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Theorem rrextust 32988
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 2733 . . . 4 (ℤMod‘𝑅) = (ℤMod‘𝑅)
41, 2, 3isrrext 32980 . . 3 (𝑅 ∈ ℝExt ↔ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ DivRing) ∧ ((ℤMod‘𝑅) ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ (𝑅 ∈ CUnifSp ∧ (UnifSt‘𝑅) = (metUnif‘𝐷))))
54simp3bi 1148 . 2 (𝑅 ∈ ℝExt → (𝑅 ∈ CUnifSp ∧ (UnifSt‘𝑅) = (metUnif‘𝐷)))
65simprd 497 1 (𝑅 ∈ ℝExt → (UnifSt‘𝑅) = (metUnif‘𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107   × cxp 5675  cres 5679  cfv 6544  0cc0 11110  Basecbs 17144  distcds 17206  DivRingcdr 20357  metUnifcmetu 20935  ℤModczlm 21050  chrcchr 21051  UnifStcuss 23758  CUnifSpccusp 23802  NrmRingcnrg 24088  NrmModcnlm 24089   ℝExt crrext 32974
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-xp 5683  df-res 5689  df-iota 6496  df-fv 6552  df-rrext 32979
This theorem is referenced by:  rrhfe  32992  rrhcne  32993  sitgclg  33341
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