Theorem ablpropd

Description: If two structures have the same group components (properties), one is an Abelian group iff the other one is. (Contributed by NM, 6-Dec-2014)

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

Proof

Step	Hyp	Ref	Expression
1		ablpropd.1	|- ( ph -> B = ( Base ` K ) )
2		ablpropd.2	|- ( ph -> B = ( Base ` L ) )
3		ablpropd.3	|- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` K ) y ) = ( x ( +g ` L ) y ) )
4	1 2 3	grppropd	|- ( ph -> ( K e. Grp <-> L e. Grp ) )
5	1 2 3	cmnpropd	|- ( ph -> ( K e. CMnd <-> L e. CMnd ) )
6	4 5	anbi12d	|- ( ph -> ( ( K e. Grp /\ K e. CMnd ) <-> ( L e. Grp /\ L e. CMnd ) ) )
7		isabl	|- ( K e. Abel <-> ( K e. Grp /\ K e. CMnd ) )
8		isabl	|- ( L e. Abel <-> ( L e. Grp /\ L e. CMnd ) )
9	6 7 8	3bitr4g	|- ( ph -> ( K e. Abel <-> L e. Abel ) )