Theorem rrexttps 33948

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 33943	. 2 ⊢ (𝑅 ∈ ℝExt → 𝑅 ∈ NrmRing)
2		nrgngp 24638	. 2 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ NrmGrp)
3		ngpxms 24577	. 2 ⊢ (𝑅 ∈ NrmGrp → 𝑅 ∈ ∞MetSp)
4		xmstps 24427	. 2 ⊢ (𝑅 ∈ ∞MetSp → 𝑅 ∈ TopSp)
5	1, 2, 3, 4	4syl 19	1 ⊢ (𝑅 ∈ ℝExt → 𝑅 ∈ TopSp)

Syntax hints:   → wi 4   ∈ wcel 2107  TopSpctps 22905  ∞MetSpcxms 24291  NrmGrpcngp 24553  NrmRingcnrg 24555   ℝExt crrext 33936

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-rab 3421  df-v 3466  df-dif 3936  df-un 3938  df-in 3940  df-ss 3950  df-nul 4316  df-if 4508  df-sn 4609  df-pr 4611  df-op 4615  df-uni 4890  df-br 5126  df-opab 5188  df-xp 5673  df-co 5676  df-res 5679  df-iota 6495  df-fv 6550  df-xms 24294  df-ms 24295  df-ngp 24559  df-nrg 24561  df-rrext 33941

This theorem is referenced by: (None)