Theorem tngvsca

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

	Ref	Expression
Hypotheses	tngbas.t	⊢ 𝑇 = ( 𝐺 toNrmGrp 𝑁 )
	tngvsca.2	⊢ · = ( ·𝑠 ‘ 𝐺 )
Assertion	tngvsca	⊢ ( 𝑁 ∈ 𝑉 → · = ( ·𝑠 ‘ 𝑇 ) )

Proof

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