MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  xmetresbl Structured version   Visualization version   GIF version

Theorem xmetresbl 22974
Description: An extended metric restricted to any ball (in particular the infinity ball) is a proper metric. Together with xmetec 22971, this shows that any extended metric space can be "factored" into the disjoint union of proper metric spaces, with points in the same region measured by that region's metric, and points in different regions being distance +∞ from each other. (Contributed by Mario Carneiro, 23-Aug-2015.)
Hypothesis
Ref Expression
xmetresbl.1 𝐵 = (𝑃(ball‘𝐷)𝑅)
Assertion
Ref Expression
xmetresbl ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → (𝐷 ↾ (𝐵 × 𝐵)) ∈ (Met‘𝐵))

Proof of Theorem xmetresbl
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp1 1128 . . 3 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → 𝐷 ∈ (∞Met‘𝑋))
2 xmetresbl.1 . . . 4 𝐵 = (𝑃(ball‘𝐷)𝑅)
3 blssm 22955 . . . 4 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑅) ⊆ 𝑋)
42, 3eqsstrid 4012 . . 3 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → 𝐵𝑋)
5 xmetres2 22898 . . 3 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵𝑋) → (𝐷 ↾ (𝐵 × 𝐵)) ∈ (∞Met‘𝐵))
61, 4, 5syl2anc 584 . 2 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → (𝐷 ↾ (𝐵 × 𝐵)) ∈ (∞Met‘𝐵))
7 xmetf 22866 . . . . . 6 (𝐷 ∈ (∞Met‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
81, 7syl 17 . . . . 5 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
9 xpss12 5563 . . . . . 6 ((𝐵𝑋𝐵𝑋) → (𝐵 × 𝐵) ⊆ (𝑋 × 𝑋))
104, 4, 9syl2anc 584 . . . . 5 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → (𝐵 × 𝐵) ⊆ (𝑋 × 𝑋))
118, 10fssresd 6538 . . . 4 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → (𝐷 ↾ (𝐵 × 𝐵)):(𝐵 × 𝐵)⟶ℝ*)
1211ffnd 6508 . . 3 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → (𝐷 ↾ (𝐵 × 𝐵)) Fn (𝐵 × 𝐵))
13 ovres 7303 . . . . . 6 ((𝑥𝐵𝑦𝐵) → (𝑥(𝐷 ↾ (𝐵 × 𝐵))𝑦) = (𝑥𝐷𝑦))
1413adantl 482 . . . . 5 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(𝐷 ↾ (𝐵 × 𝐵))𝑦) = (𝑥𝐷𝑦))
15 simpl1 1183 . . . . . . . . 9 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑥𝐵𝑦𝐵)) → 𝐷 ∈ (∞Met‘𝑋))
16 eqid 2818 . . . . . . . . . 10 (𝐷 “ ℝ) = (𝐷 “ ℝ)
1716xmeter 22970 . . . . . . . . 9 (𝐷 ∈ (∞Met‘𝑋) → (𝐷 “ ℝ) Er 𝑋)
1815, 17syl 17 . . . . . . . 8 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑥𝐵𝑦𝐵)) → (𝐷 “ ℝ) Er 𝑋)
1916blssec 22972 . . . . . . . . . . . 12 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑅) ⊆ [𝑃](𝐷 “ ℝ))
202, 19eqsstrid 4012 . . . . . . . . . . 11 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → 𝐵 ⊆ [𝑃](𝐷 “ ℝ))
2120sselda 3964 . . . . . . . . . 10 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ 𝑥𝐵) → 𝑥 ∈ [𝑃](𝐷 “ ℝ))
2221adantrr 713 . . . . . . . . 9 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑥𝐵𝑦𝐵)) → 𝑥 ∈ [𝑃](𝐷 “ ℝ))
23 simpl2 1184 . . . . . . . . . 10 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑥𝐵𝑦𝐵)) → 𝑃𝑋)
24 elecg 8321 . . . . . . . . . 10 ((𝑥 ∈ [𝑃](𝐷 “ ℝ) ∧ 𝑃𝑋) → (𝑥 ∈ [𝑃](𝐷 “ ℝ) ↔ 𝑃(𝐷 “ ℝ)𝑥))
2522, 23, 24syl2anc 584 . . . . . . . . 9 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑥𝐵𝑦𝐵)) → (𝑥 ∈ [𝑃](𝐷 “ ℝ) ↔ 𝑃(𝐷 “ ℝ)𝑥))
2622, 25mpbid 233 . . . . . . . 8 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑥𝐵𝑦𝐵)) → 𝑃(𝐷 “ ℝ)𝑥)
2720sselda 3964 . . . . . . . . . 10 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ 𝑦𝐵) → 𝑦 ∈ [𝑃](𝐷 “ ℝ))
2827adantrl 712 . . . . . . . . 9 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑥𝐵𝑦𝐵)) → 𝑦 ∈ [𝑃](𝐷 “ ℝ))
29 elecg 8321 . . . . . . . . . 10 ((𝑦 ∈ [𝑃](𝐷 “ ℝ) ∧ 𝑃𝑋) → (𝑦 ∈ [𝑃](𝐷 “ ℝ) ↔ 𝑃(𝐷 “ ℝ)𝑦))
3028, 23, 29syl2anc 584 . . . . . . . . 9 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑥𝐵𝑦𝐵)) → (𝑦 ∈ [𝑃](𝐷 “ ℝ) ↔ 𝑃(𝐷 “ ℝ)𝑦))
3128, 30mpbid 233 . . . . . . . 8 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑥𝐵𝑦𝐵)) → 𝑃(𝐷 “ ℝ)𝑦)
3218, 26, 31ertr3d 8296 . . . . . . 7 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑥𝐵𝑦𝐵)) → 𝑥(𝐷 “ ℝ)𝑦)
3316xmeterval 22969 . . . . . . . 8 (𝐷 ∈ (∞Met‘𝑋) → (𝑥(𝐷 “ ℝ)𝑦 ↔ (𝑥𝑋𝑦𝑋 ∧ (𝑥𝐷𝑦) ∈ ℝ)))
3415, 33syl 17 . . . . . . 7 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(𝐷 “ ℝ)𝑦 ↔ (𝑥𝑋𝑦𝑋 ∧ (𝑥𝐷𝑦) ∈ ℝ)))
3532, 34mpbid 233 . . . . . 6 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑥𝐵𝑦𝐵)) → (𝑥𝑋𝑦𝑋 ∧ (𝑥𝐷𝑦) ∈ ℝ))
3635simp3d 1136 . . . . 5 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑥𝐵𝑦𝐵)) → (𝑥𝐷𝑦) ∈ ℝ)
3714, 36eqeltrd 2910 . . . 4 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(𝐷 ↾ (𝐵 × 𝐵))𝑦) ∈ ℝ)
3837ralrimivva 3188 . . 3 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → ∀𝑥𝐵𝑦𝐵 (𝑥(𝐷 ↾ (𝐵 × 𝐵))𝑦) ∈ ℝ)
39 ffnov 7267 . . 3 ((𝐷 ↾ (𝐵 × 𝐵)):(𝐵 × 𝐵)⟶ℝ ↔ ((𝐷 ↾ (𝐵 × 𝐵)) Fn (𝐵 × 𝐵) ∧ ∀𝑥𝐵𝑦𝐵 (𝑥(𝐷 ↾ (𝐵 × 𝐵))𝑦) ∈ ℝ))
4012, 38, 39sylanbrc 583 . 2 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → (𝐷 ↾ (𝐵 × 𝐵)):(𝐵 × 𝐵)⟶ℝ)
41 ismet2 22870 . 2 ((𝐷 ↾ (𝐵 × 𝐵)) ∈ (Met‘𝐵) ↔ ((𝐷 ↾ (𝐵 × 𝐵)) ∈ (∞Met‘𝐵) ∧ (𝐷 ↾ (𝐵 × 𝐵)):(𝐵 × 𝐵)⟶ℝ))
426, 40, 41sylanbrc 583 1 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → (𝐷 ↾ (𝐵 × 𝐵)) ∈ (Met‘𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  wral 3135  wss 3933   class class class wbr 5057   × cxp 5546  ccnv 5547  cres 5550  cima 5551   Fn wfn 6343  wf 6344  cfv 6348  (class class class)co 7145   Er wer 8275  [cec 8276  cr 10524  *cxr 10662  ∞Metcxmet 20458  Metcmet 20459  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  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  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-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  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-po 5467  df-so 5468  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-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7678  df-2nd 7679  df-er 8278  df-ec 8280  df-map 8397  df-en 8498  df-dom 8499  df-sdom 8500  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-div 11286  df-2 11688  df-rp 12378  df-xneg 12495  df-xadd 12496  df-xmul 12497  df-psmet 20465  df-xmet 20466  df-met 20467  df-bl 20468
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator