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 e. RRExt -> ( RRHom ` R ) e. ( 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 ) X. ( Base ` R ) ) ) = ( ( dist ` R ) |` ( ( Base ` R ) X. ( Base ` R ) ) )
4		eqid	|- ( Base ` R ) = ( Base ` R )
5		eqid	|- ( ZMod ` R ) = ( ZMod ` R )
6		rrextdrg	|- ( R e. RRExt -> R e. DivRing )
7		rrextnrg	|- ( R e. RRExt -> R e. NrmRing )
8	5	rrextnlm	|- ( R e. RRExt -> ( ZMod ` R ) e. NrmMod )
9		rrextchr	|- ( R e. RRExt -> ( chr ` R ) = 0 )
10		rrextcusp	|- ( R e. RRExt -> R e. CUnifSp )
11	4 3	rrextust	|- ( R e. RRExt -> ( UnifSt ` R ) = ( metUnif ` ( ( dist ` R ) |` ( ( Base ` R ) X. ( Base ` R ) ) ) ) )
12	3 1 4 2 5 6 7 8 9 10 11	rrhcn	|- ( R e. RRExt -> ( RRHom ` R ) e. ( J Cn K ) )