Theorem tnggrpr

Description: If a structure equipped with a norm is a normed group, the structure itself must be a group. (Contributed by AV, 15-Oct-2021)

	Ref	Expression
Hypothesis	tngngp3.t	⊢ 𝑇 = ( 𝐺 toNrmGrp 𝑁 )
Assertion	tnggrpr	⊢ ( ( 𝑁 ∈ 𝑉 ∧ 𝑇 ∈ NrmGrp ) → 𝐺 ∈ Grp )

Proof

Step	Hyp	Ref	Expression
1		tngngp3.t	⊢ 𝑇 = ( 𝐺 toNrmGrp 𝑁 )
2		eqid	⊢ ( Base ‘ 𝐺 ) = ( Base ‘ 𝐺 )
3	1 2	tngbas	⊢ ( 𝑁 ∈ 𝑉 → ( Base ‘ 𝐺 ) = ( Base ‘ 𝑇 ) )
4		eqidd	⊢ ( 𝑁 ∈ 𝑉 → ( Base ‘ 𝐺 ) = ( Base ‘ 𝐺 ) )
5		eqid	⊢ ( +g ‘ 𝐺 ) = ( +g ‘ 𝐺 )
6	1 5	tngplusg	⊢ ( 𝑁 ∈ 𝑉 → ( +g ‘ 𝐺 ) = ( +g ‘ 𝑇 ) )
7	6	eqcomd	⊢ ( 𝑁 ∈ 𝑉 → ( +g ‘ 𝑇 ) = ( +g ‘ 𝐺 ) )
8	7	oveqdr	⊢ ( ( 𝑁 ∈ 𝑉 ∧ ( 𝑥 ∈ ( Base ‘ 𝐺 ) ∧ 𝑦 ∈ ( Base ‘ 𝐺 ) ) ) → ( 𝑥 ( +g ‘ 𝑇 ) 𝑦 ) = ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) )
9	3 4 8	grppropd	⊢ ( 𝑁 ∈ 𝑉 → ( 𝑇 ∈ Grp ↔ 𝐺 ∈ Grp ) )
10	9	biimpd	⊢ ( 𝑁 ∈ 𝑉 → ( 𝑇 ∈ Grp → 𝐺 ∈ Grp ) )
11		ngpgrp	⊢ ( 𝑇 ∈ NrmGrp → 𝑇 ∈ Grp )
12	10 11	impel	⊢ ( ( 𝑁 ∈ 𝑉 ∧ 𝑇 ∈ NrmGrp ) → 𝐺 ∈ Grp )