Theorem rrextust 31251

Description: The uniformity of an extension of ℝ 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

Step	Hyp	Ref	Expression
1		rrextust.b	. . . 4 ⊢ 𝐵 = (Base‘𝑅)
2		rrextust.d	. . . 4 ⊢ 𝐷 = ((dist‘𝑅) ↾ (𝐵 × 𝐵))
3		eqid 2823	. . . 4 ⊢ (ℤMod‘𝑅) = (ℤMod‘𝑅)
4	1, 2, 3	isrrext 31243	. . 3 ⊢ (𝑅 ∈ ℝExt ↔ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ DivRing) ∧ ((ℤMod‘𝑅) ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ (𝑅 ∈ CUnifSp ∧ (UnifSt‘𝑅) = (metUnif‘𝐷))))
5	4	simp3bi 1143	. 2 ⊢ (𝑅 ∈ ℝExt → (𝑅 ∈ CUnifSp ∧ (UnifSt‘𝑅) = (metUnif‘𝐷)))
6	5	simprd 498	1 ⊢ (𝑅 ∈ ℝExt → (UnifSt‘𝑅) = (metUnif‘𝐷))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114   × cxp 5555   ↾ cres 5559  ‘cfv 6357  0cc0 10539  Basecbs 16485  distcds 16576  DivRingcdr 19504  metUnifcmetu 20538  ℤModczlm 20650  chrcchr 20651  UnifStcuss 22864  CUnifSpccusp 22908  NrmRingcnrg 23191  NrmModcnlm 23192   ℝExt crrext 31237

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-rab 3149  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-xp 5563  df-res 5569  df-iota 6316  df-fv 6365  df-rrext 31242

This theorem is referenced by:  rrhfe  31255  rrhcne  31256  sitgclg  31602