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Theorem rrextcusp 30389
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 2771 . . . 4 (Base‘𝑅) = (Base‘𝑅)
2 eqid 2771 . . . 4 ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))) = ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅)))
3 eqid 2771 . . . 4 (ℤMod‘𝑅) = (ℤMod‘𝑅)
41, 2, 3isrrext 30384 . . 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 1141 . 2 (𝑅 ∈ ℝExt → (𝑅 ∈ CUnifSp ∧ (UnifSt‘𝑅) = (metUnif‘((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))))))
65simpld 482 1 (𝑅 ∈ ℝExt → 𝑅 ∈ CUnifSp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145   × cxp 5247  cres 5251  cfv 6031  0cc0 10138  Basecbs 16064  distcds 16158  DivRingcdr 18957  metUnifcmetu 19952  ℤModczlm 20064  chrcchr 20065  UnifStcuss 22277  CUnifSpccusp 22321  NrmRingcnrg 22604  NrmModcnlm 22605   ℝExt crrext 30378
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-xp 5255  df-res 5261  df-iota 5994  df-fv 6039  df-rrext 30383
This theorem is referenced by:  rrhfe  30396  rrhcne  30397  sitgclg  30744
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