Theorem tngbas

Description: The base set 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 )
	tngbas.2	|- B = ( Base ` G )
Assertion	tngbas	|- ( N e. V -> B = ( Base ` T ) )

Proof

Step	Hyp	Ref	Expression
1		tngbas.t	|- T = ( G toNrmGrp N )
2		tngbas.2	|- B = ( Base ` G )
3		baseid	|- Base = Slot ( Base ` ndx )
4		tsetndxnbasendx	|- ( TopSet ` ndx ) =/= ( Base ` ndx )
5	4	necomi	|- ( Base ` ndx ) =/= ( TopSet ` ndx )
6		dsndxnbasendx	|- ( dist ` ndx ) =/= ( Base ` ndx )
7	6	necomi	|- ( Base ` ndx ) =/= ( dist ` ndx )
8	1 3 5 7	tnglem	|- ( N e. V -> ( Base ` G ) = ( Base ` T ) )
9	2 8	eqtrid	|- ( N e. V -> B = ( Base ` T ) )