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Theorem rrextnrg 32451
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 2736 . . . 4 (Base‘𝑅) = (Base‘𝑅)
2 eqid 2736 . . . 4 ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))) = ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅)))
3 eqid 2736 . . . 4 (ℤMod‘𝑅) = (ℤMod‘𝑅)
41, 2, 3isrrext 32450 . . 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 1145 . 2 (𝑅 ∈ ℝExt → (𝑅 ∈ NrmRing ∧ 𝑅 ∈ DivRing))
65simpld 495 1 (𝑅 ∈ ℝExt → 𝑅 ∈ NrmRing)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106   × cxp 5629  cres 5633  cfv 6493  0cc0 11047  Basecbs 17075  distcds 17134  DivRingcdr 20170  metUnifcmetu 20772  ℤModczlm 20886  chrcchr 20887  UnifStcuss 23589  CUnifSpccusp 23633  NrmRingcnrg 23919  NrmModcnlm 23920   ℝExt crrext 32444
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-opab 5166  df-xp 5637  df-res 5643  df-iota 6445  df-fv 6501  df-rrext 32449
This theorem is referenced by:  rrexttps  32456  rrexthaus  32457  rrhfe  32462  rrhcne  32463  rrhqima  32464  sitgclg  32811
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