Metamath Proof Explorer


Theorem xmetdcn

Description: The metric function of an extended metric space is always continuous in the topology generated by it. (Contributed by Mario Carneiro, 4-Sep-2015)

Ref Expression
Hypotheses xmetdcn2.1 J = MetOpen D
xmetdcn.2 K = ordTop
Assertion xmetdcn D ∞Met X D J × t J Cn K

Proof

Step Hyp Ref Expression
1 xmetdcn2.1 J = MetOpen D
2 xmetdcn.2 K = ordTop
3 letopon ordTop TopOn *
4 2 3 eqeltri K TopOn *
5 eqid dist 𝑠 * = dist 𝑠 *
6 eqid MetOpen dist 𝑠 * = MetOpen dist 𝑠 *
7 5 6 xrsmopn ordTop MetOpen dist 𝑠 *
8 2 7 eqsstri K MetOpen dist 𝑠 *
9 5 xrsxmet dist 𝑠 * ∞Met *
10 6 mopnuni dist 𝑠 * ∞Met * * = MetOpen dist 𝑠 *
11 9 10 ax-mp * = MetOpen dist 𝑠 *
12 11 cnss2 K TopOn * K MetOpen dist 𝑠 * J × t J Cn MetOpen dist 𝑠 * J × t J Cn K
13 4 8 12 mp2an J × t J Cn MetOpen dist 𝑠 * J × t J Cn K
14 1 5 6 xmetdcn2 D ∞Met X D J × t J Cn MetOpen dist 𝑠 *
15 13 14 sselid D ∞Met X D J × t J Cn K