Theorem rrextust 31624

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 2736	. . . 4 ⊢ (ℤMod‘𝑅) = (ℤMod‘𝑅)
4	1, 2, 3	isrrext 31616	. . 3 ⊢ (𝑅 ∈ ℝExt ↔ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ DivRing) ∧ ((ℤMod‘𝑅) ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ (𝑅 ∈ CUnifSp ∧ (UnifSt‘𝑅) = (metUnif‘𝐷))))
5	4	simp3bi 1149	. 2 ⊢ (𝑅 ∈ ℝExt → (𝑅 ∈ CUnifSp ∧ (UnifSt‘𝑅) = (metUnif‘𝐷)))
6	5	simprd 499	1 ⊢ (𝑅 ∈ ℝExt → (UnifSt‘𝑅) = (metUnif‘𝐷))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1543   ∈ wcel 2112   × cxp 5534   ↾ cres 5538  ‘cfv 6358  0cc0 10694  Basecbs 16666  distcds 16758  DivRingcdr 19721  metUnifcmetu 20308  ℤModczlm 20421  chrcchr 20422  UnifStcuss 23105  CUnifSpccusp 23148  NrmRingcnrg 23431  NrmModcnlm 23432   ℝExt crrext 31610

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2073  df-clab 2715  df-cleq 2728  df-clel 2809  df-rab 3060  df-v 3400  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-if 4426  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4806  df-br 5040  df-opab 5102  df-xp 5542  df-res 5548  df-iota 6316  df-fv 6366  df-rrext 31615

This theorem is referenced by:  rrhfe  31628  rrhcne  31629  sitgclg  31975