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 )

Metamath Proof Explorer

Theorem metdsre

Proof