Theorem blbas 23037

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.

Step	Hyp	Ref	Expression
1		blin2 23036	. . . . . 6 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) ∧ (𝑥 ∈ ran (ball‘𝐷) ∧ 𝑦 ∈ ran (ball‘𝐷))) → ∃𝑟 ∈ ℝ+ (𝑧(ball‘𝐷)𝑟) ⊆ (𝑥 ∩ 𝑦))
2		simpll 766	. . . . . . 7 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) ∧ (𝑥 ∈ ran (ball‘𝐷) ∧ 𝑦 ∈ ran (ball‘𝐷))) → 𝐷 ∈ (∞Met‘𝑋))
3		elinel1 4122	. . . . . . . . . 10 ⊢ (𝑧 ∈ (𝑥 ∩ 𝑦) → 𝑧 ∈ 𝑥)
4		elunii 4805	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝑥 ∧ 𝑥 ∈ ran (ball‘𝐷)) → 𝑧 ∈ ∪ ran (ball‘𝐷))
5	3, 4	sylan 583	. . . . . . . . 9 ⊢ ((𝑧 ∈ (𝑥 ∩ 𝑦) ∧ 𝑥 ∈ ran (ball‘𝐷)) → 𝑧 ∈ ∪ ran (ball‘𝐷))
6	5	ad2ant2lr 747	. . . . . . . 8 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) ∧ (𝑥 ∈ ran (ball‘𝐷) ∧ 𝑦 ∈ ran (ball‘𝐷))) → 𝑧 ∈ ∪ ran (ball‘𝐷))
7		unirnbl 23027	. . . . . . . . 9 ⊢ (𝐷 ∈ (∞Met‘𝑋) → ∪ ran (ball‘𝐷) = 𝑋)
8	7	ad2antrr 725	. . . . . . . 8 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) ∧ (𝑥 ∈ ran (ball‘𝐷) ∧ 𝑦 ∈ ran (ball‘𝐷))) → ∪ ran (ball‘𝐷) = 𝑋)
9	6, 8	eleqtrd 2892	. . . . . . 7 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) ∧ (𝑥 ∈ ran (ball‘𝐷) ∧ 𝑦 ∈ ran (ball‘𝐷))) → 𝑧 ∈ 𝑋)
10		blssex 23034	. . . . . . 7 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑧 ∈ 𝑋) → (∃𝑏 ∈ ran (ball‘𝐷)(𝑧 ∈ 𝑏 ∧ 𝑏 ⊆ (𝑥 ∩ 𝑦)) ↔ ∃𝑟 ∈ ℝ+ (𝑧(ball‘𝐷)𝑟) ⊆ (𝑥 ∩ 𝑦)))
11	2, 9, 10	syl2anc 587	. . . . . 6 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) ∧ (𝑥 ∈ ran (ball‘𝐷) ∧ 𝑦 ∈ ran (ball‘𝐷))) → (∃𝑏 ∈ ran (ball‘𝐷)(𝑧 ∈ 𝑏 ∧ 𝑏 ⊆ (𝑥 ∩ 𝑦)) ↔ ∃𝑟 ∈ ℝ+ (𝑧(ball‘𝐷)𝑟) ⊆ (𝑥 ∩ 𝑦)))
12	1, 11	mpbird 260	. . . . 5 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) ∧ (𝑥 ∈ ran (ball‘𝐷) ∧ 𝑦 ∈ ran (ball‘𝐷))) → ∃𝑏 ∈ ran (ball‘𝐷)(𝑧 ∈ 𝑏 ∧ 𝑏 ⊆ (𝑥 ∩ 𝑦)))
13	12	ex 416	. . . 4 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) → ((𝑥 ∈ ran (ball‘𝐷) ∧ 𝑦 ∈ ran (ball‘𝐷)) → ∃𝑏 ∈ ran (ball‘𝐷)(𝑧 ∈ 𝑏 ∧ 𝑏 ⊆ (𝑥 ∩ 𝑦))))
14	13	ralrimdva 3154	. . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → ((𝑥 ∈ ran (ball‘𝐷) ∧ 𝑦 ∈ ran (ball‘𝐷)) → ∀𝑧 ∈ (𝑥 ∩ 𝑦)∃𝑏 ∈ ran (ball‘𝐷)(𝑧 ∈ 𝑏 ∧ 𝑏 ⊆ (𝑥 ∩ 𝑦))))
15	14	ralrimivv 3155	. 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → ∀𝑥 ∈ ran (ball‘𝐷)∀𝑦 ∈ ran (ball‘𝐷)∀𝑧 ∈ (𝑥 ∩ 𝑦)∃𝑏 ∈ ran (ball‘𝐷)(𝑧 ∈ 𝑏 ∧ 𝑏 ⊆ (𝑥 ∩ 𝑦)))
16		fvex 6658	. . . 4 ⊢ (ball‘𝐷) ∈ V
17	16	rnex 7599	. . 3 ⊢ ran (ball‘𝐷) ∈ V
18		isbasis2g 21553	. . 3 ⊢ (ran (ball‘𝐷) ∈ V → (ran (ball‘𝐷) ∈ TopBases ↔ ∀𝑥 ∈ ran (ball‘𝐷)∀𝑦 ∈ ran (ball‘𝐷)∀𝑧 ∈ (𝑥 ∩ 𝑦)∃𝑏 ∈ ran (ball‘𝐷)(𝑧 ∈ 𝑏 ∧ 𝑏 ⊆ (𝑥 ∩ 𝑦))))
19	17, 18	ax-mp 5	. 2 ⊢ (ran (ball‘𝐷) ∈ TopBases ↔ ∀𝑥 ∈ ran (ball‘𝐷)∀𝑦 ∈ ran (ball‘𝐷)∀𝑧 ∈ (𝑥 ∩ 𝑦)∃𝑏 ∈ ran (ball‘𝐷)(𝑧 ∈ 𝑏 ∧ 𝑏 ⊆ (𝑥 ∩ 𝑦)))
20	15, 19	sylibr 237	1 ⊢ (𝐷 ∈ (∞Met‘𝑋) → ran (ball‘𝐷) ∈ TopBases)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107  Vcvv 3441   ∩ cin 3880   ⊆ wss 3881  ∪ cuni 4800  ran crn 5520  ‘cfv 6324  (class class class)co 7135  ℝ⁺crp 12377  ∞Metcxmet 20076  ballcbl 20078  TopBasesctb 21550

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603  ax-pre-sup 10604

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-er 8272  df-map 8391  df-en 8493  df-dom 8494  df-sdom 8495  df-sup 8890  df-inf 8891  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-div 11287  df-nn 11626  df-2 11688  df-n0 11886  df-z 11970  df-uz 12232  df-q 12337  df-rp 12378  df-xneg 12495  df-xadd 12496  df-xmul 12497  df-psmet 20083  df-xmet 20084  df-bl 20086  df-bases 21551

This theorem is referenced by:  mopntopon  23046  elmopn  23049  imasf1oxms  23096  blssopn  23102  metss  23115