Theorem tnglvec

Description: Augmenting a structure with a norm conserves left vector spaces. (Contributed by Thierry Arnoux, 20-May-2023)

	Ref	Expression
Hypothesis	tnglvec.t	⊢ 𝑇 = ( 𝐺 toNrmGrp 𝑁 )
Assertion	tnglvec	⊢ ( 𝑁 ∈ 𝑉 → ( 𝐺 ∈ LVec ↔ 𝑇 ∈ LVec ) )

Proof

Step	Hyp	Ref	Expression
1		tnglvec.t	⊢ 𝑇 = ( 𝐺 toNrmGrp 𝑁 )
2		eqidd	⊢ ( 𝑁 ∈ 𝑉 → ( Base ‘ 𝐺 ) = ( Base ‘ 𝐺 ) )
3		eqid	⊢ ( Base ‘ 𝐺 ) = ( Base ‘ 𝐺 )
4	1 3	tngbas	⊢ ( 𝑁 ∈ 𝑉 → ( Base ‘ 𝐺 ) = ( Base ‘ 𝑇 ) )
5		eqid	⊢ ( +g ‘ 𝐺 ) = ( +g ‘ 𝐺 )
6	1 5	tngplusg	⊢ ( 𝑁 ∈ 𝑉 → ( +g ‘ 𝐺 ) = ( +g ‘ 𝑇 ) )
7	6	oveqdr	⊢ ( ( 𝑁 ∈ 𝑉 ∧ ( 𝑥 ∈ ( Base ‘ 𝐺 ) ∧ 𝑦 ∈ ( Base ‘ 𝐺 ) ) ) → ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) = ( 𝑥 ( +g ‘ 𝑇 ) 𝑦 ) )
8		eqidd	⊢ ( 𝑁 ∈ 𝑉 → ( Scalar ‘ 𝐺 ) = ( Scalar ‘ 𝐺 ) )
9		eqid	⊢ ( Scalar ‘ 𝐺 ) = ( Scalar ‘ 𝐺 )
10	1 9	tngsca	⊢ ( 𝑁 ∈ 𝑉 → ( Scalar ‘ 𝐺 ) = ( Scalar ‘ 𝑇 ) )
11		eqid	⊢ ( Base ‘ ( Scalar ‘ 𝐺 ) ) = ( Base ‘ ( Scalar ‘ 𝐺 ) )
12		eqid	⊢ ( ·𝑠 ‘ 𝐺 ) = ( ·𝑠 ‘ 𝐺 )
13	1 12	tngvsca	⊢ ( 𝑁 ∈ 𝑉 → ( ·𝑠 ‘ 𝐺 ) = ( ·𝑠 ‘ 𝑇 ) )
14	13	oveqdr	⊢ ( ( 𝑁 ∈ 𝑉 ∧ ( 𝑥 ∈ ( Base ‘ ( Scalar ‘ 𝐺 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐺 ) ) ) → ( 𝑥 ( ·𝑠 ‘ 𝐺 ) 𝑦 ) = ( 𝑥 ( ·𝑠 ‘ 𝑇 ) 𝑦 ) )
15	2 4 7 8 10 11 14	lvecpropd	⊢ ( 𝑁 ∈ 𝑉 → ( 𝐺 ∈ LVec ↔ 𝑇 ∈ LVec ) )