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	⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝐾 ) )
	ablpropd.2	⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝐿 ) )
	ablpropd.3	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑥 ( +g ‘ 𝐾 ) 𝑦 ) = ( 𝑥 ( +g ‘ 𝐿 ) 𝑦 ) )
Assertion	ablpropd	⊢ ( 𝜑 → ( 𝐾 ∈ Abel ↔ 𝐿 ∈ Abel ) )

Proof

Step	Hyp	Ref	Expression
1		ablpropd.1	⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝐾 ) )
2		ablpropd.2	⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝐿 ) )
3		ablpropd.3	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑥 ( +g ‘ 𝐾 ) 𝑦 ) = ( 𝑥 ( +g ‘ 𝐿 ) 𝑦 ) )
4	1 2 3	grppropd	⊢ ( 𝜑 → ( 𝐾 ∈ Grp ↔ 𝐿 ∈ Grp ) )
5	1 2 3	cmnpropd	⊢ ( 𝜑 → ( 𝐾 ∈ CMnd ↔ 𝐿 ∈ CMnd ) )
6	4 5	anbi12d	⊢ ( 𝜑 → ( ( 𝐾 ∈ Grp ∧ 𝐾 ∈ CMnd ) ↔ ( 𝐿 ∈ Grp ∧ 𝐿 ∈ CMnd ) ) )
7		isabl	⊢ ( 𝐾 ∈ Abel ↔ ( 𝐾 ∈ Grp ∧ 𝐾 ∈ CMnd ) )
8		isabl	⊢ ( 𝐿 ∈ Abel ↔ ( 𝐿 ∈ Grp ∧ 𝐿 ∈ CMnd ) )
9	6 7 8	3bitr4g	⊢ ( 𝜑 → ( 𝐾 ∈ Abel ↔ 𝐿 ∈ Abel ) )