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Metamath Proof Explorer

Theorem mopnex

Description: The topology generated by an extended metric can also be generated by a true metric. Thus, "metrizable topologies" can equivalently be defined in terms of metrics or extended metrics. (Contributed by Mario Carneiro, 26-Aug-2015)

Ref Expression
Hypothesis mopnex.1 ⊢ 𝐽 = ( MetOpen ‘ 𝐷 )
Assertion mopnex ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → ∃ 𝑑 ∈ ( Met ‘ 𝑋 ) 𝐽 = ( MetOpen ‘ 𝑑 ) )

Proof

Step Hyp Ref Expression
1 mopnex.1 ⊢ 𝐽 = ( MetOpen ‘ 𝐷 )
2 1rp ⊢ 1 ∈ ℝ+
3 eqid ⊢ ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑋 ↦ if ( ( 𝑥 𝐷 𝑦 ) ≤ 1 , ( 𝑥 𝐷 𝑦 ) , 1 ) ) = ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑋 ↦ if ( ( 𝑥 𝐷 𝑦 ) ≤ 1 , ( 𝑥 𝐷 𝑦 ) , 1 ) )
4 3 stdbdmet ⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 1 ∈ ℝ+ ) → ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑋 ↦ if ( ( 𝑥 𝐷 𝑦 ) ≤ 1 , ( 𝑥 𝐷 𝑦 ) , 1 ) ) ∈ ( Met ‘ 𝑋 ) )
5 2 4 mpan2 ⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑋 ↦ if ( ( 𝑥 𝐷 𝑦 ) ≤ 1 , ( 𝑥 𝐷 𝑦 ) , 1 ) ) ∈ ( Met ‘ 𝑋 ) )
6 1xr ⊢ 1 ∈ ℝ*
7 0lt1 ⊢ 0 < 1
8 3 1 stdbdmopn ⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 1 ∈ ℝ* ∧ 0 < 1 ) → 𝐽 = ( MetOpen ‘ ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑋 ↦ if ( ( 𝑥 𝐷 𝑦 ) ≤ 1 , ( 𝑥 𝐷 𝑦 ) , 1 ) ) ) )
9 6 7 8 mp3an23 ⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → 𝐽 = ( MetOpen ‘ ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑋 ↦ if ( ( 𝑥 𝐷 𝑦 ) ≤ 1 , ( 𝑥 𝐷 𝑦 ) , 1 ) ) ) )
10 fveq2 ⊢ ( 𝑑 = ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑋 ↦ if ( ( 𝑥 𝐷 𝑦 ) ≤ 1 , ( 𝑥 𝐷 𝑦 ) , 1 ) ) → ( MetOpen ‘ 𝑑 ) = ( MetOpen ‘ ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑋 ↦ if ( ( 𝑥 𝐷 𝑦 ) ≤ 1 , ( 𝑥 𝐷 𝑦 ) , 1 ) ) ) )
11 10 rspceeqv ⊢ ( ( ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑋 ↦ if ( ( 𝑥 𝐷 𝑦 ) ≤ 1 , ( 𝑥 𝐷 𝑦 ) , 1 ) ) ∈ ( Met ‘ 𝑋 ) ∧ 𝐽 = ( MetOpen ‘ ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑋 ↦ if ( ( 𝑥 𝐷 𝑦 ) ≤ 1 , ( 𝑥 𝐷 𝑦 ) , 1 ) ) ) ) → ∃ 𝑑 ∈ ( Met ‘ 𝑋 ) 𝐽 = ( MetOpen ‘ 𝑑 ) )
12 5 9 11 syl2anc ⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → ∃ 𝑑 ∈ ( Met ‘ 𝑋 ) 𝐽 = ( MetOpen ‘ 𝑑 ) )