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Mirrors > Home > MPE Home > Th. List > dfsubrg  Unicode version 
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 componentwise) 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, 27Nov2014.) 
Ref  Expression 

dfsubrg  SubRing ↾_{s} 
Step  Hyp  Ref  Expression 

1  csubrg 15557  . 2 SubRing  
2  vw  . . 3  
3  crg 15353  . . 3  
4  2  cv 1631  . . . . . . 7 
5  vs  . . . . . . . 8  
6  5  cv 1631  . . . . . . 7 
7  cress 13165  . . . . . . 7 ↾_{s}  
8  4, 6, 7  co 5874  . . . . . 6 ↾_{s} 
9  8, 3  wcel 1696  . . . . 5 ↾_{s} 
10  cur 15355  . . . . . . 7  
11  4, 10  cfv 5271  . . . . . 6 
12  11, 6  wcel 1696  . . . . 5 
13  9, 12  wa 358  . . . 4 ↾_{s} 
14  cbs 13164  . . . . . 6  
15  4, 14  cfv 5271  . . . . 5 
16  15  cpw 3638  . . . 4 
17  13, 5, 16  crab 2560  . . 3 ↾_{s} 
18  2, 3, 17  cmpt 4093  . 2 ↾_{s} 
19  1, 18  wceq 1632  1 SubRing ↾_{s} 
Colors of variables: wff set class 
This definition is referenced by: issubrg 15561 
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