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Theorem blvalps 23722
Description: The ball around a point 𝑃 is the set of all points whose distance from 𝑃 is less than the ball's radius 𝑅. (Contributed by NM, 31-Aug-2006.) (Revised by Mario Carneiro, 11-Nov-2013.) (Revised by Thierry Arnoux, 11-Mar-2018.)
Assertion
Ref Expression
blvalps ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑅) = {𝑥𝑋 ∣ (𝑃𝐷𝑥) < 𝑅})
Distinct variable groups:   𝑥,𝑃   𝑥,𝐷   𝑥,𝑅   𝑥,𝑋

Proof of Theorem blvalps
Dummy variables 𝑟 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 blfvalps 23720 . . 3 (𝐷 ∈ (PsMet‘𝑋) → (ball‘𝐷) = (𝑦𝑋, 𝑟 ∈ ℝ* ↦ {𝑥𝑋 ∣ (𝑦𝐷𝑥) < 𝑟}))
213ad2ant1 1133 . 2 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → (ball‘𝐷) = (𝑦𝑋, 𝑟 ∈ ℝ* ↦ {𝑥𝑋 ∣ (𝑦𝐷𝑥) < 𝑟}))
3 simprl 769 . . . . 5 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑦 = 𝑃𝑟 = 𝑅)) → 𝑦 = 𝑃)
43oveq1d 7368 . . . 4 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑦 = 𝑃𝑟 = 𝑅)) → (𝑦𝐷𝑥) = (𝑃𝐷𝑥))
5 simprr 771 . . . 4 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑦 = 𝑃𝑟 = 𝑅)) → 𝑟 = 𝑅)
64, 5breq12d 5116 . . 3 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑦 = 𝑃𝑟 = 𝑅)) → ((𝑦𝐷𝑥) < 𝑟 ↔ (𝑃𝐷𝑥) < 𝑅))
76rabbidv 3413 . 2 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑦 = 𝑃𝑟 = 𝑅)) → {𝑥𝑋 ∣ (𝑦𝐷𝑥) < 𝑟} = {𝑥𝑋 ∣ (𝑃𝐷𝑥) < 𝑅})
8 simp2 1137 . 2 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → 𝑃𝑋)
9 simp3 1138 . 2 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → 𝑅 ∈ ℝ*)
10 elfvdm 6876 . . . 4 (𝐷 ∈ (PsMet‘𝑋) → 𝑋 ∈ dom PsMet)
11103ad2ant1 1133 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → 𝑋 ∈ dom PsMet)
12 rabexg 5286 . . 3 (𝑋 ∈ dom PsMet → {𝑥𝑋 ∣ (𝑃𝐷𝑥) < 𝑅} ∈ V)
1311, 12syl 17 . 2 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → {𝑥𝑋 ∣ (𝑃𝐷𝑥) < 𝑅} ∈ V)
142, 7, 8, 9, 13ovmpod 7503 1 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑅) = {𝑥𝑋 ∣ (𝑃𝐷𝑥) < 𝑅})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087   = wceq 1541  wcel 2106  {crab 3405  Vcvv 3443   class class class wbr 5103  dom cdm 5631  cfv 6493  (class class class)co 7353  cmpo 7355  *cxr 11184   < clt 11185  PsMetcpsmet 20765  ballcbl 20768
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7668  ax-cnex 11103  ax-resscn 11104
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-fv 6501  df-ov 7356  df-oprab 7357  df-mpo 7358  df-1st 7917  df-2nd 7918  df-map 8763  df-xr 11189  df-psmet 20773  df-bl 20776
This theorem is referenced by:  elblps  23724  blval2  23902
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