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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 ⊢ ( 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 ) )