Definition df-sub 8043

Description: Define subtraction. Theorem subval 8062 shows its value (and describes how this definition works), Theorem subaddi 8157 relates it to addition, and Theorems subcli 8146 and resubcli 8133 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 8041	. 2 class -
2		vx	. . 3 setvar x
3		vy	. . 3 setvar y
4		cc 7725	. . 3 class CC
5	3	cv 1334	. . . . . 6 class y
6		vz	. . . . . . 7 setvar z
7	6	cv 1334	. . . . . 6 class z
8		caddc 7730	. . . . . 6 class +
9	5, 7, 8	co 5821	. . . . 5 class ( y + z )
10	2	cv 1334	. . . . 5 class x
11	9, 10	wceq 1335	. . . 4 wff ( y + z ) = x
12	11, 6, 4	crio 5776	. . 3 class ( iota_ z e. CC ( y + z ) = x )
13	2, 3, 4, 4, 12	cmpo 5823	. 2 class ( x e. CC , y e. CC |-> ( iota_ z e. CC ( y + z ) = x ) )
14	1, 13	wceq 1335	1 wff - = ( x e. CC , y e. CC |-> ( iota_ z e. CC ( y + z ) = x ) )

This definition is referenced by:  subval  8062  subf  8072