Definition df-subrg 18778

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 componentwise) 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 ) | ( ( w ↾s s ) e. Ring /\ ( 1r ` w ) e. s ) } )

Distinct variable group:   w, s

Detailed syntax breakdown of Definition df-subrg

Step	Hyp	Ref	Expression
1		csubrg 18776	. 2 class SubRing
2		vw	. . 3 setvar w
3		crg 18547	. . 3 class Ring
4	2	cv 1482	. . . . . . 7 class w
5		vs	. . . . . . . 8 setvar s
6	5	cv 1482	. . . . . . 7 class s
7		cress 15858	. . . . . . 7 class ↾s
8	4, 6, 7	co 6650	. . . . . 6 class ( w ↾s s )
9	8, 3	wcel 1990	. . . . 5 wff ( w ↾s s ) e. Ring
10		cur 18501	. . . . . . 7 class 1r
11	4, 10	cfv 5888	. . . . . 6 class ( 1r ` w )
12	11, 6	wcel 1990	. . . . 5 wff ( 1r ` w ) e. s
13	9, 12	wa 384	. . . 4 wff ( ( w ↾s s ) e. Ring /\ ( 1r ` w ) e. s )
14		cbs 15857	. . . . . 6 class Base
15	4, 14	cfv 5888	. . . . 5 class ( Base ` w )
16	15	cpw 4158	. . . 4 class ~P ( Base ` w )
17	13, 5, 16	crab 2916	. . 3 class { s e. ~P ( Base ` w ) | ( ( w ↾s s ) e. Ring /\ ( 1r ` w ) e. s ) }
18	2, 3, 17	cmpt 4729	. 2 class ( w e. Ring |-> { s e. ~P ( Base ` w ) | ( ( w ↾s s ) e. Ring /\ ( 1r ` w ) e. s ) } )
19	1, 18	wceq 1483	1 wff SubRing = ( w e. Ring |-> { s e. ~P ( Base ` w ) | ( ( w ↾s s ) e. Ring /\ ( 1r ` w ) e. s ) } )

This definition is referenced by:  issubrg  18780