Theorem rrexttps 34004

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 33999	. 2 ⊢ (𝑅 ∈ ℝExt → 𝑅 ∈ NrmRing)
2		nrgngp 24556	. 2 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ NrmGrp)
3		ngpxms 24495	. 2 ⊢ (𝑅 ∈ NrmGrp → 𝑅 ∈ ∞MetSp)
4		xmstps 24347	. 2 ⊢ (𝑅 ∈ ∞MetSp → 𝑅 ∈ TopSp)
5	1, 2, 3, 4	4syl 19	1 ⊢ (𝑅 ∈ ℝExt → 𝑅 ∈ TopSp)

Syntax hints:   → wi 4   ∈ wcel 2109  TopSpctps 22825  ∞MetSpcxms 24211  NrmGrpcngp 24471  NrmRingcnrg 24473   ℝExt crrext 33992

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-rab 3412  df-v 3457  df-dif 3925  df-un 3927  df-in 3929  df-ss 3939  df-nul 4305  df-if 4497  df-sn 4598  df-pr 4600  df-op 4604  df-uni 4880  df-br 5116  df-opab 5178  df-xp 5652  df-co 5655  df-res 5658  df-iota 6472  df-fv 6527  df-xms 24214  df-ms 24215  df-ngp 24477  df-nrg 24479  df-rrext 33997

This theorem is referenced by: (None)