Theorem rrexttps 31965

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

Assertion

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

Proof of Theorem rrexttps

Step	Hyp	Ref	Expression
1		rrextnrg 31960	. . 3 ⊢ (𝑅 ∈ ℝExt → 𝑅 ∈ NrmRing)
2		nrgngp 23837	. . 3 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ NrmGrp)
3		ngpxms 23768	. . 3 ⊢ (𝑅 ∈ NrmGrp → 𝑅 ∈ ∞MetSp)
4	1, 2, 3	3syl 18	. 2 ⊢ (𝑅 ∈ ℝExt → 𝑅 ∈ ∞MetSp)
5		xmstps 23617	. 2 ⊢ (𝑅 ∈ ∞MetSp → 𝑅 ∈ TopSp)
6	4, 5	syl 17	1 ⊢ (𝑅 ∈ ℝExt → 𝑅 ∈ TopSp)

Syntax hints:   → wi 4   ∈ wcel 2110  TopSpctps 22092  ∞MetSpcxms 23481  NrmGrpcngp 23744  NrmRingcnrg 23746   ℝExt crrext 31953

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-ext 2711

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-rab 3075  df-v 3433  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-br 5080  df-opab 5142  df-xp 5596  df-co 5599  df-res 5602  df-iota 6390  df-fv 6440  df-xms 23484  df-ms 23485  df-ngp 23750  df-nrg 23752  df-rrext 31958

This theorem is referenced by: (None)