MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-subrg Structured version   Unicode version

Definition df-subrg 16989
Description: Define a subring of a ring as a set of elements that is a ring in its own right and contains the multiplicative identity.

The additional constraint is necessary because the multiplicative identity of a ring, unlike the additive identity of a ring/group or the multiplicative identity of a field, cannot be identified by a local property. Thus, it is possible for a subset of a ring to be a ring while not containing the true identity if it contains a false identity. For instance, the subset  ( ZZ  X.  {
0 } ) of  ( ZZ  X.  ZZ ) (where multiplication is component-wise) contains the false identity  <. 1 ,  0 >. which preserves every element of the subset and thus appears to be the identity of the subset, but is not the identity of the larger ring. (Contributed by Stefan O'Rear, 27-Nov-2014.)

Assertion
Ref Expression
df-subrg  |- SubRing  =  ( w  e.  Ring  |->  { s  e.  ~P ( Base `  w )  |  ( ( ws  s )  e. 
Ring  /\  ( 1r `  w )  e.  s ) } )
Distinct variable group:    w, s

Detailed syntax breakdown of Definition df-subrg
StepHypRef Expression
1 csubrg 16987 . 2  class SubRing
2 vw . . 3  setvar  w
3 crg 16771 . . 3  class  Ring
42cv 1369 . . . . . . 7  class  w
5 vs . . . . . . . 8  setvar  s
65cv 1369 . . . . . . 7  class  s
7 cress 14296 . . . . . . 7  classs
84, 6, 7co 6203 . . . . . 6  class  ( ws  s )
98, 3wcel 1758 . . . . 5  wff  ( ws  s )  e.  Ring
10 cur 16728 . . . . . . 7  class  1r
114, 10cfv 5529 . . . . . 6  class  ( 1r
`  w )
1211, 6wcel 1758 . . . . 5  wff  ( 1r
`  w )  e.  s
139, 12wa 369 . . . 4  wff  ( ( ws  s )  e.  Ring  /\  ( 1r `  w
)  e.  s )
14 cbs 14295 . . . . . 6  class  Base
154, 14cfv 5529 . . . . 5  class  ( Base `  w )
1615cpw 3971 . . . 4  class  ~P ( Base `  w )
1713, 5, 16crab 2803 . . 3  class  { s  e.  ~P ( Base `  w )  |  ( ( ws  s )  e. 
Ring  /\  ( 1r `  w )  e.  s ) }
182, 3, 17cmpt 4461 . 2  class  ( w  e.  Ring  |->  { s  e.  ~P ( Base `  w )  |  ( ( ws  s )  e. 
Ring  /\  ( 1r `  w )  e.  s ) } )
191, 18wceq 1370 1  wff SubRing  =  ( w  e.  Ring  |->  { s  e.  ~P ( Base `  w )  |  ( ( ws  s )  e. 
Ring  /\  ( 1r `  w )  e.  s ) } )
Colors of variables: wff setvar class
This definition is referenced by:  issubrg  16991
  Copyright terms: Public domain W3C validator