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Metamath Proof Explorer

Theorem metdsre

Description: The distance from a point to a nonempty set in a proper metric space is a real number. (Contributed by Mario Carneiro, 5-Sep-2015)

Ref Expression
Hypothesis metdscn.f ⊢ 𝐹 = ( 𝑥 ∈ 𝑋 ↦ inf ( ran ( 𝑦 ∈ 𝑆 ↦ ( 𝑥 𝐷 𝑦 ) ) , ℝ* , < ) )
Assertion metdsre ( ( 𝐷 ∈ ( Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅ ) → 𝐹 : 𝑋 ⟶ ℝ )

Proof

Step Hyp Ref Expression
1 metdscn.f ⊢ 𝐹 = ( 𝑥 ∈ 𝑋 ↦ inf ( ran ( 𝑦 ∈ 𝑆 ↦ ( 𝑥 𝐷 𝑦 ) ) , ℝ* , < ) )
2 n0 ⊢ ( 𝑆 ≠ ∅ ↔ ∃ 𝑧 𝑧 ∈ 𝑆 )
3 metxmet ⊢ ( 𝐷 ∈ ( Met ‘ 𝑋 ) → 𝐷 ∈ ( ∞Met ‘ 𝑋 ) )
4 1 metdsf ⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ) → 𝐹 : 𝑋 ⟶ ( 0 [,] +∞ ) )
5 3 4 sylan ⊢ ( ( 𝐷 ∈ ( Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ) → 𝐹 : 𝑋 ⟶ ( 0 [,] +∞ ) )
6 5 adantr ⊢ ( ( ( 𝐷 ∈ ( Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ) ∧ 𝑧 ∈ 𝑆 ) → 𝐹 : 𝑋 ⟶ ( 0 [,] +∞ ) )
7 6 ffnd ⊢ ( ( ( 𝐷 ∈ ( Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ) ∧ 𝑧 ∈ 𝑆 ) → 𝐹 Fn 𝑋 )
8 5 adantr ⊢ ( ( ( 𝐷 ∈ ( Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ) ∧ ( 𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋 ) ) → 𝐹 : 𝑋 ⟶ ( 0 [,] +∞ ) )
9 simprr ⊢ ( ( ( 𝐷 ∈ ( Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ) ∧ ( 𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋 ) ) → 𝑤 ∈ 𝑋 )
10 8 9 ffvelcdmd ⊢ ( ( ( 𝐷 ∈ ( Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ) ∧ ( 𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋 ) ) → ( 𝐹 ‘ 𝑤 ) ∈ ( 0 [,] +∞ ) )
11 eliccxr ⊢ ( ( 𝐹 ‘ 𝑤 ) ∈ ( 0 [,] +∞ ) → ( 𝐹 ‘ 𝑤 ) ∈ ℝ* )
12 10 11 syl ⊢ ( ( ( 𝐷 ∈ ( Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ) ∧ ( 𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋 ) ) → ( 𝐹 ‘ 𝑤 ) ∈ ℝ* )
13 simpll ⊢ ( ( ( 𝐷 ∈ ( Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ) ∧ ( 𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋 ) ) → 𝐷 ∈ ( Met ‘ 𝑋 ) )
14 simpr ⊢ ( ( 𝐷 ∈ ( Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ) → 𝑆 ⊆ 𝑋 )
15 14 sselda ⊢ ( ( ( 𝐷 ∈ ( Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ) ∧ 𝑧 ∈ 𝑆 ) → 𝑧 ∈ 𝑋 )
16 15 adantrr ⊢ ( ( ( 𝐷 ∈ ( Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ) ∧ ( 𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋 ) ) → 𝑧 ∈ 𝑋 )
17 metcl ⊢ ( ( 𝐷 ∈ ( Met ‘ 𝑋 ) ∧ 𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋 ) → ( 𝑧 𝐷 𝑤 ) ∈ ℝ )
18 13 16 9 17 syl3anc ⊢ ( ( ( 𝐷 ∈ ( Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ) ∧ ( 𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋 ) ) → ( 𝑧 𝐷 𝑤 ) ∈ ℝ )
19 elxrge0 ⊢ ( ( 𝐹 ‘ 𝑤 ) ∈ ( 0 [,] +∞ ) ↔ ( ( 𝐹 ‘ 𝑤 ) ∈ ℝ* ∧ 0 ≤ ( 𝐹 ‘ 𝑤 ) ) )
20 19 simprbi ⊢ ( ( 𝐹 ‘ 𝑤 ) ∈ ( 0 [,] +∞ ) → 0 ≤ ( 𝐹 ‘ 𝑤 ) )
21 10 20 syl ⊢ ( ( ( 𝐷 ∈ ( Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ) ∧ ( 𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋 ) ) → 0 ≤ ( 𝐹 ‘ 𝑤 ) )
22 1 metdsle ⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ) ∧ ( 𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋 ) ) → ( 𝐹 ‘ 𝑤 ) ≤ ( 𝑧 𝐷 𝑤 ) )
23 3 22 sylanl1 ⊢ ( ( ( 𝐷 ∈ ( Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ) ∧ ( 𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋 ) ) → ( 𝐹 ‘ 𝑤 ) ≤ ( 𝑧 𝐷 𝑤 ) )
24 xrrege0 ⊢ ( ( ( ( 𝐹 ‘ 𝑤 ) ∈ ℝ* ∧ ( 𝑧 𝐷 𝑤 ) ∈ ℝ ) ∧ ( 0 ≤ ( 𝐹 ‘ 𝑤 ) ∧ ( 𝐹 ‘ 𝑤 ) ≤ ( 𝑧 𝐷 𝑤 ) ) ) → ( 𝐹 ‘ 𝑤 ) ∈ ℝ )
25 12 18 21 23 24 syl22anc ⊢ ( ( ( 𝐷 ∈ ( Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ) ∧ ( 𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋 ) ) → ( 𝐹 ‘ 𝑤 ) ∈ ℝ )
26 25 anassrs ⊢ ( ( ( ( 𝐷 ∈ ( Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ) ∧ 𝑧 ∈ 𝑆 ) ∧ 𝑤 ∈ 𝑋 ) → ( 𝐹 ‘ 𝑤 ) ∈ ℝ )
27 26 ralrimiva ⊢ ( ( ( 𝐷 ∈ ( Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ) ∧ 𝑧 ∈ 𝑆 ) → ∀ 𝑤 ∈ 𝑋 ( 𝐹 ‘ 𝑤 ) ∈ ℝ )
28 ffnfv ⊢ ( 𝐹 : 𝑋 ⟶ ℝ ↔ ( 𝐹 Fn 𝑋 ∧ ∀ 𝑤 ∈ 𝑋 ( 𝐹 ‘ 𝑤 ) ∈ ℝ ) )
29 7 27 28 sylanbrc ⊢ ( ( ( 𝐷 ∈ ( Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ) ∧ 𝑧 ∈ 𝑆 ) → 𝐹 : 𝑋 ⟶ ℝ )
30 29 ex ⊢ ( ( 𝐷 ∈ ( Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ) → ( 𝑧 ∈ 𝑆 → 𝐹 : 𝑋 ⟶ ℝ ) )
31 30 exlimdv ⊢ ( ( 𝐷 ∈ ( Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ) → ( ∃ 𝑧 𝑧 ∈ 𝑆 → 𝐹 : 𝑋 ⟶ ℝ ) )
32 2 31 biimtrid ⊢ ( ( 𝐷 ∈ ( Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ) → ( 𝑆 ≠ ∅ → 𝐹 : 𝑋 ⟶ ℝ ) )
33 32 3impia ⊢ ( ( 𝐷 ∈ ( Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅ ) → 𝐹 : 𝑋 ⟶ ℝ )