Metamath Proof Explorer

Theorem setsxms

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 setsxms ( 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 setsmstopn ⊢ ( 𝜑 → ( MetOpen ‘ 𝐷 ) = ( TopOpen ‘ 𝐾 ) )
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 eqtrd ⊢ ( 𝜑 → 𝐷 = ( ( dist ‘ 𝐾 ) ↾ ( ( Base ‘ 𝐾 ) × ( Base ‘ 𝐾 ) ) ) )
11 10 fveq2d ⊢ ( 𝜑 → ( MetOpen ‘ 𝐷 ) = ( MetOpen ‘ ( ( dist ‘ 𝐾 ) ↾ ( ( Base ‘ 𝐾 ) × ( Base ‘ 𝐾 ) ) ) ) )
12 5 11 eqtr3d ⊢ ( 𝜑 → ( TopOpen ‘ 𝐾 ) = ( MetOpen ‘ ( ( dist ‘ 𝐾 ) ↾ ( ( Base ‘ 𝐾 ) × ( Base ‘ 𝐾 ) ) ) ) )
13 eqid ⊢ ( TopOpen ‘ 𝐾 ) = ( TopOpen ‘ 𝐾 )
14 eqid ⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
15 eqid ⊢ ( ( dist ‘ 𝐾 ) ↾ ( ( Base ‘ 𝐾 ) × ( Base ‘ 𝐾 ) ) ) = ( ( dist ‘ 𝐾 ) ↾ ( ( Base ‘ 𝐾 ) × ( Base ‘ 𝐾 ) ) )
16 13 14 15 isxms2 ⊢ ( 𝐾 ∈ ∞MetSp ↔ ( ( ( dist ‘ 𝐾 ) ↾ ( ( Base ‘ 𝐾 ) × ( Base ‘ 𝐾 ) ) ) ∈ ( ∞Met ‘ ( Base ‘ 𝐾 ) ) ∧ ( TopOpen ‘ 𝐾 ) = ( MetOpen ‘ ( ( dist ‘ 𝐾 ) ↾ ( ( Base ‘ 𝐾 ) × ( Base ‘ 𝐾 ) ) ) ) ) )
17 16 rbaib ⊢ ( ( TopOpen ‘ 𝐾 ) = ( MetOpen ‘ ( ( dist ‘ 𝐾 ) ↾ ( ( Base ‘ 𝐾 ) × ( Base ‘ 𝐾 ) ) ) ) → ( 𝐾 ∈ ∞MetSp ↔ ( ( dist ‘ 𝐾 ) ↾ ( ( Base ‘ 𝐾 ) × ( Base ‘ 𝐾 ) ) ) ∈ ( ∞Met ‘ ( Base ‘ 𝐾 ) ) ) )
18 12 17 syl ⊢ ( 𝜑 → ( 𝐾 ∈ ∞MetSp ↔ ( ( dist ‘ 𝐾 ) ↾ ( ( Base ‘ 𝐾 ) × ( Base ‘ 𝐾 ) ) ) ∈ ( ∞Met ‘ ( Base ‘ 𝐾 ) ) ) )
19 7 fveq2d ⊢ ( 𝜑 → ( ∞Met ‘ 𝑋 ) = ( ∞Met ‘ ( Base ‘ 𝐾 ) ) )
20 10 19 eleq12d ⊢ ( 𝜑 → ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ↔ ( ( dist ‘ 𝐾 ) ↾ ( ( Base ‘ 𝐾 ) × ( Base ‘ 𝐾 ) ) ) ∈ ( ∞Met ‘ ( Base ‘ 𝐾 ) ) ) )
21 18 20 bitr4d ⊢ ( 𝜑 → ( 𝐾 ∈ ∞MetSp ↔ 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ) )