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

Definition df-subrg 15858
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 15856 . 2  class SubRing
2 vw . . 3  set  w
3 crg 15652 . . 3  class  Ring
42cv 1651 . . . . . . 7  class  w
5 vs . . . . . . . 8  set  s
65cv 1651 . . . . . . 7  class  s
7 cress 13462 . . . . . . 7  classs
84, 6, 7co 6073 . . . . . 6  class  ( ws  s )
98, 3wcel 1725 . . . . 5  wff  ( ws  s )  e.  Ring
10 cur 15654 . . . . . . 7  class  1r
114, 10cfv 5446 . . . . . 6  class  ( 1r
`  w )
1211, 6wcel 1725 . . . . 5  wff  ( 1r
`  w )  e.  s
139, 12wa 359 . . . 4  wff  ( ( ws  s )  e.  Ring  /\  ( 1r `  w
)  e.  s )
14 cbs 13461 . . . . . 6  class  Base
154, 14cfv 5446 . . . . 5  class  ( Base `  w )
1615cpw 3791 . . . 4  class  ~P ( Base `  w )
1713, 5, 16crab 2701 . . 3  class  { s  e.  ~P ( Base `  w )  |  ( ( ws  s )  e. 
Ring  /\  ( 1r `  w )  e.  s ) }
182, 3, 17cmpt 4258 . 2  class  ( w  e.  Ring  |->  { s  e.  ~P ( Base `  w )  |  ( ( ws  s )  e. 
Ring  /\  ( 1r `  w )  e.  s ) } )
191, 18wceq 1652 1  wff SubRing  =  ( w  e.  Ring  |->  { s  e.  ~P ( Base `  w )  |  ( ( ws  s )  e. 
Ring  /\  ( 1r `  w )  e.  s ) } )
Colors of variables: wff set class
This definition is referenced by:  issubrg  15860
  Copyright terms: Public domain W3C validator