Theorem tngsca

Description: The scalar ring of a structure augmented with a norm. (Contributed by Mario Carneiro, 2-Oct-2015)

	Ref	Expression
Hypotheses	tngbas.t	⊢ 𝑇 = ( 𝐺 toNrmGrp 𝑁 )
	tngsca.2	⊢ 𝐹 = ( Scalar ‘ 𝐺 )
Assertion	tngsca	⊢ ( 𝑁 ∈ 𝑉 → 𝐹 = ( Scalar ‘ 𝑇 ) )

Proof

Step	Hyp	Ref	Expression
1		tngbas.t	⊢ 𝑇 = ( 𝐺 toNrmGrp 𝑁 )
2		tngsca.2	⊢ 𝐹 = ( Scalar ‘ 𝐺 )
3		df-sca	⊢ Scalar = Slot 5
4		5nn	⊢ 5 ∈ ℕ
5		5lt9	⊢ 5 < 9
6	1 3 4 5	tnglem	⊢ ( 𝑁 ∈ 𝑉 → ( Scalar ‘ 𝐺 ) = ( Scalar ‘ 𝑇 ) )
7	2 6	syl5eq	⊢ ( 𝑁 ∈ 𝑉 → 𝐹 = ( Scalar ‘ 𝑇 ) )