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Metamath Proof Explorer

Theorem isms

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 isms ⊢ K ∈ MetSp ↔ K ∈ ∞MetSp ∧ D ∈ Met ⁡ X

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 fveq2 ⊢ f = K → dist ⁡ f = dist ⁡ K
5 fveq2 ⊢ f = K → Base f = Base K
6 5 2 eqtr4di ⊢ f = K → Base f = X
7 6 sqxpeqd ⊢ f = K → Base f × Base f = X × X
8 4 7 reseq12d ⊢ f = K → dist ⁡ f ↾ Base f × Base f = dist ⁡ K ↾ X × X
9 8 3 eqtr4di ⊢ f = K → dist ⁡ f ↾ Base f × Base f = D
10 6 fveq2d ⊢ f = K → Met ⁡ Base f = Met ⁡ X
11 9 10 eleq12d ⊢ f = K → dist ⁡ f ↾ Base f × Base f ∈ Met ⁡ Base f ↔ D ∈ Met ⁡ X
12 df-ms ⊢ MetSp = f ∈ ∞MetSp | dist ⁡ f ↾ Base f × Base f ∈ Met ⁡ Base f
13 11 12 elrab2 ⊢ K ∈ MetSp ↔ K ∈ ∞MetSp ∧ D ∈ Met ⁡ X