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Definition df-subrg 16841
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 16839 . 2  class SubRing
2 vw . . 3  setvar  w
3 crg 16633 . . 3  class  Ring
42cv 1368 . . . . . . 7  class  w
5 vs . . . . . . . 8  setvar  s
65cv 1368 . . . . . . 7  class  s
7 cress 14167 . . . . . . 7  classs
84, 6, 7co 6086 . . . . . 6  class  ( ws  s )
98, 3wcel 1756 . . . . 5  wff  ( ws  s )  e.  Ring
10 cur 16591 . . . . . . 7  class  1r
114, 10cfv 5413 . . . . . 6  class  ( 1r
`  w )
1211, 6wcel 1756 . . . . 5  wff  ( 1r
`  w )  e.  s
139, 12wa 369 . . . 4  wff  ( ( ws  s )  e.  Ring  /\  ( 1r `  w
)  e.  s )
14 cbs 14166 . . . . . 6  class  Base
154, 14cfv 5413 . . . . 5  class  ( Base `  w )
1615cpw 3855 . . . 4  class  ~P ( Base `  w )
1713, 5, 16crab 2714 . . 3  class  { s  e.  ~P ( Base `  w )  |  ( ( ws  s )  e. 
Ring  /\  ( 1r `  w )  e.  s ) }
182, 3, 17cmpt 4345 . 2  class  ( w  e.  Ring  |->  { s  e.  ~P ( Base `  w )  |  ( ( ws  s )  e. 
Ring  /\  ( 1r `  w )  e.  s ) } )
191, 18wceq 1369 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  16843
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