Theorem rrexttps

Description: An extension of RR is a topological space. (Contributed by Thierry Arnoux, 7-Sep-2018)

	Ref	Expression
Assertion	rrexttps	⊢ ( 𝑅 ∈ ℝExt → 𝑅 ∈ TopSp )

Proof

Step	Hyp	Ref	Expression
1		rrextnrg	⊢ ( 𝑅 ∈ ℝExt → 𝑅 ∈ NrmRing )
2		nrgngp	⊢ ( 𝑅 ∈ NrmRing → 𝑅 ∈ NrmGrp )
3		ngpxms	⊢ ( 𝑅 ∈ NrmGrp → 𝑅 ∈ ∞MetSp )
4	1 2 3	3syl	⊢ ( 𝑅 ∈ ℝExt → 𝑅 ∈ ∞MetSp )
5		xmstps	⊢ ( 𝑅 ∈ ∞MetSp → 𝑅 ∈ TopSp )
6	4 5	syl	⊢ ( 𝑅 ∈ ℝExt → 𝑅 ∈ TopSp )