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	$$⊢T=G\mathrm{toNrmGrp}N$$
	tngmulr.2	$$⊢\underset{\dot{}}{·}={\cdot }_G$$
Assertion	tngmulr	$$⊢N\in V\to \underset{\dot{}}{·}={\cdot }_T$$

Proof

Step	Hyp	Ref	Expression
1		tngbas.t	$$⊢T=G\mathrm{toNrmGrp}N$$
2		tngmulr.2	$$⊢\underset{\dot{}}{·}={\cdot }_G$$
3		mulridx	$$⊢{\cdot }_{𝑟}=\mathrm{Slot}{\cdot }_{\mathrm{ndx}}$$
4		tsetndxnmulrndx	$$⊢\mathrm{TopSet}\left(\mathrm{ndx}\right)\ne {\cdot }_{\mathrm{ndx}}$$
5	4	necomi	$$⊢{\cdot }_{\mathrm{ndx}}\ne \mathrm{TopSet}\left(\mathrm{ndx}\right)$$
6		dsndxnmulrndx	$$⊢\mathrm{dist}\left(\mathrm{ndx}\right)\ne {\cdot }_{\mathrm{ndx}}$$
7	6	necomi	$$⊢{\cdot }_{\mathrm{ndx}}\ne \mathrm{dist}\left(\mathrm{ndx}\right)$$
8	1 3 5 7	tnglem	$$⊢N\in V\to {\cdot }_G={\cdot }_T$$
9	2 8	eqtrid	$$⊢N\in V\to \underset{\dot{}}{·}={\cdot }_T$$