Mathbox for Thierry Arnoux < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rrexttps Structured version   Visualization version   GIF version

Theorem rrexttps 31251
 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
StepHypRef Expression
1 rrextnrg 31246 . . 3 (𝑅 ∈ ℝExt → 𝑅 ∈ NrmRing)
2 nrgngp 23243 . . 3 (𝑅 ∈ NrmRing → 𝑅 ∈ NrmGrp)
3 ngpxms 23182 . . 3 (𝑅 ∈ NrmGrp → 𝑅 ∈ ∞MetSp)
41, 2, 33syl 18 . 2 (𝑅 ∈ ℝExt → 𝑅 ∈ ∞MetSp)
5 xmstps 23035 . 2 (𝑅 ∈ ∞MetSp → 𝑅 ∈ TopSp)
64, 5syl 17 1 (𝑅 ∈ ℝExt → 𝑅 ∈ TopSp)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2114  TopSpctps 21512  ∞MetSpcxms 22899  NrmGrpcngp 23159  NrmRingcnrg 23161   ℝExt crrext 31239 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-rab 3134  df-v 3472  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4811  df-br 5039  df-opab 5101  df-xp 5533  df-co 5536  df-res 5539  df-iota 6286  df-fv 6335  df-xms 22902  df-ms 22903  df-ngp 23165  df-nrg 23167  df-rrext 31244 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator