Theorem rrexttps 33952

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 33947	. 2 ⊢ (𝑅 ∈ ℝExt → 𝑅 ∈ NrmRing)
2		nrgngp 24704	. 2 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ NrmGrp)
3		ngpxms 24635	. 2 ⊢ (𝑅 ∈ NrmGrp → 𝑅 ∈ ∞MetSp)
4		xmstps 24484	. 2 ⊢ (𝑅 ∈ ∞MetSp → 𝑅 ∈ TopSp)
5	1, 2, 3, 4	4syl 19	1 ⊢ (𝑅 ∈ ℝExt → 𝑅 ∈ TopSp)

Syntax hints:   → wi 4   ∈ wcel 2108  TopSpctps 22959  ∞MetSpcxms 24348  NrmGrpcngp 24611  NrmRingcnrg 24613   ℝExt crrext 33940

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-xp 5706  df-co 5709  df-res 5712  df-iota 6525  df-fv 6581  df-xms 24351  df-ms 24352  df-ngp 24617  df-nrg 24619  df-rrext 33945

This theorem is referenced by: (None)