Theorem tngplusg

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

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

Proof

Step	Hyp	Ref	Expression
1		tngbas.t	⊢ 𝑇 = ( 𝐺 toNrmGrp 𝑁 )
2		tngplusg.2	⊢ + = ( +g ‘ 𝐺 )
3		df-plusg	⊢ +g = Slot 2
4		2nn	⊢ 2 ∈ ℕ
5		2lt9	⊢ 2 < 9
6	1 3 4 5	tnglem	⊢ ( 𝑁 ∈ 𝑉 → ( +g ‘ 𝐺 ) = ( +g ‘ 𝑇 ) )
7	2 6	syl5eq	⊢ ( 𝑁 ∈ 𝑉 → + = ( +g ‘ 𝑇 ) )