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Definition df-submnd 14777
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
StepHypRef Expression
1 csubmnd 14775 . 2  class SubMnd
2 vs . . 3  set  s
3 cmnd 14722 . . 3  class  Mnd
42cv 1653 . . . . . . 7  class  s
5 c0g 13761 . . . . . . 7  class  0g
64, 5cfv 5489 . . . . . 6  class  ( 0g
`  s )
7 vt . . . . . . 7  set  t
87cv 1653 . . . . . 6  class  t
96, 8wcel 1728 . . . . 5  wff  ( 0g
`  s )  e.  t
10 vx . . . . . . . . . 10  set  x
1110cv 1653 . . . . . . . . 9  class  x
12 vy . . . . . . . . . 10  set  y
1312cv 1653 . . . . . . . . 9  class  y
14 cplusg 13567 . . . . . . . . . 10  class  +g
154, 14cfv 5489 . . . . . . . . 9  class  ( +g  `  s )
1611, 13, 15co 6117 . . . . . . . 8  class  ( x ( +g  `  s
) y )
1716, 8wcel 1728 . . . . . . 7  wff  ( x ( +g  `  s
) y )  e.  t
1817, 12, 8wral 2712 . . . . . 6  wff  A. y  e.  t  ( x
( +g  `  s ) y )  e.  t
1918, 10, 8wral 2712 . . . . 5  wff  A. x  e.  t  A. y  e.  t  ( x
( +g  `  s ) y )  e.  t
209, 19wa 360 . . . 4  wff  ( ( 0g `  s )  e.  t  /\  A. x  e.  t  A. y  e.  t  (
x ( +g  `  s
) y )  e.  t )
21 cbs 13507 . . . . . 6  class  Base
224, 21cfv 5489 . . . . 5  class  ( Base `  s )
2322cpw 3828 . . . 4  class  ~P ( Base `  s )
2420, 7, 23crab 2716 . . 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 ) }
252, 3, 24cmpt 4297 . 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 ) } )
261, 25wceq 1654 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 ) } )
Colors of variables: wff set class
This definition is referenced by:  submrcl  14785  issubm  14786
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