Definition df-bj-mgmhom 37885

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 37884	. 2 class Mgm⟶
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cmgm 18734	. . 3 class Mgm
5		vu	. . . . . . . . . 10 setvar 𝑢
6	5	cv 1569	. . . . . . . . 9 class 𝑢
7		vv	. . . . . . . . . 10 setvar 𝑣
8	7	cv 1569	. . . . . . . . 9 class 𝑣
9	2	cv 1569	. . . . . . . . . 10 class 𝑥
10		cplusg 17348	. . . . . . . . . 10 class +g
11	9, 10	cfv 6537	. . . . . . . . 9 class (+g‘𝑥)
12	6, 8, 11	co 7417	. . . . . . . 8 class (𝑢(+g‘𝑥)𝑣)
13		vf	. . . . . . . . 9 setvar 𝑓
14	13	cv 1569	. . . . . . . 8 class 𝑓
15	12, 14	cfv 6537	. . . . . . 7 class (𝑓‘(𝑢(+g‘𝑥)𝑣))
16	6, 14	cfv 6537	. . . . . . . 8 class (𝑓‘𝑢)
17	8, 14	cfv 6537	. . . . . . . 8 class (𝑓‘𝑣)
18	3	cv 1569	. . . . . . . . 9 class 𝑦
19	18, 10	cfv 6537	. . . . . . . 8 class (+g‘𝑦)
20	16, 17, 19	co 7417	. . . . . . 7 class ((𝑓‘𝑢)(+g‘𝑦)(𝑓‘𝑣))
21	15, 20	wceq 1570	. . . . . 6 wff (𝑓‘(𝑢(+g‘𝑥)𝑣)) = ((𝑓‘𝑢)(+g‘𝑦)(𝑓‘𝑣))
22		cbs 17307	. . . . . . 7 class Base
23	9, 22	cfv 6537	. . . . . 6 class (Base‘𝑥)
24	21, 7, 23	wral 3078	. . . . 5 wff ∀𝑣 ∈ (Base‘𝑥)(𝑓‘(𝑢(+g‘𝑥)𝑣)) = ((𝑓‘𝑢)(+g‘𝑦)(𝑓‘𝑣))
25	24, 5, 23	wral 3078	. . . 4 wff ∀𝑢 ∈ (Base‘𝑥)∀𝑣 ∈ (Base‘𝑥)(𝑓‘(𝑢(+g‘𝑥)𝑣)) = ((𝑓‘𝑢)(+g‘𝑦)(𝑓‘𝑣))
26	18, 22	cfv 6537	. . . . 5 class (Base‘𝑦)
27		csethom 37880	. . . . 5 class Set⟶
28	23, 26, 27	co 7417	. . . 4 class ((Base‘𝑥) Set⟶ (Base‘𝑦))
29	25, 13, 28	crab 3414	. . 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 7419	. 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 1570	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 used by: (None)