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

Definition df-subrg 15871
 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 of (where multiplication is component-wise) contains the false identity 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 s
Distinct variable group:   ,

Detailed syntax breakdown of Definition df-subrg
StepHypRef Expression
1 csubrg 15869 . 2 SubRing
2 vw . . 3
3 crg 15665 . . 3
42cv 1652 . . . . . . 7
5 vs . . . . . . . 8
65cv 1652 . . . . . . 7
7 cress 13475 . . . . . . 7 s
84, 6, 7co 6084 . . . . . 6 s
98, 3wcel 1726 . . . . 5 s
10 cur 15667 . . . . . . 7
114, 10cfv 5457 . . . . . 6
1211, 6wcel 1726 . . . . 5
139, 12wa 360 . . . 4 s
14 cbs 13474 . . . . . 6
154, 14cfv 5457 . . . . 5
1615cpw 3801 . . . 4
1713, 5, 16crab 2711 . . 3 s
182, 3, 17cmpt 4269 . 2 s
191, 18wceq 1653 1 SubRing s
 Colors of variables: wff set class This definition is referenced by:  issubrg  15873
 Copyright terms: Public domain W3C validator