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Theorem rrextcusp 31356
Description: An extension of is a complete uniform space. (Contributed by Thierry Arnoux, 2-May-2018.)
Assertion
Ref Expression
rrextcusp (𝑅 ∈ ℝExt → 𝑅 ∈ CUnifSp)

Proof of Theorem rrextcusp
StepHypRef Expression
1 eqid 2798 . . . 4 (Base‘𝑅) = (Base‘𝑅)
2 eqid 2798 . . . 4 ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))) = ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅)))
3 eqid 2798 . . . 4 (ℤMod‘𝑅) = (ℤMod‘𝑅)
41, 2, 3isrrext 31351 . . 3 (𝑅 ∈ ℝExt ↔ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ DivRing) ∧ ((ℤMod‘𝑅) ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ (𝑅 ∈ CUnifSp ∧ (UnifSt‘𝑅) = (metUnif‘((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅)))))))
54simp3bi 1144 . 2 (𝑅 ∈ ℝExt → (𝑅 ∈ CUnifSp ∧ (UnifSt‘𝑅) = (metUnif‘((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))))))
65simpld 498 1 (𝑅 ∈ ℝExt → 𝑅 ∈ CUnifSp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2111   × cxp 5517  cres 5521  cfv 6324  0cc0 10526  Basecbs 16475  distcds 16566  DivRingcdr 19495  metUnifcmetu 20082  ℤModczlm 20194  chrcchr 20195  UnifStcuss 22859  CUnifSpccusp 22903  NrmRingcnrg 23186  NrmModcnlm 23187   ℝExt crrext 31345
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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-rab 3115  df-v 3443  df-un 3886  df-in 3888  df-ss 3898  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-xp 5525  df-res 5531  df-iota 6283  df-fv 6332  df-rrext 31350
This theorem is referenced by:  rrhfe  31363  rrhcne  31364  sitgclg  31710
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