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Theorem xmetresbl 24434
Description: An extended metric restricted to any ball (in particular the infinity ball) is a proper metric. Together with xmetec 24431, 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 1133 . . 3 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → 𝐷 ∈ (∞Met‘𝑋))
2 xmetresbl.1 . . . 4 𝐵 = (𝑃(ball‘𝐷)𝑅)
3 blssm 24415 . . . 4 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑅) ⊆ 𝑋)
42, 3eqsstrid 4028 . . 3 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → 𝐵𝑋)
5 xmetres2 24358 . . 3 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵𝑋) → (𝐷 ↾ (𝐵 × 𝐵)) ∈ (∞Met‘𝐵))
61, 4, 5syl2anc 582 . 2 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → (𝐷 ↾ (𝐵 × 𝐵)) ∈ (∞Met‘𝐵))
7 xmetf 24326 . . . . . 6 (𝐷 ∈ (∞Met‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
81, 7syl 17 . . . . 5 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
9 xpss12 5697 . . . . . 6 ((𝐵𝑋𝐵𝑋) → (𝐵 × 𝐵) ⊆ (𝑋 × 𝑋))
104, 4, 9syl2anc 582 . . . . 5 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → (𝐵 × 𝐵) ⊆ (𝑋 × 𝑋))
118, 10fssresd 6769 . . . 4 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → (𝐷 ↾ (𝐵 × 𝐵)):(𝐵 × 𝐵)⟶ℝ*)
1211ffnd 6729 . . 3 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → (𝐷 ↾ (𝐵 × 𝐵)) Fn (𝐵 × 𝐵))
13 ovres 7592 . . . . . 6 ((𝑥𝐵𝑦𝐵) → (𝑥(𝐷 ↾ (𝐵 × 𝐵))𝑦) = (𝑥𝐷𝑦))
1413adantl 480 . . . . 5 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(𝐷 ↾ (𝐵 × 𝐵))𝑦) = (𝑥𝐷𝑦))
15 simpl1 1188 . . . . . . . . 9 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑥𝐵𝑦𝐵)) → 𝐷 ∈ (∞Met‘𝑋))
16 eqid 2726 . . . . . . . . . 10 (𝐷 “ ℝ) = (𝐷 “ ℝ)
1716xmeter 24430 . . . . . . . . 9 (𝐷 ∈ (∞Met‘𝑋) → (𝐷 “ ℝ) Er 𝑋)
1815, 17syl 17 . . . . . . . 8 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑥𝐵𝑦𝐵)) → (𝐷 “ ℝ) Er 𝑋)
1916blssec 24432 . . . . . . . . . . . 12 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑅) ⊆ [𝑃](𝐷 “ ℝ))
202, 19eqsstrid 4028 . . . . . . . . . . 11 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → 𝐵 ⊆ [𝑃](𝐷 “ ℝ))
2120sselda 3979 . . . . . . . . . 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 1189 . . . . . . . . . 10 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑥𝐵𝑦𝐵)) → 𝑃𝑋)
24 elecg 8778 . . . . . . . . . 10 ((𝑥 ∈ [𝑃](𝐷 “ ℝ) ∧ 𝑃𝑋) → (𝑥 ∈ [𝑃](𝐷 “ ℝ) ↔ 𝑃(𝐷 “ ℝ)𝑥))
2522, 23, 24syl2anc 582 . . . . . . . . 9 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑥𝐵𝑦𝐵)) → (𝑥 ∈ [𝑃](𝐷 “ ℝ) ↔ 𝑃(𝐷 “ ℝ)𝑥))
2622, 25mpbid 231 . . . . . . . 8 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑥𝐵𝑦𝐵)) → 𝑃(𝐷 “ ℝ)𝑥)
2720sselda 3979 . . . . . . . . . 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 8778 . . . . . . . . . 10 ((𝑦 ∈ [𝑃](𝐷 “ ℝ) ∧ 𝑃𝑋) → (𝑦 ∈ [𝑃](𝐷 “ ℝ) ↔ 𝑃(𝐷 “ ℝ)𝑦))
3028, 23, 29syl2anc 582 . . . . . . . . 9 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑥𝐵𝑦𝐵)) → (𝑦 ∈ [𝑃](𝐷 “ ℝ) ↔ 𝑃(𝐷 “ ℝ)𝑦))
3128, 30mpbid 231 . . . . . . . 8 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑥𝐵𝑦𝐵)) → 𝑃(𝐷 “ ℝ)𝑦)
3218, 26, 31ertr3d 8752 . . . . . . 7 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑥𝐵𝑦𝐵)) → 𝑥(𝐷 “ ℝ)𝑦)
3316xmeterval 24429 . . . . . . . 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 231 . . . . . 6 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑥𝐵𝑦𝐵)) → (𝑥𝑋𝑦𝑋 ∧ (𝑥𝐷𝑦) ∈ ℝ))
3635simp3d 1141 . . . . 5 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑥𝐵𝑦𝐵)) → (𝑥𝐷𝑦) ∈ ℝ)
3714, 36eqeltrd 2826 . . . 4 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(𝐷 ↾ (𝐵 × 𝐵))𝑦) ∈ ℝ)
3837ralrimivva 3191 . . 3 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → ∀𝑥𝐵𝑦𝐵 (𝑥(𝐷 ↾ (𝐵 × 𝐵))𝑦) ∈ ℝ)
39 ffnov 7552 . . 3 ((𝐷 ↾ (𝐵 × 𝐵)):(𝐵 × 𝐵)⟶ℝ ↔ ((𝐷 ↾ (𝐵 × 𝐵)) Fn (𝐵 × 𝐵) ∧ ∀𝑥𝐵𝑦𝐵 (𝑥(𝐷 ↾ (𝐵 × 𝐵))𝑦) ∈ ℝ))
4012, 38, 39sylanbrc 581 . 2 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → (𝐷 ↾ (𝐵 × 𝐵)):(𝐵 × 𝐵)⟶ℝ)
41 ismet2 24330 . 2 ((𝐷 ↾ (𝐵 × 𝐵)) ∈ (Met‘𝐵) ↔ ((𝐷 ↾ (𝐵 × 𝐵)) ∈ (∞Met‘𝐵) ∧ (𝐷 ↾ (𝐵 × 𝐵)):(𝐵 × 𝐵)⟶ℝ))
426, 40, 41sylanbrc 581 1 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → (𝐷 ↾ (𝐵 × 𝐵)) ∈ (Met‘𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1084   = wceq 1534  wcel 2099  wral 3051  wss 3947   class class class wbr 5153   × cxp 5680  ccnv 5681  cres 5684  cima 5685   Fn wfn 6549  wf 6550  cfv 6554  (class class class)co 7424   Er wer 8731  [cec 8732  cr 11157  *cxr 11297  ∞Metcxmet 21328  Metcmet 21329  ballcbl 21330
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11214  ax-resscn 11215  ax-1cn 11216  ax-icn 11217  ax-addcl 11218  ax-addrcl 11219  ax-mulcl 11220  ax-mulrcl 11221  ax-mulcom 11222  ax-addass 11223  ax-mulass 11224  ax-distr 11225  ax-i2m1 11226  ax-1ne0 11227  ax-1rid 11228  ax-rnegex 11229  ax-rrecex 11230  ax-cnre 11231  ax-pre-lttri 11232  ax-pre-lttrn 11233  ax-pre-ltadd 11234  ax-pre-mulgt0 11235
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-po 5594  df-so 5595  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-1st 8003  df-2nd 8004  df-er 8734  df-ec 8736  df-map 8857  df-en 8975  df-dom 8976  df-sdom 8977  df-pnf 11300  df-mnf 11301  df-xr 11302  df-ltxr 11303  df-le 11304  df-sub 11496  df-neg 11497  df-div 11922  df-2 12327  df-rp 13029  df-xneg 13146  df-xadd 13147  df-xmul 13148  df-psmet 21335  df-xmet 21336  df-met 21337  df-bl 21338
This theorem is referenced by: (None)
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