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Definition df-subrg 18084
 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 18082 . 2 SubRing
2 vw . . 3
3 crg 17858 . . 3
42cv 1451 . . . . . . 7
5 vs . . . . . . . 8
65cv 1451 . . . . . . 7
7 cress 15200 . . . . . . 7 s
84, 6, 7co 6308 . . . . . 6 s
98, 3wcel 1904 . . . . 5 s
10 cur 17813 . . . . . . 7
114, 10cfv 5589 . . . . . 6
1211, 6wcel 1904 . . . . 5
139, 12wa 376 . . . 4 s
14 cbs 15199 . . . . . 6
154, 14cfv 5589 . . . . 5
1615cpw 3942 . . . 4
1713, 5, 16crab 2760 . . 3 s
182, 3, 17cmpt 4454 . 2 s
191, 18wceq 1452 1 SubRing s
 Colors of variables: wff setvar class This definition is referenced by:  issubrg  18086
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