Definition df-gdiv 30016

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 ∈ GrpOp ↦ (𝑥 ∈ ran 𝑔, 𝑦 ∈ ran 𝑔 ↦ (𝑥𝑔((inv‘𝑔)‘𝑦))))

Distinct variable group:   𝑥,𝑔,𝑦

Detailed syntax breakdown of Definition df-gdiv

Step	Hyp	Ref	Expression
1		cgs 30012	. 2 class /𝑔
2		vg	. . 3 setvar 𝑔
3		cgr 30009	. . 3 class GrpOp
4		vx	. . . 4 setvar 𝑥
5		vy	. . . 4 setvar 𝑦
6	2	cv 1538	. . . . 5 class 𝑔
7	6	crn 5676	. . . 4 class ran 𝑔
8	4	cv 1538	. . . . 5 class 𝑥
9	5	cv 1538	. . . . . 6 class 𝑦
10		cgn 30011	. . . . . . 7 class inv
11	6, 10	cfv 6542	. . . . . 6 class (inv‘𝑔)
12	9, 11	cfv 6542	. . . . 5 class ((inv‘𝑔)‘𝑦)
13	8, 12, 6	co 7411	. . . 4 class (𝑥𝑔((inv‘𝑔)‘𝑦))
14	4, 5, 7, 7, 13	cmpo 7413	. . 3 class (𝑥 ∈ ran 𝑔, 𝑦 ∈ ran 𝑔 ↦ (𝑥𝑔((inv‘𝑔)‘𝑦)))
15	2, 3, 14	cmpt 5230	. 2 class (𝑔 ∈ GrpOp ↦ (𝑥 ∈ ran 𝑔, 𝑦 ∈ ran 𝑔 ↦ (𝑥𝑔((inv‘𝑔)‘𝑦))))
16	1, 15	wceq 1539	1 wff /𝑔 = (𝑔 ∈ GrpOp ↦ (𝑥 ∈ ran 𝑔, 𝑦 ∈ ran 𝑔 ↦ (𝑥𝑔((inv‘𝑔)‘𝑦))))

This definition is referenced by:  grpodivfval  30054  vsfval  30153