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 ∈ 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 ∈ 𝑉 → ( Base ‘ 𝐺 ) = ( Base ‘ 𝑇 ) )
9	2 8	eqtrid	⊢ ( 𝑁 ∈ 𝑉 → 𝐵 = ( Base ‘ 𝑇 ) )