Definition df-subrg 13300

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 13298	. 2 class SubRing
2		vw	. . 3 setvar w
3		crg 13132	. . 3 class Ring
4	2	cv 1352	. . . . . . 7 class w
5		vs	. . . . . . . 8 setvar s
6	5	cv 1352	. . . . . . 7 class s
7		cress 12457	. . . . . . 7 class ↾s
8	4, 6, 7	co 5874	. . . . . 6 class ( w ↾s s )
9	8, 3	wcel 2148	. . . . 5 wff ( w ↾s s ) e. Ring
10		cur 13095	. . . . . . 7 class 1r
11	4, 10	cfv 5216	. . . . . 6 class ( 1r ` w )
12	11, 6	wcel 2148	. . . . 5 wff ( 1r ` w ) e. s
13	9, 12	wa 104	. . . 4 wff ( ( w ↾s s ) e. Ring /\ ( 1r ` w ) e. s )
14		cbs 12456	. . . . . 6 class Base
15	4, 14	cfv 5216	. . . . 5 class ( Base ` w )
16	15	cpw 3575	. . . 4 class ~P ( Base ` w )
17	13, 5, 16	crab 2459	. . 3 class { s e. ~P ( Base ` w ) | ( ( w ↾s s ) e. Ring /\ ( 1r ` w ) e. s ) }
18	2, 3, 17	cmpt 4064	. 2 class ( w e. Ring |-> { s e. ~P ( Base ` w ) | ( ( w ↾s s ) e. Ring /\ ( 1r ` w ) e. s ) } )
19	1, 18	wceq 1353	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  13302