Definition df-sub 8448

Description: Define subtraction. Theorem subval 8467 shows its value (and describes how this definition works), Theorem subaddi 8562 relates it to addition, and Theorems subcli 8551 and resubcli 8538 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 8446	. 2 class -
2		vx	. . 3 setvar x
3		vy	. . 3 setvar y
4		cc 8127	. . 3 class CC
5	3	cv 1397	. . . . . 6 class y
6		vz	. . . . . . 7 setvar z
7	6	cv 1397	. . . . . 6 class z
8		caddc 8132	. . . . . 6 class +
9	5, 7, 8	co 6052	. . . . 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 6004	. . 3 class ( iota_ z e. CC ( y + z ) = x )
13	2, 3, 4, 4, 12	cmpo 6054	. 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  8467  subf  8477  cndsex  14718