Definition df-submnd 18797

Description: A submonoid is a subset of a monoid which contains the identity and is closed under the operation. Such subsets are themselves monoids with the same identity. (Contributed by Mario Carneiro, 7-Mar-2015.)

Assertion

Ref	Expression
df-submnd	⊢ SubMnd = (𝑠 ∈ Mnd ↦ {𝑡 ∈ 𝒫 (Base‘𝑠) ∣ ((0g‘𝑠) ∈ 𝑡 ∧ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥(+g‘𝑠)𝑦) ∈ 𝑡)})

Distinct variable group:   𝑡,𝑠,𝑥,𝑦

Detailed syntax breakdown of Definition df-submnd

Step	Hyp	Ref	Expression
1		csubmnd 18795	. 2 class SubMnd
2		vs	. . 3 setvar 𝑠
3		cmnd 18747	. . 3 class Mnd
4	2	cv 1539	. . . . . . 7 class 𝑠
5		c0g 17484	. . . . . . 7 class 0g
6	4, 5	cfv 6561	. . . . . 6 class (0g‘𝑠)
7		vt	. . . . . . 7 setvar 𝑡
8	7	cv 1539	. . . . . 6 class 𝑡
9	6, 8	wcel 2108	. . . . 5 wff (0g‘𝑠) ∈ 𝑡
10		vx	. . . . . . . . . 10 setvar 𝑥
11	10	cv 1539	. . . . . . . . 9 class 𝑥
12		vy	. . . . . . . . . 10 setvar 𝑦
13	12	cv 1539	. . . . . . . . 9 class 𝑦
14		cplusg 17297	. . . . . . . . . 10 class +g
15	4, 14	cfv 6561	. . . . . . . . 9 class (+g‘𝑠)
16	11, 13, 15	co 7431	. . . . . . . 8 class (𝑥(+g‘𝑠)𝑦)
17	16, 8	wcel 2108	. . . . . . 7 wff (𝑥(+g‘𝑠)𝑦) ∈ 𝑡
18	17, 12, 8	wral 3061	. . . . . 6 wff ∀𝑦 ∈ 𝑡 (𝑥(+g‘𝑠)𝑦) ∈ 𝑡
19	18, 10, 8	wral 3061	. . . . 5 wff ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥(+g‘𝑠)𝑦) ∈ 𝑡
20	9, 19	wa 395	. . . 4 wff ((0g‘𝑠) ∈ 𝑡 ∧ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥(+g‘𝑠)𝑦) ∈ 𝑡)
21		cbs 17247	. . . . . 6 class Base
22	4, 21	cfv 6561	. . . . 5 class (Base‘𝑠)
23	22	cpw 4600	. . . 4 class 𝒫 (Base‘𝑠)
24	20, 7, 23	crab 3436	. . 3 class {𝑡 ∈ 𝒫 (Base‘𝑠) ∣ ((0g‘𝑠) ∈ 𝑡 ∧ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥(+g‘𝑠)𝑦) ∈ 𝑡)}
25	2, 3, 24	cmpt 5225	. 2 class (𝑠 ∈ Mnd ↦ {𝑡 ∈ 𝒫 (Base‘𝑠) ∣ ((0g‘𝑠) ∈ 𝑡 ∧ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥(+g‘𝑠)𝑦) ∈ 𝑡)})
26	1, 25	wceq 1540	1 wff SubMnd = (𝑠 ∈ Mnd ↦ {𝑡 ∈ 𝒫 (Base‘𝑠) ∣ ((0g‘𝑠) ∈ 𝑡 ∧ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥(+g‘𝑠)𝑦) ∈ 𝑡)})

This definition is referenced by:  submrcl  18815  issubm  18816