Theorem blbas 24779

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 24778	. . . . . 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 779	. . . . . . 7 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) ∧ (𝑥 ∈ ran (ball‘𝐷) ∧ 𝑦 ∈ ran (ball‘𝐷))) → 𝐷 ∈ (∞Met‘𝑋))
3		elinel1 4101	. . . . . . . . . 10 ⊢ (𝑧 ∈ (𝑥 ∩ 𝑦) → 𝑧 ∈ 𝑥)
4		elunii 4875	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝑥 ∧ 𝑥 ∈ ran (ball‘𝐷)) → 𝑧 ∈ ∪ ran (ball‘𝐷))
5	3, 4	sylan 592	. . . . . . . . 9 ⊢ ((𝑧 ∈ (𝑥 ∩ 𝑦) ∧ 𝑥 ∈ ran (ball‘𝐷)) → 𝑧 ∈ ∪ ran (ball‘𝐷))
6	5	ad2ant2lr 761	. . . . . . . 8 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) ∧ (𝑥 ∈ ran (ball‘𝐷) ∧ 𝑦 ∈ ran (ball‘𝐷))) → 𝑧 ∈ ∪ ran (ball‘𝐷))
7		unirnbl 24769	. . . . . . . . 9 ⊢ (𝐷 ∈ (∞Met‘𝑋) → ∪ ran (ball‘𝐷) = 𝑋)
8	7	ad2antrr 739	. . . . . . . 8 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) ∧ (𝑥 ∈ ran (ball‘𝐷) ∧ 𝑦 ∈ ran (ball‘𝐷))) → ∪ ran (ball‘𝐷) = 𝑋)
9	6, 8	eleqtrd 2866	. . . . . . 7 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) ∧ (𝑥 ∈ ran (ball‘𝐷) ∧ 𝑦 ∈ ran (ball‘𝐷))) → 𝑧 ∈ 𝑋)
10		blssex 24776	. . . . . . 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 596	. . . . . 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 418	. . . 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 3166	. . 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 3207	. 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 6921	. . . 4 ⊢ (ball‘𝐷) ∈ V
17	16	rnex 7946	. . 3 ⊢ ran (ball‘𝐷) ∈ V
18		isbasis2g 23296	. . 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)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090  Vcvv 3454   ⊆ wss 3900   ∩ cin 4092  ∪ cuni 4870  ran crn 5656  ‘cfv 6558  (class class class)co 7443  ℝ⁺crp 13147  ∞Metcxmet 21693  ballcbl 21695  TopBasesctb 23293

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775  ax-cnex 11283  ax-resscn 11284  ax-1cn 11285  ax-icn 11286  ax-addcl 11287  ax-addrcl 11288  ax-mulcl 11289  ax-mulrcl 11290  ax-mulcom 11291  ax-addass 11292  ax-mulass 11293  ax-distr 11294  ax-i2m1 11295  ax-1ne0 11296  ax-1rid 11297  ax-rnegex 11298  ax-rrecex 11299  ax-cnre 11300  ax-pre-lttri 11301  ax-pre-lttrn 11302  ax-pre-ltadd 11303  ax-pre-mulgt0 11304  ax-pre-sup 11305

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-riota 7400  df-ov 7446  df-oprab 7447  df-mpo 7448  df-om 7902  df-1st 8025  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-rdg 8446  df-er 8745  df-map 8877  df-en 9002  df-dom 9003  df-sdom 9004  df-sup 9461  df-inf 9462  df-pnf 11372  df-mnf 11373  df-xr 11374  df-ltxr 11375  df-le 11376  df-sub 11570  df-neg 11571  df-div 12001  df-nn 12363  df-2 12432  df-n0 12634  df-z 12721  df-uz 12993  df-q 13103  df-rp 13148  df-xneg 13268  df-xadd 13269  df-xmul 13270  df-psmet 21700  df-xmet 21701  df-bl 21703  df-bases 23294

This theorem is used by:  mopntopon  24788  elmopn  24791  imasf1oxms  24838  blssopn  24844  metss  24857