Definition df-sub 8492

Description: Define subtraction. Theorem subval 8511 shows its value (and describes how this definition works), Theorem subaddi 8606 relates it to addition, and Theorems subcli 8595 and resubcli 8582 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 8490	. 2 class -
2		vx	. . 3 setvar x
3		vy	. . 3 setvar y
4		cc 8170	. . 3 class CC
5	3	cv 1401	. . . . . 6 class y
6		vz	. . . . . . 7 setvar z
7	6	cv 1401	. . . . . 6 class z
8		caddc 8175	. . . . . 6 class +
9	5, 7, 8	co 6078	. . . . 5 class ( y + z )
10	2	cv 1401	. . . . 5 class x
11	9, 10	wceq 1402	. . . 4 wff ( y + z ) = x
12	11, 6, 4	crio 6030	. . 3 class ( iota_ z e. CC ( y + z ) = x )
13	2, 3, 4, 4, 12	cmpo 6080	. 2 class ( x e. CC , y e. CC |-> ( iota_ z e. CC ( y + z ) = x ) )
14	1, 13	wceq 1402	1 wff - = ( x e. CC , y e. CC |-> ( iota_ z e. CC ( y + z ) = x ) )

This definition is referenced by:  subval  8511  subf  8521  cndsex  14865