Definition df-plusf 17845

Description: Define group addition function. Usually we will use +_g directly instead of +_𝑓, and they have the same behavior in most cases. The main advantage of +_𝑓 for any magma is that it is a guaranteed function (mgmplusf 17856), while +_g only has closure (mgmcl 17849). (Contributed by Mario Carneiro, 14-Aug-2015.)

Assertion

Ref	Expression
df-plusf	⊢ +𝑓 = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+g‘𝑔)𝑦)))

Distinct variable group:   𝑥,𝑔,𝑦

Detailed syntax breakdown of Definition df-plusf

Step	Hyp	Ref	Expression
1		cplusf 17843	. 2 class +𝑓
2		vg	. . 3 setvar 𝑔
3		cvv 3494	. . 3 class V
4		vx	. . . 4 setvar 𝑥
5		vy	. . . 4 setvar 𝑦
6	2	cv 1532	. . . . 5 class 𝑔
7		cbs 16477	. . . . 5 class Base
8	6, 7	cfv 6349	. . . 4 class (Base‘𝑔)
9	4	cv 1532	. . . . 5 class 𝑥
10	5	cv 1532	. . . . 5 class 𝑦
11		cplusg 16559	. . . . . 6 class +g
12	6, 11	cfv 6349	. . . . 5 class (+g‘𝑔)
13	9, 10, 12	co 7150	. . . 4 class (𝑥(+g‘𝑔)𝑦)
14	4, 5, 8, 8, 13	cmpo 7152	. . 3 class (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+g‘𝑔)𝑦))
15	2, 3, 14	cmpt 5138	. 2 class (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+g‘𝑔)𝑦)))
16	1, 15	wceq 1533	1 wff +𝑓 = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+g‘𝑔)𝑦)))

This definition is referenced by:  plusffval  17852