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Theorem rrextcusp 31145
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 2818 . . . 4 (Base‘𝑅) = (Base‘𝑅)
2 eqid 2818 . . . 4 ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))) = ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅)))
3 eqid 2818 . . . 4 (ℤMod‘𝑅) = (ℤMod‘𝑅)
41, 2, 3isrrext 31140 . . 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 1139 . 2 (𝑅 ∈ ℝExt → (𝑅 ∈ CUnifSp ∧ (UnifSt‘𝑅) = (metUnif‘((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))))))
65simpld 495 1 (𝑅 ∈ ℝExt → 𝑅 ∈ CUnifSp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105   × cxp 5546  cres 5550  cfv 6348  0cc0 10525  Basecbs 16471  distcds 16562  DivRingcdr 19431  metUnifcmetu 20464  ℤModczlm 20576  chrcchr 20577  UnifStcuss 22789  CUnifSpccusp 22833  NrmRingcnrg 23116  NrmModcnlm 23117   ℝExt crrext 31134
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-xp 5554  df-res 5560  df-iota 6307  df-fv 6356  df-rrext 31139
This theorem is referenced by:  rrhfe  31152  rrhcne  31153  sitgclg  31499
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