Metamath Proof Explorer


Theorem blbas

Description: The balls of a metric space form a basis for a topology. (Contributed by NM, 12-Sep-2006) (Revised by Mario Carneiro, 15-Jan-2014)

Ref Expression
Assertion blbas ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → ran ( ball ‘ 𝐷 ) ∈ TopBases )

Proof

Step Hyp Ref Expression
1 blin2 ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑧 ∈ ( 𝑥𝑦 ) ) ∧ ( 𝑥 ∈ ran ( ball ‘ 𝐷 ) ∧ 𝑦 ∈ ran ( ball ‘ 𝐷 ) ) ) → ∃ 𝑟 ∈ ℝ+ ( 𝑧 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ ( 𝑥𝑦 ) )
2 simpll ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑧 ∈ ( 𝑥𝑦 ) ) ∧ ( 𝑥 ∈ ran ( ball ‘ 𝐷 ) ∧ 𝑦 ∈ ran ( ball ‘ 𝐷 ) ) ) → 𝐷 ∈ ( ∞Met ‘ 𝑋 ) )
3 elinel1 ( 𝑧 ∈ ( 𝑥𝑦 ) → 𝑧𝑥 )
4 elunii ( ( 𝑧𝑥𝑥 ∈ ran ( ball ‘ 𝐷 ) ) → 𝑧 ran ( ball ‘ 𝐷 ) )
5 3 4 sylan ( ( 𝑧 ∈ ( 𝑥𝑦 ) ∧ 𝑥 ∈ ran ( ball ‘ 𝐷 ) ) → 𝑧 ran ( ball ‘ 𝐷 ) )
6 5 ad2ant2lr ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑧 ∈ ( 𝑥𝑦 ) ) ∧ ( 𝑥 ∈ ran ( ball ‘ 𝐷 ) ∧ 𝑦 ∈ ran ( ball ‘ 𝐷 ) ) ) → 𝑧 ran ( ball ‘ 𝐷 ) )
7 unirnbl ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → ran ( ball ‘ 𝐷 ) = 𝑋 )
8 7 ad2antrr ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑧 ∈ ( 𝑥𝑦 ) ) ∧ ( 𝑥 ∈ ran ( ball ‘ 𝐷 ) ∧ 𝑦 ∈ ran ( ball ‘ 𝐷 ) ) ) → ran ( ball ‘ 𝐷 ) = 𝑋 )
9 6 8 eleqtrd ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑧 ∈ ( 𝑥𝑦 ) ) ∧ ( 𝑥 ∈ ran ( ball ‘ 𝐷 ) ∧ 𝑦 ∈ ran ( ball ‘ 𝐷 ) ) ) → 𝑧𝑋 )
10 blssex ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑧𝑋 ) → ( ∃ 𝑏 ∈ ran ( ball ‘ 𝐷 ) ( 𝑧𝑏𝑏 ⊆ ( 𝑥𝑦 ) ) ↔ ∃ 𝑟 ∈ ℝ+ ( 𝑧 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ ( 𝑥𝑦 ) ) )
11 2 9 10 syl2anc ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑧 ∈ ( 𝑥𝑦 ) ) ∧ ( 𝑥 ∈ ran ( ball ‘ 𝐷 ) ∧ 𝑦 ∈ ran ( ball ‘ 𝐷 ) ) ) → ( ∃ 𝑏 ∈ ran ( ball ‘ 𝐷 ) ( 𝑧𝑏𝑏 ⊆ ( 𝑥𝑦 ) ) ↔ ∃ 𝑟 ∈ ℝ+ ( 𝑧 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ ( 𝑥𝑦 ) ) )
12 1 11 mpbird ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑧 ∈ ( 𝑥𝑦 ) ) ∧ ( 𝑥 ∈ ran ( ball ‘ 𝐷 ) ∧ 𝑦 ∈ ran ( ball ‘ 𝐷 ) ) ) → ∃ 𝑏 ∈ ran ( ball ‘ 𝐷 ) ( 𝑧𝑏𝑏 ⊆ ( 𝑥𝑦 ) ) )
13 12 ex ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑧 ∈ ( 𝑥𝑦 ) ) → ( ( 𝑥 ∈ ran ( ball ‘ 𝐷 ) ∧ 𝑦 ∈ ran ( ball ‘ 𝐷 ) ) → ∃ 𝑏 ∈ ran ( ball ‘ 𝐷 ) ( 𝑧𝑏𝑏 ⊆ ( 𝑥𝑦 ) ) ) )
14 13 ralrimdva ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → ( ( 𝑥 ∈ ran ( ball ‘ 𝐷 ) ∧ 𝑦 ∈ ran ( ball ‘ 𝐷 ) ) → ∀ 𝑧 ∈ ( 𝑥𝑦 ) ∃ 𝑏 ∈ ran ( ball ‘ 𝐷 ) ( 𝑧𝑏𝑏 ⊆ ( 𝑥𝑦 ) ) ) )
15 14 ralrimivv ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → ∀ 𝑥 ∈ ran ( ball ‘ 𝐷 ) ∀ 𝑦 ∈ ran ( ball ‘ 𝐷 ) ∀ 𝑧 ∈ ( 𝑥𝑦 ) ∃ 𝑏 ∈ ran ( ball ‘ 𝐷 ) ( 𝑧𝑏𝑏 ⊆ ( 𝑥𝑦 ) ) )
16 fvex ( ball ‘ 𝐷 ) ∈ V
17 16 rnex ran ( ball ‘ 𝐷 ) ∈ V
18 isbasis2g ( ran ( ball ‘ 𝐷 ) ∈ V → ( ran ( ball ‘ 𝐷 ) ∈ TopBases ↔ ∀ 𝑥 ∈ ran ( ball ‘ 𝐷 ) ∀ 𝑦 ∈ ran ( ball ‘ 𝐷 ) ∀ 𝑧 ∈ ( 𝑥𝑦 ) ∃ 𝑏 ∈ ran ( ball ‘ 𝐷 ) ( 𝑧𝑏𝑏 ⊆ ( 𝑥𝑦 ) ) ) )
19 17 18 ax-mp ( ran ( ball ‘ 𝐷 ) ∈ TopBases ↔ ∀ 𝑥 ∈ ran ( ball ‘ 𝐷 ) ∀ 𝑦 ∈ ran ( ball ‘ 𝐷 ) ∀ 𝑧 ∈ ( 𝑥𝑦 ) ∃ 𝑏 ∈ ran ( ball ‘ 𝐷 ) ( 𝑧𝑏𝑏 ⊆ ( 𝑥𝑦 ) ) )
20 15 19 sylibr ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → ran ( ball ‘ 𝐷 ) ∈ TopBases )