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Definition df-subrg 15559
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 15557 . 2  class SubRing
2 vw . . 3  set  w
3 crg 15353 . . 3  class  Ring
42cv 1631 . . . . . . 7  class  w
5 vs . . . . . . . 8  set  s
65cv 1631 . . . . . . 7  class  s
7 cress 13165 . . . . . . 7  classs
84, 6, 7co 5874 . . . . . 6  class  ( ws  s )
98, 3wcel 1696 . . . . 5  wff  ( ws  s )  e.  Ring
10 cur 15355 . . . . . . 7  class  1r
114, 10cfv 5271 . . . . . 6  class  ( 1r
`  w )
1211, 6wcel 1696 . . . . 5  wff  ( 1r
`  w )  e.  s
139, 12wa 358 . . . 4  wff  ( ( ws  s )  e.  Ring  /\  ( 1r `  w
)  e.  s )
14 cbs 13164 . . . . . 6  class  Base
154, 14cfv 5271 . . . . 5  class  ( Base `  w )
1615cpw 3638 . . . 4  class  ~P ( Base `  w )
1713, 5, 16crab 2560 . . 3  class  { s  e.  ~P ( Base `  w )  |  ( ( ws  s )  e. 
Ring  /\  ( 1r `  w )  e.  s ) }
182, 3, 17cmpt 4093 . 2  class  ( w  e.  Ring  |->  { s  e.  ~P ( Base `  w )  |  ( ( ws  s )  e. 
Ring  /\  ( 1r `  w )  e.  s ) } )
191, 18wceq 1632 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  15561
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