Theorem nzrpropd

Description: If two structures have the same components (properties), one is a nonzero ring iff the other one is. (Contributed by SN, 21-Jun-2025)

	Ref	Expression
Hypotheses	nzrpropd.1	|- ( ph -> B = ( Base ` K ) )
	nzrpropd.2	|- ( ph -> B = ( Base ` L ) )
	nzrpropd.3	|- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` K ) y ) = ( x ( +g ` L ) y ) )
	nzrpropd.4	|- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( .r ` K ) y ) = ( x ( .r ` L ) y ) )
Assertion	nzrpropd	|- ( ph -> ( K e. NzRing <-> L e. NzRing ) )

Proof

Step	Hyp	Ref	Expression
1		nzrpropd.1	|- ( ph -> B = ( Base ` K ) )
2		nzrpropd.2	|- ( ph -> B = ( Base ` L ) )
3		nzrpropd.3	|- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` K ) y ) = ( x ( +g ` L ) y ) )
4		nzrpropd.4	|- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( .r ` K ) y ) = ( x ( .r ` L ) y ) )
5	1 2 3 4	ringpropd	|- ( ph -> ( K e. Ring <-> L e. Ring ) )
6	1 2 4	rngidpropd	|- ( ph -> ( 1r ` K ) = ( 1r ` L ) )
7	1 2 3	grpidpropd	|- ( ph -> ( 0g ` K ) = ( 0g ` L ) )
8	6 7	neeq12d	|- ( ph -> ( ( 1r ` K ) =/= ( 0g ` K ) <-> ( 1r ` L ) =/= ( 0g ` L ) ) )
9	5 8	anbi12d	|- ( ph -> ( ( K e. Ring /\ ( 1r ` K ) =/= ( 0g ` K ) ) <-> ( L e. Ring /\ ( 1r ` L ) =/= ( 0g ` L ) ) ) )
10		eqid	|- ( 1r ` K ) = ( 1r ` K )
11		eqid	|- ( 0g ` K ) = ( 0g ` K )
12	10 11	isnzr	|- ( K e. NzRing <-> ( K e. Ring /\ ( 1r ` K ) =/= ( 0g ` K ) ) )
13		eqid	|- ( 1r ` L ) = ( 1r ` L )
14		eqid	|- ( 0g ` L ) = ( 0g ` L )
15	13 14	isnzr	|- ( L e. NzRing <-> ( L e. Ring /\ ( 1r ` L ) =/= ( 0g ` L ) ) )
16	9 12 15	3bitr4g	|- ( ph -> ( K e. NzRing <-> L e. NzRing ) )