Theorem resv0g

Description: 0g is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018)

	Ref	Expression
Hypotheses	resvbas.1	⊢ 𝐻 = ( 𝐺 ↾v 𝐴 )
	resv0g.2	⊢ 0 = ( 0g ‘ 𝐺 )
Assertion	resv0g	⊢ ( 𝐴 ∈ 𝑉 → 0 = ( 0g ‘ 𝐻 ) )

Proof

Step	Hyp	Ref	Expression
1		resvbas.1	⊢ 𝐻 = ( 𝐺 ↾v 𝐴 )
2		resv0g.2	⊢ 0 = ( 0g ‘ 𝐺 )
3		eqidd	⊢ ( 𝐴 ∈ 𝑉 → ( Base ‘ 𝐺 ) = ( Base ‘ 𝐺 ) )
4		eqid	⊢ ( Base ‘ 𝐺 ) = ( Base ‘ 𝐺 )
5	1 4	resvbas	⊢ ( 𝐴 ∈ 𝑉 → ( Base ‘ 𝐺 ) = ( Base ‘ 𝐻 ) )
6		eqid	⊢ ( +g ‘ 𝐺 ) = ( +g ‘ 𝐺 )
7	1 6	resvplusg	⊢ ( 𝐴 ∈ 𝑉 → ( +g ‘ 𝐺 ) = ( +g ‘ 𝐻 ) )
8	7	oveqdr	⊢ ( ( 𝐴 ∈ 𝑉 ∧ ( 𝑥 ∈ ( Base ‘ 𝐺 ) ∧ 𝑦 ∈ ( Base ‘ 𝐺 ) ) ) → ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) = ( 𝑥 ( +g ‘ 𝐻 ) 𝑦 ) )
9	3 5 8	grpidpropd	⊢ ( 𝐴 ∈ 𝑉 → ( 0g ‘ 𝐺 ) = ( 0g ‘ 𝐻 ) )
10	2 9	eqtrid	⊢ ( 𝐴 ∈ 𝑉 → 0 = ( 0g ‘ 𝐻 ) )