Theorem idompropd

Description: If two structures have the same components (properties), one is a integral domain iff the other one is. See also domnpropd . (Contributed by Thierry Arnoux, 13-Oct-2025)

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

Proof

Step	Hyp	Ref	Expression
1		domnpropd.1	|- ( ph -> B = ( Base ` K ) )
2		domnpropd.2	|- ( ph -> B = ( Base ` L ) )
3		domnpropd.3	|- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` K ) y ) = ( x ( +g ` L ) y ) )
4		domnpropd.4	|- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( .r ` K ) y ) = ( x ( .r ` L ) y ) )
5	1 2 3 4	crngpropd	|- ( ph -> ( K e. CRing <-> L e. CRing ) )
6	1 2 3 4	domnpropd	|- ( ph -> ( K e. Domn <-> L e. Domn ) )
7	5 6	anbi12d	|- ( ph -> ( ( K e. CRing /\ K e. Domn ) <-> ( L e. CRing /\ L e. Domn ) ) )
8		isidom	|- ( K e. IDomn <-> ( K e. CRing /\ K e. Domn ) )
9		isidom	|- ( L e. IDomn <-> ( L e. CRing /\ L e. Domn ) )
10	7 8 9	3bitr4g	|- ( ph -> ( K e. IDomn <-> L e. IDomn ) )