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

Definition df-subrg 15543
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 15541 . 2  class SubRing
2 vw . . 3  set  w
3 crg 15337 . . 3  class  Ring
42cv 1622 . . . . . . 7  class  w
5 vs . . . . . . . 8  set  s
65cv 1622 . . . . . . 7  class  s
7 cress 13149 . . . . . . 7  classs
84, 6, 7co 5858 . . . . . 6  class  ( ws  s )
98, 3wcel 1684 . . . . 5  wff  ( ws  s )  e.  Ring
10 cur 15339 . . . . . . 7  class  1r
114, 10cfv 5255 . . . . . 6  class  ( 1r
`  w )
1211, 6wcel 1684 . . . . 5  wff  ( 1r
`  w )  e.  s
139, 12wa 358 . . . 4  wff  ( ( ws  s )  e.  Ring  /\  ( 1r `  w
)  e.  s )
14 cbs 13148 . . . . . 6  class  Base
154, 14cfv 5255 . . . . 5  class  ( Base `  w )
1615cpw 3625 . . . 4  class  ~P ( Base `  w )
1713, 5, 16crab 2547 . . 3  class  { s  e.  ~P ( Base `  w )  |  ( ( ws  s )  e. 
Ring  /\  ( 1r `  w )  e.  s ) }
182, 3, 17cmpt 4077 . 2  class  ( w  e.  Ring  |->  { s  e.  ~P ( Base `  w )  |  ( ( ws  s )  e. 
Ring  /\  ( 1r `  w )  e.  s ) } )
191, 18wceq 1623 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  15545
  Copyright terms: Public domain W3C validator