Definition df-bj-mgmhom 37309

Description: Define the set of magma morphisms between two magmas. If domain and codomain are semigroups, monoids, or groups, then one obtains the set of morphisms of these structures. (Contributed by BJ, 10-Feb-2022.)

Assertion

Ref	Expression
df-bj-mgmhom	⊢ Mgm⟶ = (𝑥 ∈ Mgm, 𝑦 ∈ Mgm ↦ {𝑓 ∈ ((Base‘𝑥) Set⟶ (Base‘𝑦)) ∣ ∀𝑢 ∈ (Base‘𝑥)∀𝑣 ∈ (Base‘𝑥)(𝑓‘(𝑢(+g‘𝑥)𝑣)) = ((𝑓‘𝑢)(+g‘𝑦)(𝑓‘𝑣))})

Distinct variable group:   𝑥,𝑓,𝑦,𝑢,𝑣

Detailed syntax breakdown of Definition df-bj-mgmhom

Step	Hyp	Ref	Expression
1		cmgmhom 37308	. 2 class Mgm⟶
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cmgm 18567	. . 3 class Mgm
5		vu	. . . . . . . . . 10 setvar 𝑢
6	5	cv 1541	. . . . . . . . 9 class 𝑢
7		vv	. . . . . . . . . 10 setvar 𝑣
8	7	cv 1541	. . . . . . . . 9 class 𝑣
9	2	cv 1541	. . . . . . . . . 10 class 𝑥
10		cplusg 17181	. . . . . . . . . 10 class +g
11	9, 10	cfv 6493	. . . . . . . . 9 class (+g‘𝑥)
12	6, 8, 11	co 7360	. . . . . . . 8 class (𝑢(+g‘𝑥)𝑣)
13		vf	. . . . . . . . 9 setvar 𝑓
14	13	cv 1541	. . . . . . . 8 class 𝑓
15	12, 14	cfv 6493	. . . . . . 7 class (𝑓‘(𝑢(+g‘𝑥)𝑣))
16	6, 14	cfv 6493	. . . . . . . 8 class (𝑓‘𝑢)
17	8, 14	cfv 6493	. . . . . . . 8 class (𝑓‘𝑣)
18	3	cv 1541	. . . . . . . . 9 class 𝑦
19	18, 10	cfv 6493	. . . . . . . 8 class (+g‘𝑦)
20	16, 17, 19	co 7360	. . . . . . 7 class ((𝑓‘𝑢)(+g‘𝑦)(𝑓‘𝑣))
21	15, 20	wceq 1542	. . . . . 6 wff (𝑓‘(𝑢(+g‘𝑥)𝑣)) = ((𝑓‘𝑢)(+g‘𝑦)(𝑓‘𝑣))
22		cbs 17140	. . . . . . 7 class Base
23	9, 22	cfv 6493	. . . . . 6 class (Base‘𝑥)
24	21, 7, 23	wral 3052	. . . . 5 wff ∀𝑣 ∈ (Base‘𝑥)(𝑓‘(𝑢(+g‘𝑥)𝑣)) = ((𝑓‘𝑢)(+g‘𝑦)(𝑓‘𝑣))
25	24, 5, 23	wral 3052	. . . 4 wff ∀𝑢 ∈ (Base‘𝑥)∀𝑣 ∈ (Base‘𝑥)(𝑓‘(𝑢(+g‘𝑥)𝑣)) = ((𝑓‘𝑢)(+g‘𝑦)(𝑓‘𝑣))
26	18, 22	cfv 6493	. . . . 5 class (Base‘𝑦)
27		csethom 37304	. . . . 5 class Set⟶
28	23, 26, 27	co 7360	. . . 4 class ((Base‘𝑥) Set⟶ (Base‘𝑦))
29	25, 13, 28	crab 3400	. . 3 class {𝑓 ∈ ((Base‘𝑥) Set⟶ (Base‘𝑦)) ∣ ∀𝑢 ∈ (Base‘𝑥)∀𝑣 ∈ (Base‘𝑥)(𝑓‘(𝑢(+g‘𝑥)𝑣)) = ((𝑓‘𝑢)(+g‘𝑦)(𝑓‘𝑣))}
30	2, 3, 4, 4, 29	cmpo 7362	. 2 class (𝑥 ∈ Mgm, 𝑦 ∈ Mgm ↦ {𝑓 ∈ ((Base‘𝑥) Set⟶ (Base‘𝑦)) ∣ ∀𝑢 ∈ (Base‘𝑥)∀𝑣 ∈ (Base‘𝑥)(𝑓‘(𝑢(+g‘𝑥)𝑣)) = ((𝑓‘𝑢)(+g‘𝑦)(𝑓‘𝑣))})
31	1, 30	wceq 1542	1 wff Mgm⟶ = (𝑥 ∈ Mgm, 𝑦 ∈ Mgm ↦ {𝑓 ∈ ((Base‘𝑥) Set⟶ (Base‘𝑦)) ∣ ∀𝑢 ∈ (Base‘𝑥)∀𝑣 ∈ (Base‘𝑥)(𝑓‘(𝑢(+g‘𝑥)𝑣)) = ((𝑓‘𝑢)(+g‘𝑦)(𝑓‘𝑣))})

This definition is referenced by: (None)