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Metamath Proof Explorer

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 ) )