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Metamath Proof Explorer

Theorem pwsabl

Description: The structure power on an Abelian group is Abelian. (Contributed by Mario Carneiro, 21-Jan-2015)

Ref Expression
Hypothesis pwscmn.y ⊢ 𝑌 = ( 𝑅 ↑s 𝐼 )
Assertion pwsabl ( ( 𝑅 ∈ Abel ∧ 𝐼 ∈ 𝑉 ) → 𝑌 ∈ Abel )

Proof

Step Hyp Ref Expression
1 pwscmn.y ⊢ 𝑌 = ( 𝑅 ↑s 𝐼 )
2 eqid ⊢ ( Scalar ‘ 𝑅 ) = ( Scalar ‘ 𝑅 )
3 1 2 pwsval ⊢ ( ( 𝑅 ∈ Abel ∧ 𝐼 ∈ 𝑉 ) → 𝑌 = ( ( Scalar ‘ 𝑅 ) Xs ( 𝐼 × { 𝑅 } ) ) )
4 eqid ⊢ ( ( Scalar ‘ 𝑅 ) Xs ( 𝐼 × { 𝑅 } ) ) = ( ( Scalar ‘ 𝑅 ) Xs ( 𝐼 × { 𝑅 } ) )
5 simpr ⊢ ( ( 𝑅 ∈ Abel ∧ 𝐼 ∈ 𝑉 ) → 𝐼 ∈ 𝑉 )
6 fvexd ⊢ ( ( 𝑅 ∈ Abel ∧ 𝐼 ∈ 𝑉 ) → ( Scalar ‘ 𝑅 ) ∈ V )
7 fconst6g ⊢ ( 𝑅 ∈ Abel → ( 𝐼 × { 𝑅 } ) : 𝐼 ⟶ Abel )
8 7 adantr ⊢ ( ( 𝑅 ∈ Abel ∧ 𝐼 ∈ 𝑉 ) → ( 𝐼 × { 𝑅 } ) : 𝐼 ⟶ Abel )
9 4 5 6 8 prdsabld ⊢ ( ( 𝑅 ∈ Abel ∧ 𝐼 ∈ 𝑉 ) → ( ( Scalar ‘ 𝑅 ) Xs ( 𝐼 × { 𝑅 } ) ) ∈ Abel )
10 3 9 eqeltrd ⊢ ( ( 𝑅 ∈ Abel ∧ 𝐼 ∈ 𝑉 ) → 𝑌 ∈ Abel )