Theorem tngsca

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

	Ref	Expression
Hypotheses	tngbas.t	|- T = ( G toNrmGrp N )
	tngsca.2	|- F = ( Scalar ` G )
Assertion	tngsca	|- ( N e. V -> F = ( Scalar ` T ) )

Proof

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