Theorem rrextust

Description: The uniformity of an extension of RR is the uniformity generated by its distance. (Contributed by Thierry Arnoux, 2-May-2018)

	Ref	Expression
Hypotheses	rrextust.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
	rrextust.d	⊢ 𝐷 = ( ( dist ‘ 𝑅 ) ↾ ( 𝐵 × 𝐵 ) )
Assertion	rrextust	⊢ ( 𝑅 ∈ ℝExt → ( UnifSt ‘ 𝑅 ) = ( metUnif ‘ 𝐷 ) )

Proof

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