Theorem blbas 15671

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 15670	. . . . . 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 531	. . . . . . 7 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) ∧ (𝑥 ∈ ran (ball‘𝐷) ∧ 𝑦 ∈ ran (ball‘𝐷))) → 𝐷 ∈ (∞Met‘𝑋))
3		elinel1 3415	. . . . . . . . . 10 ⊢ (𝑧 ∈ (𝑥 ∩ 𝑦) → 𝑧 ∈ 𝑥)
4		elunii 3940	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝑥 ∧ 𝑥 ∈ ran (ball‘𝐷)) → 𝑧 ∈ ∪ ran (ball‘𝐷))
5	3, 4	sylan 283	. . . . . . . . 9 ⊢ ((𝑧 ∈ (𝑥 ∩ 𝑦) ∧ 𝑥 ∈ ran (ball‘𝐷)) → 𝑧 ∈ ∪ ran (ball‘𝐷))
6	5	ad2ant2lr 514	. . . . . . . 8 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) ∧ (𝑥 ∈ ran (ball‘𝐷) ∧ 𝑦 ∈ ran (ball‘𝐷))) → 𝑧 ∈ ∪ ran (ball‘𝐷))
7		unirnbl 15661	. . . . . . . . 9 ⊢ (𝐷 ∈ (∞Met‘𝑋) → ∪ ran (ball‘𝐷) = 𝑋)
8	7	ad2antrr 492	. . . . . . . 8 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) ∧ (𝑥 ∈ ran (ball‘𝐷) ∧ 𝑦 ∈ ran (ball‘𝐷))) → ∪ ran (ball‘𝐷) = 𝑋)
9	6, 8	eleqtrd 2317	. . . . . . 7 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) ∧ (𝑥 ∈ ran (ball‘𝐷) ∧ 𝑦 ∈ ran (ball‘𝐷))) → 𝑧 ∈ 𝑋)
10		blssex 15668	. . . . . . 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 415	. . . . . 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 167	. . . . 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 115	. . . 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 2630	. . 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 2631	. 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		blex 15625	. . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (ball‘𝐷) ∈ V)
17		rnexg 5049	. . 3 ⊢ ((ball‘𝐷) ∈ V → ran (ball‘𝐷) ∈ V)
18		isbasis2g 15283	. . 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	16, 17, 18	3syl 17	. 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (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	mpbird 167	1 ⊢ (𝐷 ∈ (∞Met‘𝑋) → ran (ball‘𝐷) ∈ TopBases)

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1402   ∈ wcel 2209  ∀wral 2528  ∃wrex 2529  Vcvv 2821   ∩ cin 3219   ⊆ wss 3220  ∪ cuni 3935  ran crn 4777  ‘cfv 5380  (class class class)co 6088  ℝ⁺crp 10072  ∞Metcxmet 14983  ballcbl 14985  TopBasesctb 15280

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8277  ax-resscn 8278  ax-1cn 8279  ax-1re 8280  ax-icn 8281  ax-addcl 8282  ax-addrcl 8283  ax-mulcl 8284  ax-mulrcl 8285  ax-addcom 8286  ax-mulcom 8287  ax-addass 8288  ax-mulass 8289  ax-distr 8290  ax-i2m1 8291  ax-0lt1 8292  ax-1rid 8293  ax-0id 8294  ax-rnegex 8295  ax-precex 8296  ax-cnre 8297  ax-pre-ltirr 8298  ax-pre-ltwlin 8299  ax-pre-lttrn 8300  ax-pre-apti 8301  ax-pre-ltadd 8302  ax-pre-mulgt0 8303  ax-pre-mulext 8304  ax-arch 8305  ax-caucvg 8306

This proof depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-po 4443  df-iso 4444  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-isom 5389  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-frec 6666  df-map 6928  df-sup 7331  df-inf 7332  df-pnf 8369  df-mnf 8370  df-xr 8371  df-ltxr 8372  df-le 8373  df-sub 8507  df-neg 8508  df-reap 8912  df-ap 8919  df-div 9012  df-inn 9315  df-2 9373  df-3 9374  df-4 9375  df-n0 9576  df-z 9657  df-uz 9939  df-q 10037  df-rp 10073  df-xneg 10192  df-xadd 10193  df-seqfrec 10907  df-exp 10998  df-cj 11630  df-re 11631  df-im 11632  df-rsqrt 11787  df-abs 11788  df-psmet 14990  df-xmet 14991  df-bl 14993  df-bases 15281

This theorem is used by:  mopnval  15680  mopntopon  15681  elmopn  15684  blssopn  15723  metss  15732