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Theorem xblcntrps 22947
Description: A ball contains its center. (Contributed by NM, 2-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.) (Revised by Thierry Arnoux, 11-Mar-2018.)
Assertion
Ref Expression
xblcntrps ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃𝑋 ∧ (𝑅 ∈ ℝ* ∧ 0 < 𝑅)) → 𝑃 ∈ (𝑃(ball‘𝐷)𝑅))

Proof of Theorem xblcntrps
StepHypRef Expression
1 simp2 1129 . 2 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃𝑋 ∧ (𝑅 ∈ ℝ* ∧ 0 < 𝑅)) → 𝑃𝑋)
2 psmet0 22845 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃𝑋) → (𝑃𝐷𝑃) = 0)
323adant3 1124 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃𝑋 ∧ (𝑅 ∈ ℝ* ∧ 0 < 𝑅)) → (𝑃𝐷𝑃) = 0)
4 simp3r 1194 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃𝑋 ∧ (𝑅 ∈ ℝ* ∧ 0 < 𝑅)) → 0 < 𝑅)
53, 4eqbrtrd 5079 . 2 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃𝑋 ∧ (𝑅 ∈ ℝ* ∧ 0 < 𝑅)) → (𝑃𝐷𝑃) < 𝑅)
6 elblps 22924 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → (𝑃 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝑃𝑋 ∧ (𝑃𝐷𝑃) < 𝑅)))
763adant3r 1173 . 2 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃𝑋 ∧ (𝑅 ∈ ℝ* ∧ 0 < 𝑅)) → (𝑃 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝑃𝑋 ∧ (𝑃𝐷𝑃) < 𝑅)))
81, 5, 7mpbir2and 709 1 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃𝑋 ∧ (𝑅 ∈ ℝ* ∧ 0 < 𝑅)) → 𝑃 ∈ (𝑃(ball‘𝐷)𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1079   = wceq 1528  wcel 2105   class class class wbr 5057  cfv 6348  (class class class)co 7145  0cc0 10525  *cxr 10662   < clt 10663  PsMetcpsmet 20457  ballcbl 20460
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7678  df-2nd 7679  df-map 8397  df-xr 10667  df-psmet 20465  df-bl 20468
This theorem is referenced by:  blcntrps  22949
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