Theorem rrhfe

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

	Ref	Expression
Hypothesis	rrhfe.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
Assertion	rrhfe	⊢ ( 𝑅 ∈ ℝExt → ( ℝHom ‘ 𝑅 ) : ℝ ⟶ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		rrhfe.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
2		eqid	⊢ ( ( dist ‘ 𝑅 ) ↾ ( 𝐵 × 𝐵 ) ) = ( ( dist ‘ 𝑅 ) ↾ ( 𝐵 × 𝐵 ) )
3		eqid	⊢ ( topGen ‘ ran (,) ) = ( topGen ‘ ran (,) )
4		eqid	⊢ ( TopOpen ‘ 𝑅 ) = ( TopOpen ‘ 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	1 2	rrextust	⊢ ( 𝑅 ∈ ℝExt → ( UnifSt ‘ 𝑅 ) = ( metUnif ‘ ( ( dist ‘ 𝑅 ) ↾ ( 𝐵 × 𝐵 ) ) ) )
12	2 3 1 4 5 6 7 8 9 10 11	rrhf	⊢ ( 𝑅 ∈ ℝExt → ( ℝHom ‘ 𝑅 ) : ℝ ⟶ 𝐵 )