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Theorem rrextchr 29376
 Description: The ring characteristic of an extension of ℝ is zero. (Contributed by Thierry Arnoux, 2-May-2018.)
Assertion
Ref Expression
rrextchr (𝑅 ∈ ℝExt → (chr‘𝑅) = 0)

Proof of Theorem rrextchr
StepHypRef Expression
1 eqid 2610 . . . 4 (Base‘𝑅) = (Base‘𝑅)
2 eqid 2610 . . . 4 ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))) = ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅)))
3 eqid 2610 . . . 4 (ℤMod‘𝑅) = (ℤMod‘𝑅)
41, 2, 3isrrext 29372 . . 3 (𝑅 ∈ ℝExt ↔ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ DivRing) ∧ ((ℤMod‘𝑅) ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ (𝑅 ∈ CUnifSp ∧ (UnifSt‘𝑅) = (metUnif‘((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅)))))))
54simp2bi 1070 . 2 (𝑅 ∈ ℝExt → ((ℤMod‘𝑅) ∈ NrmMod ∧ (chr‘𝑅) = 0))
65simprd 478 1 (𝑅 ∈ ℝExt → (chr‘𝑅) = 0)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   × cxp 5036   ↾ cres 5040  ‘cfv 5804  0cc0 9815  Basecbs 15695  distcds 15777  DivRingcdr 18570  metUnifcmetu 19558  ℤModczlm 19668  chrcchr 19669  UnifStcuss 21867  CUnifSpccusp 21911  NrmRingcnrg 22194  NrmModcnlm 22195   ℝExt crrext 29366 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-xp 5044  df-res 5050  df-iota 5768  df-fv 5812  df-rrext 29371 This theorem is referenced by:  rrhfe  29384  rrhcne  29385  rrhqima  29386  rrh0  29387  sitgclg  29731
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