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Definition df-subrg 17999
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 17997 . 2  class SubRing
2 vw . . 3  setvar  w
3 crg 17773 . . 3  class  Ring
42cv 1437 . . . . . . 7  class  w
5 vs . . . . . . . 8  setvar  s
65cv 1437 . . . . . . 7  class  s
7 cress 15115 . . . . . . 7  classs
84, 6, 7co 6303 . . . . . 6  class  ( ws  s )
98, 3wcel 1869 . . . . 5  wff  ( ws  s )  e.  Ring
10 cur 17728 . . . . . . 7  class  1r
114, 10cfv 5599 . . . . . 6  class  ( 1r
`  w )
1211, 6wcel 1869 . . . . 5  wff  ( 1r
`  w )  e.  s
139, 12wa 371 . . . 4  wff  ( ( ws  s )  e.  Ring  /\  ( 1r `  w
)  e.  s )
14 cbs 15114 . . . . . 6  class  Base
154, 14cfv 5599 . . . . 5  class  ( Base `  w )
1615cpw 3980 . . . 4  class  ~P ( Base `  w )
1713, 5, 16crab 2780 . . 3  class  { s  e.  ~P ( Base `  w )  |  ( ( ws  s )  e. 
Ring  /\  ( 1r `  w )  e.  s ) }
182, 3, 17cmpt 4480 . 2  class  ( w  e.  Ring  |->  { s  e.  ~P ( Base `  w )  |  ( ( ws  s )  e. 
Ring  /\  ( 1r `  w )  e.  s ) } )
191, 18wceq 1438 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  18001
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