Theorem rrextust 34519

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 2760	. . . 4 ⊢ (ℤMod‘𝑅) = (ℤMod‘𝑅)
4	1, 2, 3	isrrext 34511	. . 3 ⊢ (𝑅 ∈ ℝExt ↔ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ DivRing) ∧ ((ℤMod‘𝑅) ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ (𝑅 ∈ CUnifSp ∧ (UnifSt‘𝑅) = (metUnif‘𝐷))))
5	4	simp3bi 1165	. 2 ⊢ (𝑅 ∈ ℝExt → (𝑅 ∈ CUnifSp ∧ (UnifSt‘𝑅) = (metUnif‘𝐷)))
6	5	simprd 501	1 ⊢ (𝑅 ∈ ℝExt → (UnifSt‘𝑅) = (metUnif‘𝐷))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145   × cxp 5653   ↾ cres 5657  ‘cfv 6533  0cc0 11125  Basecbs 17302  distcds 17352  DivRingcdr 20891  metUnifcmetu 21577  ℤModczlm 21714  chrcchr 21715  UnifStcuss 24480  CUnifSpccusp 24523  NrmRingcnrg 24806  NrmModcnlm 24807   ℝExt crrext 34505

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-xp 5661  df-res 5667  df-iota 6489  df-fv 6541  df-rrext 34510

This theorem is used by:  rrhfe  34523  rrhcne  34524  sitgclg  34854