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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 ) ) )