Definition df-sub 8319

Description: Define subtraction. Theorem subval 8338 shows its value (and describes how this definition works), Theorem subaddi 8433 relates it to addition, and Theorems subcli 8422 and resubcli 8409 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 8317	. 2 class -
2		vx	. . 3 setvar x
3		vy	. . 3 setvar y
4		cc 7997	. . 3 class CC
5	3	cv 1394	. . . . . 6 class y
6		vz	. . . . . . 7 setvar z
7	6	cv 1394	. . . . . 6 class z
8		caddc 8002	. . . . . 6 class +
9	5, 7, 8	co 6001	. . . . 5 class ( y + z )
10	2	cv 1394	. . . . 5 class x
11	9, 10	wceq 1395	. . . 4 wff ( y + z ) = x
12	11, 6, 4	crio 5953	. . 3 class ( iota_ z e. CC ( y + z ) = x )
13	2, 3, 4, 4, 12	cmpo 6003	. 2 class ( x e. CC , y e. CC |-> ( iota_ z e. CC ( y + z ) = x ) )
14	1, 13	wceq 1395	1 wff - = ( x e. CC , y e. CC |-> ( iota_ z e. CC ( y + z ) = x ) )

This definition is referenced by:  subval  8338  subf  8348  cndsex  14517