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 ) )