Definition df-sub 8342

Description: Define subtraction. Theorem subval 8361 shows its value (and describes how this definition works), Theorem subaddi 8456 relates it to addition, and Theorems subcli 8445 and resubcli 8432 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 8340	. 2 class -
2		vx	. . 3 setvar x
3		vy	. . 3 setvar y
4		cc 8020	. . 3 class CC
5	3	cv 1394	. . . . . 6 class y
6		vz	. . . . . . 7 setvar z
7	6	cv 1394	. . . . . 6 class z
8		caddc 8025	. . . . . 6 class +
9	5, 7, 8	co 6013	. . . . 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 5965	. . 3 class ( iota_ z e. CC ( y + z ) = x )
13	2, 3, 4, 4, 12	cmpo 6015	. 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  8361  subf  8371  cndsex  14557