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Theorem rrextust 33286
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 2730 . . . 4 (ℤMod‘𝑅) = (ℤMod‘𝑅)
41, 2, 3isrrext 33278 . . 3 (𝑅 ∈ ℝExt ↔ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ DivRing) ∧ ((ℤMod‘𝑅) ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ (𝑅 ∈ CUnifSp ∧ (UnifSt‘𝑅) = (metUnif‘𝐷))))
54simp3bi 1145 . 2 (𝑅 ∈ ℝExt → (𝑅 ∈ CUnifSp ∧ (UnifSt‘𝑅) = (metUnif‘𝐷)))
65simprd 494 1 (𝑅 ∈ ℝExt → (UnifSt‘𝑅) = (metUnif‘𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1539  wcel 2104   × cxp 5673  cres 5677  cfv 6542  0cc0 11112  Basecbs 17148  distcds 17210  DivRingcdr 20500  metUnifcmetu 21135  ℤModczlm 21269  chrcchr 21270  UnifStcuss 23978  CUnifSpccusp 24022  NrmRingcnrg 24308  NrmModcnlm 24309   ℝExt crrext 33272
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-xp 5681  df-res 5687  df-iota 6494  df-fv 6550  df-rrext 33277
This theorem is referenced by:  rrhfe  33290  rrhcne  33291  sitgclg  33639
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