Theorem tng0

Description: The group identity of a structure augmented with a norm. (Contributed by Mario Carneiro, 4-Oct-2015) (Revised by AV, 31-Oct-2024)

	Ref	Expression
Hypotheses	tngbas.t	⊢ 𝑇 = ( 𝐺 toNrmGrp 𝑁 )
	tng0.2	⊢ 0 = ( 0g ‘ 𝐺 )
Assertion	tng0	⊢ ( 𝑁 ∈ 𝑉 → 0 = ( 0g ‘ 𝑇 ) )

Proof

Step	Hyp	Ref	Expression
1		tngbas.t	⊢ 𝑇 = ( 𝐺 toNrmGrp 𝑁 )
2		tng0.2	⊢ 0 = ( 0g ‘ 𝐺 )
3		eqidd	⊢ ( 𝑁 ∈ 𝑉 → ( Base ‘ 𝐺 ) = ( Base ‘ 𝐺 ) )
4		eqid	⊢ ( Base ‘ 𝐺 ) = ( Base ‘ 𝐺 )
5	1 4	tngbas	⊢ ( 𝑁 ∈ 𝑉 → ( Base ‘ 𝐺 ) = ( Base ‘ 𝑇 ) )
6		eqid	⊢ ( +g ‘ 𝐺 ) = ( +g ‘ 𝐺 )
7	1 6	tngplusg	⊢ ( 𝑁 ∈ 𝑉 → ( +g ‘ 𝐺 ) = ( +g ‘ 𝑇 ) )
8	7	oveqdr	⊢ ( ( 𝑁 ∈ 𝑉 ∧ ( 𝑥 ∈ ( Base ‘ 𝐺 ) ∧ 𝑦 ∈ ( Base ‘ 𝐺 ) ) ) → ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) = ( 𝑥 ( +g ‘ 𝑇 ) 𝑦 ) )
9	3 5 8	grpidpropd	⊢ ( 𝑁 ∈ 𝑉 → ( 0g ‘ 𝐺 ) = ( 0g ‘ 𝑇 ) )
10	2 9	eqtrid	⊢ ( 𝑁 ∈ 𝑉 → 0 = ( 0g ‘ 𝑇 ) )