Theorem rrexttps 31956

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 31951	. . 3 ⊢ (𝑅 ∈ ℝExt → 𝑅 ∈ NrmRing)
2		nrgngp 23826	. . 3 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ NrmGrp)
3		ngpxms 23757	. . 3 ⊢ (𝑅 ∈ NrmGrp → 𝑅 ∈ ∞MetSp)
4	1, 2, 3	3syl 18	. 2 ⊢ (𝑅 ∈ ℝExt → 𝑅 ∈ ∞MetSp)
5		xmstps 23606	. 2 ⊢ (𝑅 ∈ ∞MetSp → 𝑅 ∈ TopSp)
6	4, 5	syl 17	1 ⊢ (𝑅 ∈ ℝExt → 𝑅 ∈ TopSp)

Syntax hints:   → wi 4   ∈ wcel 2106  TopSpctps 22081  ∞MetSpcxms 23470  NrmGrpcngp 23733  NrmRingcnrg 23735   ℝExt crrext 31944

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-xp 5595  df-co 5598  df-res 5601  df-iota 6391  df-fv 6441  df-xms 23473  df-ms 23474  df-ngp 23739  df-nrg 23741  df-rrext 31949

This theorem is referenced by: (None)