Theorem rrexttps 33969

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 33964	. 2 ⊢ (𝑅 ∈ ℝExt → 𝑅 ∈ NrmRing)
2		nrgngp 24526	. 2 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ NrmGrp)
3		ngpxms 24465	. 2 ⊢ (𝑅 ∈ NrmGrp → 𝑅 ∈ ∞MetSp)
4		xmstps 24317	. 2 ⊢ (𝑅 ∈ ∞MetSp → 𝑅 ∈ TopSp)
5	1, 2, 3, 4	4syl 19	1 ⊢ (𝑅 ∈ ℝExt → 𝑅 ∈ TopSp)

Syntax hints:   → wi 4   ∈ wcel 2109  TopSpctps 22795  ∞MetSpcxms 24181  NrmGrpcngp 24441  NrmRingcnrg 24443   ℝExt crrext 33957

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 2701

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 2708  df-cleq 2721  df-clel 2803  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-xp 5637  df-co 5640  df-res 5643  df-iota 6452  df-fv 6507  df-xms 24184  df-ms 24185  df-ngp 24447  df-nrg 24449  df-rrext 33962

This theorem is referenced by: (None)