Definition df-sub 8411

Description: Define subtraction. Theorem subval 8430 shows its value (and describes how this definition works), Theorem subaddi 8525 relates it to addition, and Theorems subcli 8514 and resubcli 8501 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 8409	. 2 class -
2		vx	. . 3 setvar x
3		vy	. . 3 setvar y
4		cc 8090	. . 3 class CC
5	3	cv 1397	. . . . . 6 class y
6		vz	. . . . . . 7 setvar z
7	6	cv 1397	. . . . . 6 class z
8		caddc 8095	. . . . . 6 class +
9	5, 7, 8	co 6028	. . . . 5 class ( y + z )
10	2	cv 1397	. . . . 5 class x
11	9, 10	wceq 1398	. . . 4 wff ( y + z ) = x
12	11, 6, 4	crio 5980	. . 3 class ( iota_ z e. CC ( y + z ) = x )
13	2, 3, 4, 4, 12	cmpo 6030	. 2 class ( x e. CC , y e. CC |-> ( iota_ z e. CC ( y + z ) = x ) )
14	1, 13	wceq 1398	1 wff - = ( x e. CC , y e. CC |-> ( iota_ z e. CC ( y + z ) = x ) )

This definition is referenced by:  subval  8430  subf  8440  cndsex  14649