Theorem nrgtgp

Description: A normed ring is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015)

	Ref	Expression
Assertion	nrgtgp	⊢ ( 𝑅 ∈ NrmRing → 𝑅 ∈ TopGrp )

Proof

Step	Hyp	Ref	Expression
1		nrgngp	⊢ ( 𝑅 ∈ NrmRing → 𝑅 ∈ NrmGrp )
2		nrgring	⊢ ( 𝑅 ∈ NrmRing → 𝑅 ∈ Ring )
3		ringabl	⊢ ( 𝑅 ∈ Ring → 𝑅 ∈ Abel )
4	2 3	syl	⊢ ( 𝑅 ∈ NrmRing → 𝑅 ∈ Abel )
5		ngptgp	⊢ ( ( 𝑅 ∈ NrmGrp ∧ 𝑅 ∈ Abel ) → 𝑅 ∈ TopGrp )
6	1 4 5	syl2anc	⊢ ( 𝑅 ∈ NrmRing → 𝑅 ∈ TopGrp )