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	|- .x. = ( .r ` G )
Assertion	tngmulr	|- ( N e. V -> .x. = ( .r ` T ) )

Proof

Step	Hyp	Ref	Expression
1		tngbas.t	|- T = ( G toNrmGrp N )
2		tngmulr.2	|- .x. = ( .r ` G )
3		df-mulr	|- .r = Slot 3
4		3nn	|- 3 e. NN
5		3lt9	|- 3 < 9
6	1 3 4 5	tnglem	|- ( N e. V -> ( .r ` G ) = ( .r ` T ) )
7	2 6	syl5eq	|- ( N e. V -> .x. = ( .r ` T ) )