Theorem resvcmn

Description: Scalar restriction preserves commutative monoids. (Contributed by Thierry Arnoux, 6-Sep-2018)

	Ref	Expression
Hypothesis	resvbas.1	⊢ 𝐻 = ( 𝐺 ↾v 𝐴 )
Assertion	resvcmn	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐺 ∈ CMnd ↔ 𝐻 ∈ CMnd ) )

Proof

Step	Hyp	Ref	Expression
1		resvbas.1	⊢ 𝐻 = ( 𝐺 ↾v 𝐴 )
2		eqidd	⊢ ( 𝐴 ∈ 𝑉 → ( Base ‘ 𝐺 ) = ( Base ‘ 𝐺 ) )
3		eqid	⊢ ( Base ‘ 𝐺 ) = ( Base ‘ 𝐺 )
4	1 3	resvbas	⊢ ( 𝐴 ∈ 𝑉 → ( Base ‘ 𝐺 ) = ( Base ‘ 𝐻 ) )
5		eqid	⊢ ( +g ‘ 𝐺 ) = ( +g ‘ 𝐺 )
6	1 5	resvplusg	⊢ ( 𝐴 ∈ 𝑉 → ( +g ‘ 𝐺 ) = ( +g ‘ 𝐻 ) )
7	6	oveqdr	⊢ ( ( 𝐴 ∈ 𝑉 ∧ ( 𝑥 ∈ ( Base ‘ 𝐺 ) ∧ 𝑦 ∈ ( Base ‘ 𝐺 ) ) ) → ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) = ( 𝑥 ( +g ‘ 𝐻 ) 𝑦 ) )
8	2 4 7	cmnpropd	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐺 ∈ CMnd ↔ 𝐻 ∈ CMnd ) )