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

Theorem rrextnrg 33964
Description: An extension of is a normed ring. (Contributed by Thierry Arnoux, 2-May-2018.)
Assertion
Ref Expression
rrextnrg (𝑅 ∈ ℝExt → 𝑅 ∈ NrmRing)

Proof of Theorem rrextnrg
StepHypRef Expression
1 eqid 2735 . . . 4 (Base‘𝑅) = (Base‘𝑅)
2 eqid 2735 . . . 4 ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))) = ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅)))
3 eqid 2735 . . . 4 (ℤMod‘𝑅) = (ℤMod‘𝑅)
41, 2, 3isrrext 33963 . . 3 (𝑅 ∈ ℝExt ↔ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ DivRing) ∧ ((ℤMod‘𝑅) ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ (𝑅 ∈ CUnifSp ∧ (UnifSt‘𝑅) = (metUnif‘((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅)))))))
54simp1bi 1144 . 2 (𝑅 ∈ ℝExt → (𝑅 ∈ NrmRing ∧ 𝑅 ∈ DivRing))
65simpld 494 1 (𝑅 ∈ ℝExt → 𝑅 ∈ NrmRing)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106   × cxp 5687  cres 5691  cfv 6563  0cc0 11153  Basecbs 17245  distcds 17307  DivRingcdr 20746  metUnifcmetu 21373  ℤModczlm 21529  chrcchr 21530  UnifStcuss 24278  CUnifSpccusp 24322  NrmRingcnrg 24608  NrmModcnlm 24609   ℝExt crrext 33957
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-xp 5695  df-res 5701  df-iota 6516  df-fv 6571  df-rrext 33962
This theorem is referenced by:  rrexttps  33969  rrexthaus  33970  rrhfe  33975  rrhcne  33976  rrhqima  33977  sitgclg  34324
  Copyright terms: Public domain W3C validator