Theorem rrhcne

Description: If R is an extension of RR , then the canonical homomorphism of RR into R is continuous. (Contributed by Thierry Arnoux, 2-May-2018)

	Ref	Expression
Hypotheses	rrhcne.j	⊢ 𝐽 = ( topGen ‘ ran (,) )
	rrhcne.k	⊢ 𝐾 = ( TopOpen ‘ 𝑅 )
Assertion	rrhcne	⊢ ( 𝑅 ∈ ℝExt → ( ℝHom ‘ 𝑅 ) ∈ ( 𝐽 Cn 𝐾 ) )

Proof

Step	Hyp	Ref	Expression
1		rrhcne.j	⊢ 𝐽 = ( topGen ‘ ran (,) )
2		rrhcne.k	⊢ 𝐾 = ( TopOpen ‘ 𝑅 )
3		eqid	⊢ ( ( dist ‘ 𝑅 ) ↾ ( ( Base ‘ 𝑅 ) × ( Base ‘ 𝑅 ) ) ) = ( ( dist ‘ 𝑅 ) ↾ ( ( Base ‘ 𝑅 ) × ( Base ‘ 𝑅 ) ) )
4		eqid	⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 )
5		eqid	⊢ ( ℤMod ‘ 𝑅 ) = ( ℤMod ‘ 𝑅 )
6		rrextdrg	⊢ ( 𝑅 ∈ ℝExt → 𝑅 ∈ DivRing )
7		rrextnrg	⊢ ( 𝑅 ∈ ℝExt → 𝑅 ∈ NrmRing )
8	5	rrextnlm	⊢ ( 𝑅 ∈ ℝExt → ( ℤMod ‘ 𝑅 ) ∈ NrmMod )
9		rrextchr	⊢ ( 𝑅 ∈ ℝExt → ( chr ‘ 𝑅 ) = 0 )
10		rrextcusp	⊢ ( 𝑅 ∈ ℝExt → 𝑅 ∈ CUnifSp )
11	4 3	rrextust	⊢ ( 𝑅 ∈ ℝExt → ( UnifSt ‘ 𝑅 ) = ( metUnif ‘ ( ( dist ‘ 𝑅 ) ↾ ( ( Base ‘ 𝑅 ) × ( Base ‘ 𝑅 ) ) ) ) )
12	3 1 4 2 5 6 7 8 9 10 11	rrhcn	⊢ ( 𝑅 ∈ ℝExt → ( ℝHom ‘ 𝑅 ) ∈ ( 𝐽 Cn 𝐾 ) )