# Metamath Proof Explorer

## Theorem isms2

Description: Express the predicate " <. X , D >. is a metric space" with underlying set X and distance function D . (Contributed by NM, 27-Aug-2006) (Revised by Mario Carneiro, 24-Aug-2015)

Ref Expression
Hypotheses isms.j ${⊢}{J}=\mathrm{TopOpen}\left({K}\right)$
isms.x ${⊢}{X}={\mathrm{Base}}_{{K}}$
isms.d ${⊢}{D}={\mathrm{dist}\left({K}\right)↾}_{\left({X}×{X}\right)}$
Assertion isms2 ${⊢}{K}\in \mathrm{MetSp}↔\left({D}\in \mathrm{Met}\left({X}\right)\wedge {J}=\mathrm{MetOpen}\left({D}\right)\right)$

### Proof

Step Hyp Ref Expression
1 isms.j ${⊢}{J}=\mathrm{TopOpen}\left({K}\right)$
2 isms.x ${⊢}{X}={\mathrm{Base}}_{{K}}$
3 isms.d ${⊢}{D}={\mathrm{dist}\left({K}\right)↾}_{\left({X}×{X}\right)}$
4 1 2 3 isxms2 ${⊢}{K}\in \mathrm{\infty MetSp}↔\left({D}\in \mathrm{\infty Met}\left({X}\right)\wedge {J}=\mathrm{MetOpen}\left({D}\right)\right)$
5 4 anbi1i ${⊢}\left({K}\in \mathrm{\infty MetSp}\wedge {D}\in \mathrm{Met}\left({X}\right)\right)↔\left(\left({D}\in \mathrm{\infty Met}\left({X}\right)\wedge {J}=\mathrm{MetOpen}\left({D}\right)\right)\wedge {D}\in \mathrm{Met}\left({X}\right)\right)$
6 1 2 3 isms ${⊢}{K}\in \mathrm{MetSp}↔\left({K}\in \mathrm{\infty MetSp}\wedge {D}\in \mathrm{Met}\left({X}\right)\right)$
7 metxmet ${⊢}{D}\in \mathrm{Met}\left({X}\right)\to {D}\in \mathrm{\infty Met}\left({X}\right)$
8 7 pm4.71ri ${⊢}{D}\in \mathrm{Met}\left({X}\right)↔\left({D}\in \mathrm{\infty Met}\left({X}\right)\wedge {D}\in \mathrm{Met}\left({X}\right)\right)$
9 8 anbi1i ${⊢}\left({D}\in \mathrm{Met}\left({X}\right)\wedge {J}=\mathrm{MetOpen}\left({D}\right)\right)↔\left(\left({D}\in \mathrm{\infty Met}\left({X}\right)\wedge {D}\in \mathrm{Met}\left({X}\right)\right)\wedge {J}=\mathrm{MetOpen}\left({D}\right)\right)$
10 an32 ${⊢}\left(\left({D}\in \mathrm{\infty Met}\left({X}\right)\wedge {D}\in \mathrm{Met}\left({X}\right)\right)\wedge {J}=\mathrm{MetOpen}\left({D}\right)\right)↔\left(\left({D}\in \mathrm{\infty Met}\left({X}\right)\wedge {J}=\mathrm{MetOpen}\left({D}\right)\right)\wedge {D}\in \mathrm{Met}\left({X}\right)\right)$
11 9 10 bitri ${⊢}\left({D}\in \mathrm{Met}\left({X}\right)\wedge {J}=\mathrm{MetOpen}\left({D}\right)\right)↔\left(\left({D}\in \mathrm{\infty Met}\left({X}\right)\wedge {J}=\mathrm{MetOpen}\left({D}\right)\right)\wedge {D}\in \mathrm{Met}\left({X}\right)\right)$
12 5 6 11 3bitr4i ${⊢}{K}\in \mathrm{MetSp}↔\left({D}\in \mathrm{Met}\left({X}\right)\wedge {J}=\mathrm{MetOpen}\left({D}\right)\right)$