Theorem rrextust 34640

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 2761	. . . 4 ⊢ (ℤMod‘𝑅) = (ℤMod‘𝑅)
4	1, 2, 3	isrrext 34632	. . 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 5649   ↾ cres 5653  ‘cfv 6538  0cc0 11200  Basecbs 17387  distcds 17437  DivRingcdr 20980  metUnifcmetu 21669  ℤModczlm 21806  chrcchr 21807  UnifStcuss 24572  CUnifSpccusp 24615  NrmRingcnrg 24898  NrmModcnlm 24899   ℝExt crrext 34626

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 2733

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 2740  df-cleq 2753  df-clel 2836  df-rab 3414  df-v 3453  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 5657  df-res 5663  df-iota 6494  df-fv 6546  df-rrext 34631

This theorem is used by:  rrhfe  34644  rrhcne  34645  sitgclg  34974