Theorem rrextcusp

Description: An extension of RR is a complete uniform space. (Contributed by Thierry Arnoux, 2-May-2018)

	Ref	Expression
Assertion	rrextcusp	⊢ ( 𝑅 ∈ ℝExt → 𝑅 ∈ CUnifSp )

Proof

Step	Hyp	Ref	Expression
1		eqid	⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 )
2		eqid	⊢ ( ( dist ‘ 𝑅 ) ↾ ( ( Base ‘ 𝑅 ) × ( Base ‘ 𝑅 ) ) ) = ( ( dist ‘ 𝑅 ) ↾ ( ( Base ‘ 𝑅 ) × ( Base ‘ 𝑅 ) ) )
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 ‘ ( ( dist ‘ 𝑅 ) ↾ ( ( Base ‘ 𝑅 ) × ( Base ‘ 𝑅 ) ) ) ) ) ) )
5	4	simp3bi	⊢ ( 𝑅 ∈ ℝExt → ( 𝑅 ∈ CUnifSp ∧ ( UnifSt ‘ 𝑅 ) = ( metUnif ‘ ( ( dist ‘ 𝑅 ) ↾ ( ( Base ‘ 𝑅 ) × ( Base ‘ 𝑅 ) ) ) ) ) )
6	5	simpld	⊢ ( 𝑅 ∈ ℝExt → 𝑅 ∈ CUnifSp )