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	|- .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		mulridx	|- .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 e. V -> ( .r ` G ) = ( .r ` T ) )
9	2 8	eqtrid	|- ( N e. V -> .x. = ( .r ` T ) )