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Theorem rrextust 32256
Description: The uniformity of an extension of is the uniformity generated by its distance. (Contributed by Thierry Arnoux, 2-May-2018.)
Hypotheses
Ref Expression
rrextust.b 𝐵 = (Base‘𝑅)
rrextust.d 𝐷 = ((dist‘𝑅) ↾ (𝐵 × 𝐵))
Assertion
Ref Expression
rrextust (𝑅 ∈ ℝExt → (UnifSt‘𝑅) = (metUnif‘𝐷))

Proof of Theorem rrextust
StepHypRef Expression
1 rrextust.b . . . 4 𝐵 = (Base‘𝑅)
2 rrextust.d . . . 4 𝐷 = ((dist‘𝑅) ↾ (𝐵 × 𝐵))
3 eqid 2736 . . . 4 (ℤMod‘𝑅) = (ℤMod‘𝑅)
41, 2, 3isrrext 32248 . . 3 (𝑅 ∈ ℝExt ↔ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ DivRing) ∧ ((ℤMod‘𝑅) ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ (𝑅 ∈ CUnifSp ∧ (UnifSt‘𝑅) = (metUnif‘𝐷))))
54simp3bi 1146 . 2 (𝑅 ∈ ℝExt → (𝑅 ∈ CUnifSp ∧ (UnifSt‘𝑅) = (metUnif‘𝐷)))
65simprd 496 1 (𝑅 ∈ ℝExt → (UnifSt‘𝑅) = (metUnif‘𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1540  wcel 2105   × cxp 5618  cres 5622  cfv 6479  0cc0 10972  Basecbs 17009  distcds 17068  DivRingcdr 20093  metUnifcmetu 20694  ℤModczlm 20808  chrcchr 20809  UnifStcuss 23511  CUnifSpccusp 23555  NrmRingcnrg 23841  NrmModcnlm 23842   ℝExt crrext 32242
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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-rab 3404  df-v 3443  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-br 5093  df-opab 5155  df-xp 5626  df-res 5632  df-iota 6431  df-fv 6487  df-rrext 32247
This theorem is referenced by:  rrhfe  32260  rrhcne  32261  sitgclg  32609
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