Definition df-gdiv 27350

Description: Define a function that maps a group operation to the group's division (or subtraction) operation. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)

Assertion

Ref	Expression
df-gdiv	|- /g = ( g e. GrpOp |-> ( x e. ran g , y e. ran g |-> ( x g ( ( inv ` g ) ` y ) ) ) )

Distinct variable group:   x, g, y

Detailed syntax breakdown of Definition df-gdiv

Step	Hyp	Ref	Expression
1		cgs 27346	. 2 class /g
2		vg	. . 3 setvar g
3		cgr 27343	. . 3 class GrpOp
4		vx	. . . 4 setvar x
5		vy	. . . 4 setvar y
6	2	cv 1482	. . . . 5 class g
7	6	crn 5115	. . . 4 class ran g
8	4	cv 1482	. . . . 5 class x
9	5	cv 1482	. . . . . 6 class y
10		cgn 27345	. . . . . . 7 class inv
11	6, 10	cfv 5888	. . . . . 6 class ( inv ` g )
12	9, 11	cfv 5888	. . . . 5 class ( ( inv ` g ) ` y )
13	8, 12, 6	co 6650	. . . 4 class ( x g ( ( inv ` g ) ` y ) )
14	4, 5, 7, 7, 13	cmpt2 6652	. . 3 class ( x e. ran g , y e. ran g |-> ( x g ( ( inv ` g ) ` y ) ) )
15	2, 3, 14	cmpt 4729	. 2 class ( g e. GrpOp |-> ( x e. ran g , y e. ran g |-> ( x g ( ( inv ` g ) ` y ) ) ) )
16	1, 15	wceq 1483	1 wff /g = ( g e. GrpOp |-> ( x e. ran g , y e. ran g |-> ( x g ( ( inv ` g ) ` y ) ) ) )

This definition is referenced by:  grpodivfval  27388  vsfval  27488