MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  blbas Structured version   Visualization version   GIF version

Theorem blbas 23592
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.)
Assertion
Ref Expression
blbas (𝐷 ∈ (∞Met‘𝑋) → ran (ball‘𝐷) ∈ TopBases)

Proof of Theorem blbas
Dummy variables 𝑥 𝑟 𝑏 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 blin2 23591 . . . . . 6 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑧 ∈ (𝑥𝑦)) ∧ (𝑥 ∈ ran (ball‘𝐷) ∧ 𝑦 ∈ ran (ball‘𝐷))) → ∃𝑟 ∈ ℝ+ (𝑧(ball‘𝐷)𝑟) ⊆ (𝑥𝑦))
2 simpll 764 . . . . . . 7 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑧 ∈ (𝑥𝑦)) ∧ (𝑥 ∈ ran (ball‘𝐷) ∧ 𝑦 ∈ ran (ball‘𝐷))) → 𝐷 ∈ (∞Met‘𝑋))
3 elinel1 4130 . . . . . . . . . 10 (𝑧 ∈ (𝑥𝑦) → 𝑧𝑥)
4 elunii 4845 . . . . . . . . . 10 ((𝑧𝑥𝑥 ∈ ran (ball‘𝐷)) → 𝑧 ran (ball‘𝐷))
53, 4sylan 580 . . . . . . . . 9 ((𝑧 ∈ (𝑥𝑦) ∧ 𝑥 ∈ ran (ball‘𝐷)) → 𝑧 ran (ball‘𝐷))
65ad2ant2lr 745 . . . . . . . 8 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑧 ∈ (𝑥𝑦)) ∧ (𝑥 ∈ ran (ball‘𝐷) ∧ 𝑦 ∈ ran (ball‘𝐷))) → 𝑧 ran (ball‘𝐷))
7 unirnbl 23582 . . . . . . . . 9 (𝐷 ∈ (∞Met‘𝑋) → ran (ball‘𝐷) = 𝑋)
87ad2antrr 723 . . . . . . . 8 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑧 ∈ (𝑥𝑦)) ∧ (𝑥 ∈ ran (ball‘𝐷) ∧ 𝑦 ∈ ran (ball‘𝐷))) → ran (ball‘𝐷) = 𝑋)
96, 8eleqtrd 2842 . . . . . . 7 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑧 ∈ (𝑥𝑦)) ∧ (𝑥 ∈ ran (ball‘𝐷) ∧ 𝑦 ∈ ran (ball‘𝐷))) → 𝑧𝑋)
10 blssex 23589 . . . . . . 7 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑧𝑋) → (∃𝑏 ∈ ran (ball‘𝐷)(𝑧𝑏𝑏 ⊆ (𝑥𝑦)) ↔ ∃𝑟 ∈ ℝ+ (𝑧(ball‘𝐷)𝑟) ⊆ (𝑥𝑦)))
112, 9, 10syl2anc 584 . . . . . 6 (((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)))
121, 11mpbird 256 . . . . 5 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑧 ∈ (𝑥𝑦)) ∧ (𝑥 ∈ ran (ball‘𝐷) ∧ 𝑦 ∈ ran (ball‘𝐷))) → ∃𝑏 ∈ ran (ball‘𝐷)(𝑧𝑏𝑏 ⊆ (𝑥𝑦)))
1312ex 413 . . . 4 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑧 ∈ (𝑥𝑦)) → ((𝑥 ∈ ran (ball‘𝐷) ∧ 𝑦 ∈ ran (ball‘𝐷)) → ∃𝑏 ∈ ran (ball‘𝐷)(𝑧𝑏𝑏 ⊆ (𝑥𝑦))))
1413ralrimdva 3107 . . 3 (𝐷 ∈ (∞Met‘𝑋) → ((𝑥 ∈ ran (ball‘𝐷) ∧ 𝑦 ∈ ran (ball‘𝐷)) → ∀𝑧 ∈ (𝑥𝑦)∃𝑏 ∈ ran (ball‘𝐷)(𝑧𝑏𝑏 ⊆ (𝑥𝑦))))
1514ralrimivv 3123 . 2 (𝐷 ∈ (∞Met‘𝑋) → ∀𝑥 ∈ ran (ball‘𝐷)∀𝑦 ∈ ran (ball‘𝐷)∀𝑧 ∈ (𝑥𝑦)∃𝑏 ∈ ran (ball‘𝐷)(𝑧𝑏𝑏 ⊆ (𝑥𝑦)))
16 fvex 6796 . . . 4 (ball‘𝐷) ∈ V
1716rnex 7768 . . 3 ran (ball‘𝐷) ∈ V
18 isbasis2g 22107 . . 3 (ran (ball‘𝐷) ∈ V → (ran (ball‘𝐷) ∈ TopBases ↔ ∀𝑥 ∈ ran (ball‘𝐷)∀𝑦 ∈ ran (ball‘𝐷)∀𝑧 ∈ (𝑥𝑦)∃𝑏 ∈ ran (ball‘𝐷)(𝑧𝑏𝑏 ⊆ (𝑥𝑦))))
1917, 18ax-mp 5 . 2 (ran (ball‘𝐷) ∈ TopBases ↔ ∀𝑥 ∈ ran (ball‘𝐷)∀𝑦 ∈ ran (ball‘𝐷)∀𝑧 ∈ (𝑥𝑦)∃𝑏 ∈ ran (ball‘𝐷)(𝑧𝑏𝑏 ⊆ (𝑥𝑦)))
2015, 19sylibr 233 1 (𝐷 ∈ (∞Met‘𝑋) → ran (ball‘𝐷) ∈ TopBases)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1539  wcel 2107  wral 3065  wrex 3066  Vcvv 3433  cin 3887  wss 3888   cuni 4840  ran crn 5591  cfv 6437  (class class class)co 7284  +crp 12739  ∞Metcxmet 20591  ballcbl 20593  TopBasesctb 22104
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2710  ax-sep 5224  ax-nul 5231  ax-pow 5289  ax-pr 5353  ax-un 7597  ax-cnex 10936  ax-resscn 10937  ax-1cn 10938  ax-icn 10939  ax-addcl 10940  ax-addrcl 10941  ax-mulcl 10942  ax-mulrcl 10943  ax-mulcom 10944  ax-addass 10945  ax-mulass 10946  ax-distr 10947  ax-i2m1 10948  ax-1ne0 10949  ax-1rid 10950  ax-rnegex 10951  ax-rrecex 10952  ax-cnre 10953  ax-pre-lttri 10954  ax-pre-lttrn 10955  ax-pre-ltadd 10956  ax-pre-mulgt0 10957  ax-pre-sup 10958
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3070  df-rex 3071  df-rmo 3072  df-reu 3073  df-rab 3074  df-v 3435  df-sbc 3718  df-csb 3834  df-dif 3891  df-un 3893  df-in 3895  df-ss 3905  df-pss 3907  df-nul 4258  df-if 4461  df-pw 4536  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4841  df-iun 4927  df-br 5076  df-opab 5138  df-mpt 5159  df-tr 5193  df-id 5490  df-eprel 5496  df-po 5504  df-so 5505  df-fr 5545  df-we 5547  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-pred 6206  df-ord 6273  df-on 6274  df-lim 6275  df-suc 6276  df-iota 6395  df-fun 6439  df-fn 6440  df-f 6441  df-f1 6442  df-fo 6443  df-f1o 6444  df-fv 6445  df-riota 7241  df-ov 7287  df-oprab 7288  df-mpo 7289  df-om 7722  df-1st 7840  df-2nd 7841  df-frecs 8106  df-wrecs 8137  df-recs 8211  df-rdg 8250  df-er 8507  df-map 8626  df-en 8743  df-dom 8744  df-sdom 8745  df-sup 9210  df-inf 9211  df-pnf 11020  df-mnf 11021  df-xr 11022  df-ltxr 11023  df-le 11024  df-sub 11216  df-neg 11217  df-div 11642  df-nn 11983  df-2 12045  df-n0 12243  df-z 12329  df-uz 12592  df-q 12698  df-rp 12740  df-xneg 12857  df-xadd 12858  df-xmul 12859  df-psmet 20598  df-xmet 20599  df-bl 20601  df-bases 22105
This theorem is referenced by:  mopntopon  23601  elmopn  23604  imasf1oxms  23654  blssopn  23660  metss  23673
  Copyright terms: Public domain W3C validator