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Theorem blres 22975
Description: A ball in a restricted metric space. (Contributed by Mario Carneiro, 5-Jan-2014.)
Hypothesis
Ref Expression
blres.2 𝐶 = (𝐷 ↾ (𝑌 × 𝑌))
Assertion
Ref Expression
blres ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝑋𝑌) ∧ 𝑅 ∈ ℝ*) → (𝑃(ball‘𝐶)𝑅) = ((𝑃(ball‘𝐷)𝑅) ∩ 𝑌))

Proof of Theorem blres
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 elinel2 4177 . . . . . . . . 9 (𝑃 ∈ (𝑋𝑌) → 𝑃𝑌)
2 blres.2 . . . . . . . . . . 11 𝐶 = (𝐷 ↾ (𝑌 × 𝑌))
32oveqi 7163 . . . . . . . . . 10 (𝑃𝐶𝑥) = (𝑃(𝐷 ↾ (𝑌 × 𝑌))𝑥)
4 ovres 7308 . . . . . . . . . 10 ((𝑃𝑌𝑥𝑌) → (𝑃(𝐷 ↾ (𝑌 × 𝑌))𝑥) = (𝑃𝐷𝑥))
53, 4syl5eq 2873 . . . . . . . . 9 ((𝑃𝑌𝑥𝑌) → (𝑃𝐶𝑥) = (𝑃𝐷𝑥))
61, 5sylan 580 . . . . . . . 8 ((𝑃 ∈ (𝑋𝑌) ∧ 𝑥𝑌) → (𝑃𝐶𝑥) = (𝑃𝐷𝑥))
76breq1d 5073 . . . . . . 7 ((𝑃 ∈ (𝑋𝑌) ∧ 𝑥𝑌) → ((𝑃𝐶𝑥) < 𝑅 ↔ (𝑃𝐷𝑥) < 𝑅))
87anbi2d 628 . . . . . 6 ((𝑃 ∈ (𝑋𝑌) ∧ 𝑥𝑌) → ((𝑥𝑋 ∧ (𝑃𝐶𝑥) < 𝑅) ↔ (𝑥𝑋 ∧ (𝑃𝐷𝑥) < 𝑅)))
98pm5.32da 579 . . . . 5 (𝑃 ∈ (𝑋𝑌) → ((𝑥𝑌 ∧ (𝑥𝑋 ∧ (𝑃𝐶𝑥) < 𝑅)) ↔ (𝑥𝑌 ∧ (𝑥𝑋 ∧ (𝑃𝐷𝑥) < 𝑅))))
1093ad2ant2 1128 . . . 4 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝑋𝑌) ∧ 𝑅 ∈ ℝ*) → ((𝑥𝑌 ∧ (𝑥𝑋 ∧ (𝑃𝐶𝑥) < 𝑅)) ↔ (𝑥𝑌 ∧ (𝑥𝑋 ∧ (𝑃𝐷𝑥) < 𝑅))))
11 elin 4173 . . . . . . 7 (𝑥 ∈ (𝑋𝑌) ↔ (𝑥𝑋𝑥𝑌))
1211biancomi 463 . . . . . 6 (𝑥 ∈ (𝑋𝑌) ↔ (𝑥𝑌𝑥𝑋))
1312anbi1i 623 . . . . 5 ((𝑥 ∈ (𝑋𝑌) ∧ (𝑃𝐶𝑥) < 𝑅) ↔ ((𝑥𝑌𝑥𝑋) ∧ (𝑃𝐶𝑥) < 𝑅))
14 anass 469 . . . . 5 (((𝑥𝑌𝑥𝑋) ∧ (𝑃𝐶𝑥) < 𝑅) ↔ (𝑥𝑌 ∧ (𝑥𝑋 ∧ (𝑃𝐶𝑥) < 𝑅)))
1513, 14bitri 276 . . . 4 ((𝑥 ∈ (𝑋𝑌) ∧ (𝑃𝐶𝑥) < 𝑅) ↔ (𝑥𝑌 ∧ (𝑥𝑋 ∧ (𝑃𝐶𝑥) < 𝑅)))
16 ancom 461 . . . 4 (((𝑥𝑋 ∧ (𝑃𝐷𝑥) < 𝑅) ∧ 𝑥𝑌) ↔ (𝑥𝑌 ∧ (𝑥𝑋 ∧ (𝑃𝐷𝑥) < 𝑅)))
1710, 15, 163bitr4g 315 . . 3 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝑋𝑌) ∧ 𝑅 ∈ ℝ*) → ((𝑥 ∈ (𝑋𝑌) ∧ (𝑃𝐶𝑥) < 𝑅) ↔ ((𝑥𝑋 ∧ (𝑃𝐷𝑥) < 𝑅) ∧ 𝑥𝑌)))
18 xmetres 22908 . . . . 5 (𝐷 ∈ (∞Met‘𝑋) → (𝐷 ↾ (𝑌 × 𝑌)) ∈ (∞Met‘(𝑋𝑌)))
192, 18eqeltrid 2922 . . . 4 (𝐷 ∈ (∞Met‘𝑋) → 𝐶 ∈ (∞Met‘(𝑋𝑌)))
20 elbl 22932 . . . 4 ((𝐶 ∈ (∞Met‘(𝑋𝑌)) ∧ 𝑃 ∈ (𝑋𝑌) ∧ 𝑅 ∈ ℝ*) → (𝑥 ∈ (𝑃(ball‘𝐶)𝑅) ↔ (𝑥 ∈ (𝑋𝑌) ∧ (𝑃𝐶𝑥) < 𝑅)))
2119, 20syl3an1 1157 . . 3 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝑋𝑌) ∧ 𝑅 ∈ ℝ*) → (𝑥 ∈ (𝑃(ball‘𝐶)𝑅) ↔ (𝑥 ∈ (𝑋𝑌) ∧ (𝑃𝐶𝑥) < 𝑅)))
22 elin 4173 . . . 4 (𝑥 ∈ ((𝑃(ball‘𝐷)𝑅) ∩ 𝑌) ↔ (𝑥 ∈ (𝑃(ball‘𝐷)𝑅) ∧ 𝑥𝑌))
23 elinel1 4176 . . . . . 6 (𝑃 ∈ (𝑋𝑌) → 𝑃𝑋)
24 elbl 22932 . . . . . 6 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → (𝑥 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝑥𝑋 ∧ (𝑃𝐷𝑥) < 𝑅)))
2523, 24syl3an2 1158 . . . . 5 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝑋𝑌) ∧ 𝑅 ∈ ℝ*) → (𝑥 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝑥𝑋 ∧ (𝑃𝐷𝑥) < 𝑅)))
2625anbi1d 629 . . . 4 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝑋𝑌) ∧ 𝑅 ∈ ℝ*) → ((𝑥 ∈ (𝑃(ball‘𝐷)𝑅) ∧ 𝑥𝑌) ↔ ((𝑥𝑋 ∧ (𝑃𝐷𝑥) < 𝑅) ∧ 𝑥𝑌)))
2722, 26syl5bb 284 . . 3 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝑋𝑌) ∧ 𝑅 ∈ ℝ*) → (𝑥 ∈ ((𝑃(ball‘𝐷)𝑅) ∩ 𝑌) ↔ ((𝑥𝑋 ∧ (𝑃𝐷𝑥) < 𝑅) ∧ 𝑥𝑌)))
2817, 21, 273bitr4d 312 . 2 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝑋𝑌) ∧ 𝑅 ∈ ℝ*) → (𝑥 ∈ (𝑃(ball‘𝐶)𝑅) ↔ 𝑥 ∈ ((𝑃(ball‘𝐷)𝑅) ∩ 𝑌)))
2928eqrdv 2824 1 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝑋𝑌) ∧ 𝑅 ∈ ℝ*) → (𝑃(ball‘𝐶)𝑅) = ((𝑃(ball‘𝐷)𝑅) ∩ 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1081   = wceq 1530  wcel 2107  cin 3939   class class class wbr 5063   × cxp 5552  cres 5556  cfv 6354  (class class class)co 7150  *cxr 10668   < clt 10669  ∞Metcxmet 20465  ballcbl 20467
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455  ax-cnex 10587  ax-resscn 10588
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-fv 6362  df-ov 7153  df-oprab 7154  df-mpo 7155  df-1st 7685  df-2nd 7686  df-map 8403  df-xr 10673  df-psmet 20472  df-xmet 20473  df-bl 20475
This theorem is referenced by:  metrest  23068  xrsmopn  23354  lebnumii  23504  blssp  34918  sstotbnd2  34939  blbnd  34952  ssbnd  34953  iooabslt  41658
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