Definition df-bj-mgmhom 37112

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 37111	. 2 class Mgm⟶
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cmgm 18571	. . 3 class Mgm
5		vu	. . . . . . . . . 10 setvar 𝑢
6	5	cv 1539	. . . . . . . . 9 class 𝑢
7		vv	. . . . . . . . . 10 setvar 𝑣
8	7	cv 1539	. . . . . . . . 9 class 𝑣
9	2	cv 1539	. . . . . . . . . 10 class 𝑥
10		cplusg 17226	. . . . . . . . . 10 class +g
11	9, 10	cfv 6519	. . . . . . . . 9 class (+g‘𝑥)
12	6, 8, 11	co 7394	. . . . . . . 8 class (𝑢(+g‘𝑥)𝑣)
13		vf	. . . . . . . . 9 setvar 𝑓
14	13	cv 1539	. . . . . . . 8 class 𝑓
15	12, 14	cfv 6519	. . . . . . 7 class (𝑓‘(𝑢(+g‘𝑥)𝑣))
16	6, 14	cfv 6519	. . . . . . . 8 class (𝑓‘𝑢)
17	8, 14	cfv 6519	. . . . . . . 8 class (𝑓‘𝑣)
18	3	cv 1539	. . . . . . . . 9 class 𝑦
19	18, 10	cfv 6519	. . . . . . . 8 class (+g‘𝑦)
20	16, 17, 19	co 7394	. . . . . . 7 class ((𝑓‘𝑢)(+g‘𝑦)(𝑓‘𝑣))
21	15, 20	wceq 1540	. . . . . 6 wff (𝑓‘(𝑢(+g‘𝑥)𝑣)) = ((𝑓‘𝑢)(+g‘𝑦)(𝑓‘𝑣))
22		cbs 17185	. . . . . . 7 class Base
23	9, 22	cfv 6519	. . . . . 6 class (Base‘𝑥)
24	21, 7, 23	wral 3046	. . . . 5 wff ∀𝑣 ∈ (Base‘𝑥)(𝑓‘(𝑢(+g‘𝑥)𝑣)) = ((𝑓‘𝑢)(+g‘𝑦)(𝑓‘𝑣))
25	24, 5, 23	wral 3046	. . . 4 wff ∀𝑢 ∈ (Base‘𝑥)∀𝑣 ∈ (Base‘𝑥)(𝑓‘(𝑢(+g‘𝑥)𝑣)) = ((𝑓‘𝑢)(+g‘𝑦)(𝑓‘𝑣))
26	18, 22	cfv 6519	. . . . 5 class (Base‘𝑦)
27		csethom 37107	. . . . 5 class Set⟶
28	23, 26, 27	co 7394	. . . 4 class ((Base‘𝑥) Set⟶ (Base‘𝑦))
29	25, 13, 28	crab 3411	. . 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 7396	. 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 1540	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)