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	$⊢ (R \in Abel \land I \in V) \to Y \in Abel$

Step	Hyp	Ref	Expression
1		pwscmn.y	$⊢ Y = R ↑_{𝑠} I$
2		eqid	$⊢ Scalar (R) = Scalar (R)$
3	1 2	pwsval	$⊢ (R \in Abel \land I \in V) \to Y = Scalar (R) ⨉_{𝑠} (I \times \{R\})$
4		eqid	$⊢ Scalar (R) ⨉_{𝑠} (I \times \{R\}) = Scalar (R) ⨉_{𝑠} (I \times \{R\})$
5		simpr	$⊢ (R \in Abel \land I \in V) \to I \in V$
6		fvexd	$⊢ (R \in Abel \land I \in V) \to Scalar (R) \in V$
7		fconst6g	$⊢ R \in Abel \to (I \times \{R\}) : I ⟶ Abel$
8	7	adantr	$⊢ (R \in Abel \land I \in V) \to (I \times \{R\}) : I ⟶ Abel$
9	4 5 6 8	prdsabld	$⊢ (R \in Abel \land I \in V) \to Scalar (R) ⨉_{𝑠} (I \times \{R\}) \in Abel$
10	3 9	eqeltrd	$⊢ (R \in Abel \land I \in V) \to Y \in Abel$

Metamath Proof Explorer