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Metamath Proof Explorer

Theorem tmsms

Description: The constructed metric space is a metric space given a metric. (Contributed by Mario Carneiro, 2-Sep-2015)

Ref Expression
Hypothesis tmsbas.k ⊢ 𝐾 = ( toMetSp ‘ 𝐷 )
Assertion tmsms ( 𝐷 ∈ ( Met ‘ 𝑋 ) → 𝐾 ∈ MetSp )

Proof

Step Hyp Ref Expression
1 tmsbas.k ⊢ 𝐾 = ( toMetSp ‘ 𝐷 )
2 metxmet ⊢ ( 𝐷 ∈ ( Met ‘ 𝑋 ) → 𝐷 ∈ ( ∞Met ‘ 𝑋 ) )
3 1 tmsxms ⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → 𝐾 ∈ ∞MetSp )
4 2 3 syl ⊢ ( 𝐷 ∈ ( Met ‘ 𝑋 ) → 𝐾 ∈ ∞MetSp )
5 1 tmsds ⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → 𝐷 = ( dist ‘ 𝐾 ) )
6 2 5 syl ⊢ ( 𝐷 ∈ ( Met ‘ 𝑋 ) → 𝐷 = ( dist ‘ 𝐾 ) )
7 1 tmsbas ⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → 𝑋 = ( Base ‘ 𝐾 ) )
8 2 7 syl ⊢ ( 𝐷 ∈ ( Met ‘ 𝑋 ) → 𝑋 = ( Base ‘ 𝐾 ) )
9 8 fveq2d ⊢ ( 𝐷 ∈ ( Met ‘ 𝑋 ) → ( Met ‘ 𝑋 ) = ( Met ‘ ( Base ‘ 𝐾 ) ) )
10 6 9 eleq12d ⊢ ( 𝐷 ∈ ( Met ‘ 𝑋 ) → ( 𝐷 ∈ ( Met ‘ 𝑋 ) ↔ ( dist ‘ 𝐾 ) ∈ ( Met ‘ ( Base ‘ 𝐾 ) ) ) )
11 10 ibi ⊢ ( 𝐷 ∈ ( Met ‘ 𝑋 ) → ( dist ‘ 𝐾 ) ∈ ( Met ‘ ( Base ‘ 𝐾 ) ) )
12 ssid ⊢ ( Base ‘ 𝐾 ) ⊆ ( Base ‘ 𝐾 )
13 metres2 ⊢ ( ( ( dist ‘ 𝐾 ) ∈ ( Met ‘ ( Base ‘ 𝐾 ) ) ∧ ( Base ‘ 𝐾 ) ⊆ ( Base ‘ 𝐾 ) ) → ( ( dist ‘ 𝐾 ) ↾ ( ( Base ‘ 𝐾 ) × ( Base ‘ 𝐾 ) ) ) ∈ ( Met ‘ ( Base ‘ 𝐾 ) ) )
14 11 12 13 sylancl ⊢ ( 𝐷 ∈ ( Met ‘ 𝑋 ) → ( ( dist ‘ 𝐾 ) ↾ ( ( Base ‘ 𝐾 ) × ( Base ‘ 𝐾 ) ) ) ∈ ( Met ‘ ( Base ‘ 𝐾 ) ) )
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 4 14 18 sylanbrc ⊢ ( 𝐷 ∈ ( Met ‘ 𝑋 ) → 𝐾 ∈ MetSp )