Definition df-subrg 13718

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 13716	. 2 class SubRing
2		vw	. . 3 setvar w
3		crg 13495	. . 3 class Ring
4	2	cv 1363	. . . . . . 7 class w
5		vs	. . . . . . . 8 setvar s
6	5	cv 1363	. . . . . . 7 class s
7		cress 12622	. . . . . . 7 class ↾s
8	4, 6, 7	co 5919	. . . . . 6 class ( w ↾s s )
9	8, 3	wcel 2164	. . . . 5 wff ( w ↾s s ) e. Ring
10		cur 13458	. . . . . . 7 class 1r
11	4, 10	cfv 5255	. . . . . 6 class ( 1r ` w )
12	11, 6	wcel 2164	. . . . 5 wff ( 1r ` w ) e. s
13	9, 12	wa 104	. . . 4 wff ( ( w ↾s s ) e. Ring /\ ( 1r ` w ) e. s )
14		cbs 12621	. . . . . 6 class Base
15	4, 14	cfv 5255	. . . . 5 class ( Base ` w )
16	15	cpw 3602	. . . 4 class ~P ( Base ` w )
17	13, 5, 16	crab 2476	. . 3 class { s e. ~P ( Base ` w ) | ( ( w ↾s s ) e. Ring /\ ( 1r ` w ) e. s ) }
18	2, 3, 17	cmpt 4091	. 2 class ( w e. Ring |-> { s e. ~P ( Base ` w ) | ( ( w ↾s s ) e. Ring /\ ( 1r ` w ) e. s ) } )
19	1, 18	wceq 1364	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  13720