Theorem tngmulr

Description: The ring multiplication of a structure augmented with a norm. (Contributed by Mario Carneiro, 2-Oct-2015) (Revised by AV, 31-Oct-2024)

	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		mulrid	⊢ .r = Slot ( .r ‘ ndx )
4		tsetndxnmulrndx	⊢ ( TopSet ‘ ndx ) ≠ ( .r ‘ ndx )
5	4	necomi	⊢ ( .r ‘ ndx ) ≠ ( TopSet ‘ ndx )
6		dsndxnmulrndx	⊢ ( dist ‘ ndx ) ≠ ( .r ‘ ndx )
7	6	necomi	⊢ ( .r ‘ ndx ) ≠ ( dist ‘ ndx )
8	1 3 5 7	tnglem	⊢ ( 𝑁 ∈ 𝑉 → ( .r ‘ 𝐺 ) = ( .r ‘ 𝑇 ) )
9	2 8	syl5eq	⊢ ( 𝑁 ∈ 𝑉 → · = ( .r ‘ 𝑇 ) )