Theorem rrextust 34171

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 2737	. . . 4 ⊢ (ℤMod‘𝑅) = (ℤMod‘𝑅)
4	1, 2, 3	isrrext 34163	. . 3 ⊢ (𝑅 ∈ ℝExt ↔ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ DivRing) ∧ ((ℤMod‘𝑅) ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ (𝑅 ∈ CUnifSp ∧ (UnifSt‘𝑅) = (metUnif‘𝐷))))
5	4	simp3bi 1148	. 2 ⊢ (𝑅 ∈ ℝExt → (𝑅 ∈ CUnifSp ∧ (UnifSt‘𝑅) = (metUnif‘𝐷)))
6	5	simprd 495	1 ⊢ (𝑅 ∈ ℝExt → (UnifSt‘𝑅) = (metUnif‘𝐷))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114   × cxp 5623   ↾ cres 5627  ‘cfv 6493  0cc0 11032  Basecbs 17173  distcds 17223  DivRingcdr 20700  metUnifcmetu 21338  ℤModczlm 21493  chrcchr 21494  UnifStcuss 24231  CUnifSpccusp 24274  NrmRingcnrg 24557  NrmModcnlm 24558   ℝExt crrext 34157

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-xp 5631  df-res 5637  df-iota 6449  df-fv 6501  df-rrext 34162

This theorem is referenced by:  rrhfe  34175  rrhcne  34176  sitgclg  34505