566	. 2 class ( x e. CC , y e. CC |-> ( iota_ z e. CC ( y + z ) = x ) )
14	1, 13	wceq 1285	1 wff - = ( x e. CC , y e. CC |-> ( iota_ z e. CC ( y + z ) = x ) )

This definition is referenced by:  subval  7437  subf  7447