Definition df-submnd 12684

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 e. Mnd |-> { t e. ~P ( Base ` s ) | ( ( 0g ` s ) e. t /\ A. x e. t A. y e. t ( x ( +g ` s ) y ) e. t ) } )

Distinct variable group:   t, s, x, y

Detailed syntax breakdown of Definition df-submnd

Step	Hyp	Ref	Expression
1		csubmnd 12682	. 2 class SubMnd
2		vs	. . 3 setvar s
3		cmnd 12652	. . 3 class Mnd
4	2	cv 1347	. . . . . . 7 class s
5		c0g 12596	. . . . . . 7 class 0g
6	4, 5	cfv 5198	. . . . . 6 class ( 0g ` s )
7		vt	. . . . . . 7 setvar t
8	7	cv 1347	. . . . . 6 class t
9	6, 8	wcel 2141	. . . . 5 wff ( 0g ` s ) e. t
10		vx	. . . . . . . . . 10 setvar x
11	10	cv 1347	. . . . . . . . 9 class x
12		vy	. . . . . . . . . 10 setvar y
13	12	cv 1347	. . . . . . . . 9 class y
14		cplusg 12480	. . . . . . . . . 10 class +g
15	4, 14	cfv 5198	. . . . . . . . 9 class ( +g ` s )
16	11, 13, 15	co 5853	. . . . . . . 8 class ( x ( +g ` s ) y )
17	16, 8	wcel 2141	. . . . . . 7 wff ( x ( +g ` s ) y ) e. t
18	17, 12, 8	wral 2448	. . . . . 6 wff A. y e. t ( x ( +g ` s ) y ) e. t
19	18, 10, 8	wral 2448	. . . . 5 wff A. x e. t A. y e. t ( x ( +g ` s ) y ) e. t
20	9, 19	wa 103	. . . 4 wff ( ( 0g ` s ) e. t /\ A. x e. t A. y e. t ( x ( +g ` s ) y ) e. t )
21		cbs 12416	. . . . . 6 class Base
22	4, 21	cfv 5198	. . . . 5 class ( Base ` s )
23	22	cpw 3566	. . . 4 class ~P ( Base ` s )
24	20, 7, 23	crab 2452	. . 3 class { t e. ~P ( Base ` s ) | ( ( 0g ` s ) e. t /\ A. x e. t A. y e. t ( x ( +g ` s ) y ) e. t ) }
25	2, 3, 24	cmpt 4050	. 2 class ( s e. Mnd |-> { t e. ~P ( Base ` s ) | ( ( 0g ` s ) e. t /\ A. x e. t A. y e. t ( x ( +g ` s ) y ) e. t ) } )
26	1, 25	wceq 1348	1 wff SubMnd = ( s e. Mnd |-> { t e. ~P ( Base ` s ) | ( ( 0g ` s ) e. t /\ A. x e. t A. y e. t ( x ( +g ` s ) y ) e. t ) } )

This definition is referenced by:  submrcl  12694  issubm  12695