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Metamath Proof Explorer

Theorem setsms

Description: The constructed metric space is a metric space iff the provided distance function is a metric. (Contributed by Mario Carneiro, 28-Aug-2015)

Ref Expression
Hypotheses setsms.x ⊢ ( 𝜑 → 𝑋 = ( Base ‘ 𝑀 ) )
setsms.d ⊢ ( 𝜑 → 𝐷 = ( ( dist ‘ 𝑀 ) ↾ ( 𝑋 × 𝑋 ) ) )
setsms.k ⊢ ( 𝜑 → 𝐾 = ( 𝑀 sSet ⟨ ( TopSet ‘ ndx ) , ( MetOpen ‘ 𝐷 ) ⟩ ) )
setsms.m ⊢ ( 𝜑 → 𝑀 ∈ 𝑉 )
Assertion setsms ( 𝜑 → ( 𝐾 ∈ MetSp ↔ 𝐷 ∈ ( Met ‘ 𝑋 ) ) )

Proof

Step Hyp Ref Expression
1 setsms.x ⊢ ( 𝜑 → 𝑋 = ( Base ‘ 𝑀 ) )
2 setsms.d ⊢ ( 𝜑 → 𝐷 = ( ( dist ‘ 𝑀 ) ↾ ( 𝑋 × 𝑋 ) ) )
3 setsms.k ⊢ ( 𝜑 → 𝐾 = ( 𝑀 sSet ⟨ ( TopSet ‘ ndx ) , ( MetOpen ‘ 𝐷 ) ⟩ ) )
4 setsms.m ⊢ ( 𝜑 → 𝑀 ∈ 𝑉 )
5 1 2 3 4 setsxms ⊢ ( 𝜑 → ( 𝐾 ∈ ∞MetSp ↔ 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ) )
6 1 2 3 setsmsds ⊢ ( 𝜑 → ( dist ‘ 𝑀 ) = ( dist ‘ 𝐾 ) )
7 1 2 3 setsmsbas ⊢ ( 𝜑 → 𝑋 = ( Base ‘ 𝐾 ) )
8 7 sqxpeqd ⊢ ( 𝜑 → ( 𝑋 × 𝑋 ) = ( ( Base ‘ 𝐾 ) × ( Base ‘ 𝐾 ) ) )
9 6 8 reseq12d ⊢ ( 𝜑 → ( ( dist ‘ 𝑀 ) ↾ ( 𝑋 × 𝑋 ) ) = ( ( dist ‘ 𝐾 ) ↾ ( ( Base ‘ 𝐾 ) × ( Base ‘ 𝐾 ) ) ) )
10 2 9 eqtr2d ⊢ ( 𝜑 → ( ( dist ‘ 𝐾 ) ↾ ( ( Base ‘ 𝐾 ) × ( Base ‘ 𝐾 ) ) ) = 𝐷 )
11 7 fveq2d ⊢ ( 𝜑 → ( Met ‘ 𝑋 ) = ( Met ‘ ( Base ‘ 𝐾 ) ) )
12 11 eqcomd ⊢ ( 𝜑 → ( Met ‘ ( Base ‘ 𝐾 ) ) = ( Met ‘ 𝑋 ) )
13 10 12 eleq12d ⊢ ( 𝜑 → ( ( ( dist ‘ 𝐾 ) ↾ ( ( Base ‘ 𝐾 ) × ( Base ‘ 𝐾 ) ) ) ∈ ( Met ‘ ( Base ‘ 𝐾 ) ) ↔ 𝐷 ∈ ( Met ‘ 𝑋 ) ) )
14 5 13 anbi12d ⊢ ( 𝜑 → ( ( 𝐾 ∈ ∞MetSp ∧ ( ( dist ‘ 𝐾 ) ↾ ( ( Base ‘ 𝐾 ) × ( Base ‘ 𝐾 ) ) ) ∈ ( Met ‘ ( Base ‘ 𝐾 ) ) ) ↔ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐷 ∈ ( Met ‘ 𝑋 ) ) ) )
15 eqid ⊢ ( TopOpen ‘ 𝐾 ) = ( TopOpen ‘ 𝐾 )
16 eqid ⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
17 eqid ⊢ ( ( dist ‘ 𝐾 ) ↾ ( ( Base ‘ 𝐾 ) × ( Base ‘ 𝐾 ) ) ) = ( ( dist ‘ 𝐾 ) ↾ ( ( Base ‘ 𝐾 ) × ( Base ‘ 𝐾 ) ) )
18 15 16 17 isms ⊢ ( 𝐾 ∈ MetSp ↔ ( 𝐾 ∈ ∞MetSp ∧ ( ( dist ‘ 𝐾 ) ↾ ( ( Base ‘ 𝐾 ) × ( Base ‘ 𝐾 ) ) ) ∈ ( Met ‘ ( Base ‘ 𝐾 ) ) ) )
19 metxmet ⊢ ( 𝐷 ∈ ( Met ‘ 𝑋 ) → 𝐷 ∈ ( ∞Met ‘ 𝑋 ) )
20 19 pm4.71ri ⊢ ( 𝐷 ∈ ( Met ‘ 𝑋 ) ↔ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐷 ∈ ( Met ‘ 𝑋 ) ) )
21 14 18 20 3bitr4g ⊢ ( 𝜑 → ( 𝐾 ∈ MetSp ↔ 𝐷 ∈ ( Met ‘ 𝑋 ) ) )