Theorem tngmulr

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

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

Proof

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