Theorem imsxmet

Description: The induced metric of a normed complex vector space is an extended metric space. (Contributed by Mario Carneiro, 10-Sep-2015) (New usage is discouraged.)

	Ref	Expression
Hypotheses	imsmet.1	$$⊢X=\mathrm{BaseSet}\left(U\right)$$
	imsmet.8	$$⊢D=\mathrm{IndMet}\left(U\right)$$
Assertion	imsxmet	$$⊢U\in \mathrm{NrmCVec}\to D\in \mathrm{\infty Met}\left(X\right)$$

Proof

Step	Hyp	Ref	Expression
1		imsmet.1	$$⊢X=\mathrm{BaseSet}\left(U\right)$$
2		imsmet.8	$$⊢D=\mathrm{IndMet}\left(U\right)$$
3	1 2	imsmet	$$⊢U\in \mathrm{NrmCVec}\to D\in \mathrm{Met}\left(X\right)$$
4		metxmet	$$⊢D\in \mathrm{Met}\left(X\right)\to D\in \mathrm{\infty Met}\left(X\right)$$
5	3 4	syl	$$⊢U\in \mathrm{NrmCVec}\to D\in \mathrm{\infty Met}\left(X\right)$$