Theorem tngbas

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

	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		df-base	⊢ Base = Slot 1
4		1nn	⊢ 1 ∈ ℕ
5		1lt9	⊢ 1 < 9
6	1 3 4 5	tnglem	⊢ ( 𝑁 ∈ 𝑉 → ( Base ‘ 𝐺 ) = ( Base ‘ 𝑇 ) )
7	2 6	syl5eq	⊢ ( 𝑁 ∈ 𝑉 → 𝐵 = ( Base ‘ 𝑇 ) )