Definition df-sub 8330

Description: Define subtraction. Theorem subval 8349 shows its value (and describes how this definition works), Theorem subaddi 8444 relates it to addition, and Theorems subcli 8433 and resubcli 8420 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 8328	. 2 class -
2		vx	. . 3 setvar x
3		vy	. . 3 setvar y
4		cc 8008	. . 3 class CC
5	3	cv 1394	. . . . . 6 class y
6		vz	. . . . . . 7 setvar z
7	6	cv 1394	. . . . . 6 class z
8		caddc 8013	. . . . . 6 class +
9	5, 7, 8	co 6007	. . . . 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 5959	. . 3 class ( iota_ z e. CC ( y + z ) = x )
13	2, 3, 4, 4, 12	cmpo 6009	. 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  8349  subf  8359  cndsex  14533