Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ballss3 Structured version   Visualization version   GIF version

Theorem ballss3 44236
Description: A sufficient condition for a ball being a subset. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
Hypotheses
Ref Expression
ballss3.y 𝑥𝜑
ballss3.d (𝜑𝐷 ∈ (PsMet‘𝑋))
ballss3.p (𝜑𝑃𝑋)
ballss3.r (𝜑𝑅 ∈ ℝ*)
ballss3.a ((𝜑𝑥𝑋 ∧ (𝑃𝐷𝑥) < 𝑅) → 𝑥𝐴)
Assertion
Ref Expression
ballss3 (𝜑 → (𝑃(ball‘𝐷)𝑅) ⊆ 𝐴)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐷   𝑥,𝑃   𝑥,𝑅
Allowed substitution hints:   𝜑(𝑥)   𝑋(𝑥)

Proof of Theorem ballss3
StepHypRef Expression
1 ballss3.y . . 3 𝑥𝜑
2 simpl 482 . . . . 5 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 𝜑)
3 simpr 484 . . . . . . 7 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 𝑥 ∈ (𝑃(ball‘𝐷)𝑅))
4 ballss3.d . . . . . . . . 9 (𝜑𝐷 ∈ (PsMet‘𝑋))
5 ballss3.p . . . . . . . . 9 (𝜑𝑃𝑋)
6 ballss3.r . . . . . . . . 9 (𝜑𝑅 ∈ ℝ*)
7 elblps 24214 . . . . . . . . 9 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → (𝑥 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝑥𝑋 ∧ (𝑃𝐷𝑥) < 𝑅)))
84, 5, 6, 7syl3anc 1368 . . . . . . . 8 (𝜑 → (𝑥 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝑥𝑋 ∧ (𝑃𝐷𝑥) < 𝑅)))
98adantr 480 . . . . . . 7 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑥 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝑥𝑋 ∧ (𝑃𝐷𝑥) < 𝑅)))
103, 9mpbid 231 . . . . . 6 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑥𝑋 ∧ (𝑃𝐷𝑥) < 𝑅))
1110simpld 494 . . . . 5 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 𝑥𝑋)
1210simprd 495 . . . . 5 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑃𝐷𝑥) < 𝑅)
13 ballss3.a . . . . 5 ((𝜑𝑥𝑋 ∧ (𝑃𝐷𝑥) < 𝑅) → 𝑥𝐴)
142, 11, 12, 13syl3anc 1368 . . . 4 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 𝑥𝐴)
1514ex 412 . . 3 (𝜑 → (𝑥 ∈ (𝑃(ball‘𝐷)𝑅) → 𝑥𝐴))
161, 15ralrimi 3246 . 2 (𝜑 → ∀𝑥 ∈ (𝑃(ball‘𝐷)𝑅)𝑥𝐴)
17 dfss3 3962 . 2 ((𝑃(ball‘𝐷)𝑅) ⊆ 𝐴 ↔ ∀𝑥 ∈ (𝑃(ball‘𝐷)𝑅)𝑥𝐴)
1816, 17sylibr 233 1 (𝜑 → (𝑃(ball‘𝐷)𝑅) ⊆ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1084  wnf 1777  wcel 2098  wral 3053  wss 3940   class class class wbr 5138  cfv 6533  (class class class)co 7401  *cxr 11243   < clt 11244  PsMetcpsmet 21211  ballcbl 21214
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718  ax-cnex 11161  ax-resscn 11162
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-fv 6541  df-ov 7404  df-oprab 7405  df-mpo 7406  df-1st 7968  df-2nd 7969  df-map 8817  df-xr 11248  df-psmet 21219  df-bl 21222
This theorem is referenced by:  ioorrnopnlem  45471
  Copyright terms: Public domain W3C validator