Theorem rrexttps 31856

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 31851	. . 3 ⊢ (𝑅 ∈ ℝExt → 𝑅 ∈ NrmRing)
2		nrgngp 23732	. . 3 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ NrmGrp)
3		ngpxms 23663	. . 3 ⊢ (𝑅 ∈ NrmGrp → 𝑅 ∈ ∞MetSp)
4	1, 2, 3	3syl 18	. 2 ⊢ (𝑅 ∈ ℝExt → 𝑅 ∈ ∞MetSp)
5		xmstps 23514	. 2 ⊢ (𝑅 ∈ ∞MetSp → 𝑅 ∈ TopSp)
6	4, 5	syl 17	1 ⊢ (𝑅 ∈ ℝExt → 𝑅 ∈ TopSp)

Syntax hints:   → wi 4   ∈ wcel 2108  TopSpctps 21989  ∞MetSpcxms 23378  NrmGrpcngp 23639  NrmRingcnrg 23641   ℝExt crrext 31844

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-xp 5586  df-co 5589  df-res 5592  df-iota 6376  df-fv 6426  df-xms 23381  df-ms 23382  df-ngp 23645  df-nrg 23647  df-rrext 31849

This theorem is referenced by: (None)