Theorem rrexttps 34305

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 34300	. 2 ⊢ (𝑅 ∈ ℝExt → 𝑅 ∈ NrmRing)
2		nrgngp 24724	. 2 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ NrmGrp)
3		ngpxms 24663	. 2 ⊢ (𝑅 ∈ NrmGrp → 𝑅 ∈ ∞MetSp)
4		xmstps 24515	. 2 ⊢ (𝑅 ∈ ∞MetSp → 𝑅 ∈ TopSp)
5	1, 2, 3, 4	4syl 19	1 ⊢ (𝑅 ∈ ℝExt → 𝑅 ∈ TopSp)

Syntax hints:   → wi 4   ∈ wcel 2144  TopSpctps 22994  ∞MetSpcxms 24379  NrmGrpcngp 24639  NrmRingcnrg 24641   ℝExt crrext 34293

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5103  df-opab 5165  df-xp 5655  df-co 5658  df-res 5661  df-iota 6479  df-fv 6531  df-xms 24382  df-ms 24383  df-ngp 24645  df-nrg 24647  df-rrext 34298

This theorem is referenced by: (None)