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

Theorem xmetresbl 23023
Description: An extended metric restricted to any ball (in particular the infinity ball) is a proper metric. Together with xmetec 23020, 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 1132 . . 3 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → 𝐷 ∈ (∞Met‘𝑋))
2 xmetresbl.1 . . . 4 𝐵 = (𝑃(ball‘𝐷)𝑅)
3 blssm 23004 . . . 4 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑅) ⊆ 𝑋)
42, 3eqsstrid 3994 . . 3 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → 𝐵𝑋)
5 xmetres2 22947 . . 3 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵𝑋) → (𝐷 ↾ (𝐵 × 𝐵)) ∈ (∞Met‘𝐵))
61, 4, 5syl2anc 586 . 2 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → (𝐷 ↾ (𝐵 × 𝐵)) ∈ (∞Met‘𝐵))
7 xmetf 22915 . . . . . 6 (𝐷 ∈ (∞Met‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
81, 7syl 17 . . . . 5 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
9 xpss12 5546 . . . . . 6 ((𝐵𝑋𝐵𝑋) → (𝐵 × 𝐵) ⊆ (𝑋 × 𝑋))
104, 4, 9syl2anc 586 . . . . 5 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → (𝐵 × 𝐵) ⊆ (𝑋 × 𝑋))
118, 10fssresd 6521 . . . 4 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → (𝐷 ↾ (𝐵 × 𝐵)):(𝐵 × 𝐵)⟶ℝ*)
1211ffnd 6491 . . 3 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → (𝐷 ↾ (𝐵 × 𝐵)) Fn (𝐵 × 𝐵))
13 ovres 7292 . . . . . 6 ((𝑥𝐵𝑦𝐵) → (𝑥(𝐷 ↾ (𝐵 × 𝐵))𝑦) = (𝑥𝐷𝑦))
1413adantl 484 . . . . 5 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(𝐷 ↾ (𝐵 × 𝐵))𝑦) = (𝑥𝐷𝑦))
15 simpl1 1187 . . . . . . . . 9 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑥𝐵𝑦𝐵)) → 𝐷 ∈ (∞Met‘𝑋))
16 eqid 2820 . . . . . . . . . 10 (𝐷 “ ℝ) = (𝐷 “ ℝ)
1716xmeter 23019 . . . . . . . . 9 (𝐷 ∈ (∞Met‘𝑋) → (𝐷 “ ℝ) Er 𝑋)
1815, 17syl 17 . . . . . . . 8 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑥𝐵𝑦𝐵)) → (𝐷 “ ℝ) Er 𝑋)
1916blssec 23021 . . . . . . . . . . . 12 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑅) ⊆ [𝑃](𝐷 “ ℝ))
202, 19eqsstrid 3994 . . . . . . . . . . 11 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → 𝐵 ⊆ [𝑃](𝐷 “ ℝ))
2120sselda 3946 . . . . . . . . . 10 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ 𝑥𝐵) → 𝑥 ∈ [𝑃](𝐷 “ ℝ))
2221adantrr 715 . . . . . . . . 9 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑥𝐵𝑦𝐵)) → 𝑥 ∈ [𝑃](𝐷 “ ℝ))
23 simpl2 1188 . . . . . . . . . 10 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑥𝐵𝑦𝐵)) → 𝑃𝑋)
24 elecg 8310 . . . . . . . . . 10 ((𝑥 ∈ [𝑃](𝐷 “ ℝ) ∧ 𝑃𝑋) → (𝑥 ∈ [𝑃](𝐷 “ ℝ) ↔ 𝑃(𝐷 “ ℝ)𝑥))
2522, 23, 24syl2anc 586 . . . . . . . . 9 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑥𝐵𝑦𝐵)) → (𝑥 ∈ [𝑃](𝐷 “ ℝ) ↔ 𝑃(𝐷 “ ℝ)𝑥))
2622, 25mpbid 234 . . . . . . . 8 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑥𝐵𝑦𝐵)) → 𝑃(𝐷 “ ℝ)𝑥)
2720sselda 3946 . . . . . . . . . 10 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ 𝑦𝐵) → 𝑦 ∈ [𝑃](𝐷 “ ℝ))
2827adantrl 714 . . . . . . . . 9 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑥𝐵𝑦𝐵)) → 𝑦 ∈ [𝑃](𝐷 “ ℝ))
29 elecg 8310 . . . . . . . . . 10 ((𝑦 ∈ [𝑃](𝐷 “ ℝ) ∧ 𝑃𝑋) → (𝑦 ∈ [𝑃](𝐷 “ ℝ) ↔ 𝑃(𝐷 “ ℝ)𝑦))
3028, 23, 29syl2anc 586 . . . . . . . . 9 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑥𝐵𝑦𝐵)) → (𝑦 ∈ [𝑃](𝐷 “ ℝ) ↔ 𝑃(𝐷 “ ℝ)𝑦))
3128, 30mpbid 234 . . . . . . . 8 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑥𝐵𝑦𝐵)) → 𝑃(𝐷 “ ℝ)𝑦)
3218, 26, 31ertr3d 8285 . . . . . . 7 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑥𝐵𝑦𝐵)) → 𝑥(𝐷 “ ℝ)𝑦)
3316xmeterval 23018 . . . . . . . 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 234 . . . . . 6 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑥𝐵𝑦𝐵)) → (𝑥𝑋𝑦𝑋 ∧ (𝑥𝐷𝑦) ∈ ℝ))
3635simp3d 1140 . . . . 5 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑥𝐵𝑦𝐵)) → (𝑥𝐷𝑦) ∈ ℝ)
3714, 36eqeltrd 2911 . . . 4 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(𝐷 ↾ (𝐵 × 𝐵))𝑦) ∈ ℝ)
3837ralrimivva 3178 . . 3 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → ∀𝑥𝐵𝑦𝐵 (𝑥(𝐷 ↾ (𝐵 × 𝐵))𝑦) ∈ ℝ)
39 ffnov 7255 . . 3 ((𝐷 ↾ (𝐵 × 𝐵)):(𝐵 × 𝐵)⟶ℝ ↔ ((𝐷 ↾ (𝐵 × 𝐵)) Fn (𝐵 × 𝐵) ∧ ∀𝑥𝐵𝑦𝐵 (𝑥(𝐷 ↾ (𝐵 × 𝐵))𝑦) ∈ ℝ))
4012, 38, 39sylanbrc 585 . 2 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → (𝐷 ↾ (𝐵 × 𝐵)):(𝐵 × 𝐵)⟶ℝ)
41 ismet2 22919 . 2 ((𝐷 ↾ (𝐵 × 𝐵)) ∈ (Met‘𝐵) ↔ ((𝐷 ↾ (𝐵 × 𝐵)) ∈ (∞Met‘𝐵) ∧ (𝐷 ↾ (𝐵 × 𝐵)):(𝐵 × 𝐵)⟶ℝ))
426, 40, 41sylanbrc 585 1 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → (𝐷 ↾ (𝐵 × 𝐵)) ∈ (Met‘𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  w3a 1083   = wceq 1537  wcel 2114  wral 3125  wss 3913   class class class wbr 5042   × cxp 5529  ccnv 5530  cres 5533  cima 5534   Fn wfn 6326  wf 6327  cfv 6331  (class class class)co 7133   Er wer 8264  [cec 8265  cr 10514  *cxr 10652  ∞Metcxmet 20506  Metcmet 20507  ballcbl 20508
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306  ax-un 7439  ax-cnex 10571  ax-resscn 10572  ax-1cn 10573  ax-icn 10574  ax-addcl 10575  ax-addrcl 10576  ax-mulcl 10577  ax-mulrcl 10578  ax-mulcom 10579  ax-addass 10580  ax-mulass 10581  ax-distr 10582  ax-i2m1 10583  ax-1ne0 10584  ax-1rid 10585  ax-rnegex 10586  ax-rrecex 10587  ax-cnre 10588  ax-pre-lttri 10589  ax-pre-lttrn 10590  ax-pre-ltadd 10591  ax-pre-mulgt0 10592
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-nel 3111  df-ral 3130  df-rex 3131  df-reu 3132  df-rmo 3133  df-rab 3134  df-v 3475  df-sbc 3753  df-csb 3861  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-pw 4517  df-sn 4544  df-pr 4546  df-op 4550  df-uni 4815  df-iun 4897  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5436  df-po 5450  df-so 5451  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-iota 6290  df-fun 6333  df-fn 6334  df-f 6335  df-f1 6336  df-fo 6337  df-f1o 6338  df-fv 6339  df-riota 7091  df-ov 7136  df-oprab 7137  df-mpo 7138  df-1st 7667  df-2nd 7668  df-er 8267  df-ec 8269  df-map 8386  df-en 8488  df-dom 8489  df-sdom 8490  df-pnf 10655  df-mnf 10656  df-xr 10657  df-ltxr 10658  df-le 10659  df-sub 10850  df-neg 10851  df-div 11276  df-2 11679  df-rp 12369  df-xneg 12486  df-xadd 12487  df-xmul 12488  df-psmet 20513  df-xmet 20514  df-met 20515  df-bl 20516
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator