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 = TopOpen ⁡ K
isms.x ⊢ X = Base K
isms.d ⊢ D = dist ⁡ K ↾ X × X
Assertion isms2 ⊢ K ∈ MetSp ↔ D ∈ Met ⁡ X ∧ J = MetOpen ⁡ D

Proof

Step Hyp Ref Expression
1 isms.j ⊢ J = TopOpen ⁡ K
2 isms.x ⊢ X = Base K
3 isms.d ⊢ D = dist ⁡ K ↾ X × X
4 1 2 3 isxms2 ⊢ K ∈ ∞MetSp ↔ D ∈ ∞Met ⁡ X ∧ J = MetOpen ⁡ D
5 4 anbi1i ⊢ K ∈ ∞MetSp ∧ D ∈ Met ⁡ X ↔ D ∈ ∞Met ⁡ X ∧ J = MetOpen ⁡ D ∧ D ∈ Met ⁡ X
6 1 2 3 isms ⊢ K ∈ MetSp ↔ K ∈ ∞MetSp ∧ D ∈ Met ⁡ X
7 metxmet ⊢ D ∈ Met ⁡ X → D ∈ ∞Met ⁡ X
8 7 pm4.71ri ⊢ D ∈ Met ⁡ X ↔ D ∈ ∞Met ⁡ X ∧ D ∈ Met ⁡ X
9 8 anbi1i ⊢ D ∈ Met ⁡ X ∧ J = MetOpen ⁡ D ↔ D ∈ ∞Met ⁡ X ∧ D ∈ Met ⁡ X ∧ J = MetOpen ⁡ D
10 an32 ⊢ D ∈ ∞Met ⁡ X ∧ D ∈ Met ⁡ X ∧ J = MetOpen ⁡ D ↔ D ∈ ∞Met ⁡ X ∧ J = MetOpen ⁡ D ∧ D ∈ Met ⁡ X
11 9 10 bitri ⊢ D ∈ Met ⁡ X ∧ J = MetOpen ⁡ D ↔ D ∈ ∞Met ⁡ X ∧ J = MetOpen ⁡ D ∧ D ∈ Met ⁡ X
12 5 6 11 3bitr4i ⊢ K ∈ MetSp ↔ D ∈ Met ⁡ X ∧ J = MetOpen ⁡ D