Definition df-sub 8352

Description: Define subtraction. Theorem subval 8371 shows its value (and describes how this definition works), Theorem subaddi 8466 relates it to addition, and Theorems subcli 8455 and resubcli 8442 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 8350	. 2 class -
2		vx	. . 3 setvar x
3		vy	. . 3 setvar y
4		cc 8030	. . 3 class CC
5	3	cv 1396	. . . . . 6 class y
6		vz	. . . . . . 7 setvar z
7	6	cv 1396	. . . . . 6 class z
8		caddc 8035	. . . . . 6 class +
9	5, 7, 8	co 6018	. . . . 5 class ( y + z )
10	2	cv 1396	. . . . 5 class x
11	9, 10	wceq 1397	. . . 4 wff ( y + z ) = x
12	11, 6, 4	crio 5970	. . 3 class ( iota_ z e. CC ( y + z ) = x )
13	2, 3, 4, 4, 12	cmpo 6020	. 2 class ( x e. CC , y e. CC |-> ( iota_ z e. CC ( y + z ) = x ) )
14	1, 13	wceq 1397	1 wff - = ( x e. CC , y e. CC |-> ( iota_ z e. CC ( y + z ) = x ) )

This definition is referenced by:  subval  8371  subf  8381  cndsex  14586