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Theorem xmetresbl 12429
Description: An extended metric restricted to any ball (in particular the infinity ball) is a proper metric. Together with xmetec 12426, 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 964 . . 3 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → 𝐷 ∈ (∞Met‘𝑋))
2 xmetresbl.1 . . . 4 𝐵 = (𝑃(ball‘𝐷)𝑅)
3 blssm 12410 . . . 4 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑅) ⊆ 𝑋)
42, 3eqsstrid 3109 . . 3 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → 𝐵𝑋)
5 xmetres2 12368 . . 3 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵𝑋) → (𝐷 ↾ (𝐵 × 𝐵)) ∈ (∞Met‘𝐵))
61, 4, 5syl2anc 406 . 2 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → (𝐷 ↾ (𝐵 × 𝐵)) ∈ (∞Met‘𝐵))
7 xmetf 12339 . . . . . 6 (𝐷 ∈ (∞Met‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
81, 7syl 14 . . . . 5 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
9 xpss12 4606 . . . . . 6 ((𝐵𝑋𝐵𝑋) → (𝐵 × 𝐵) ⊆ (𝑋 × 𝑋))
104, 4, 9syl2anc 406 . . . . 5 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → (𝐵 × 𝐵) ⊆ (𝑋 × 𝑋))
118, 10fssresd 5257 . . . 4 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → (𝐷 ↾ (𝐵 × 𝐵)):(𝐵 × 𝐵)⟶ℝ*)
1211ffnd 5231 . . 3 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → (𝐷 ↾ (𝐵 × 𝐵)) Fn (𝐵 × 𝐵))
13 ovres 5864 . . . . . 6 ((𝑥𝐵𝑦𝐵) → (𝑥(𝐷 ↾ (𝐵 × 𝐵))𝑦) = (𝑥𝐷𝑦))
1413adantl 273 . . . . 5 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(𝐷 ↾ (𝐵 × 𝐵))𝑦) = (𝑥𝐷𝑦))
15 simpl1 967 . . . . . . . . 9 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑥𝐵𝑦𝐵)) → 𝐷 ∈ (∞Met‘𝑋))
16 eqid 2115 . . . . . . . . . 10 (𝐷 “ ℝ) = (𝐷 “ ℝ)
1716xmeter 12425 . . . . . . . . 9 (𝐷 ∈ (∞Met‘𝑋) → (𝐷 “ ℝ) Er 𝑋)
1815, 17syl 14 . . . . . . . 8 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑥𝐵𝑦𝐵)) → (𝐷 “ ℝ) Er 𝑋)
1916blssec 12427 . . . . . . . . . . . 12 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑅) ⊆ [𝑃](𝐷 “ ℝ))
202, 19eqsstrid 3109 . . . . . . . . . . 11 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → 𝐵 ⊆ [𝑃](𝐷 “ ℝ))
2120sselda 3063 . . . . . . . . . 10 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ 𝑥𝐵) → 𝑥 ∈ [𝑃](𝐷 “ ℝ))
2221adantrr 468 . . . . . . . . 9 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑥𝐵𝑦𝐵)) → 𝑥 ∈ [𝑃](𝐷 “ ℝ))
23 simpl2 968 . . . . . . . . . 10 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑥𝐵𝑦𝐵)) → 𝑃𝑋)
24 elecg 6421 . . . . . . . . . 10 ((𝑥 ∈ [𝑃](𝐷 “ ℝ) ∧ 𝑃𝑋) → (𝑥 ∈ [𝑃](𝐷 “ ℝ) ↔ 𝑃(𝐷 “ ℝ)𝑥))
2522, 23, 24syl2anc 406 . . . . . . . . 9 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑥𝐵𝑦𝐵)) → (𝑥 ∈ [𝑃](𝐷 “ ℝ) ↔ 𝑃(𝐷 “ ℝ)𝑥))
2622, 25mpbid 146 . . . . . . . 8 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑥𝐵𝑦𝐵)) → 𝑃(𝐷 “ ℝ)𝑥)
2720sselda 3063 . . . . . . . . . 10 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ 𝑦𝐵) → 𝑦 ∈ [𝑃](𝐷 “ ℝ))
2827adantrl 467 . . . . . . . . 9 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑥𝐵𝑦𝐵)) → 𝑦 ∈ [𝑃](𝐷 “ ℝ))
29 elecg 6421 . . . . . . . . . 10 ((𝑦 ∈ [𝑃](𝐷 “ ℝ) ∧ 𝑃𝑋) → (𝑦 ∈ [𝑃](𝐷 “ ℝ) ↔ 𝑃(𝐷 “ ℝ)𝑦))
3028, 23, 29syl2anc 406 . . . . . . . . 9 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑥𝐵𝑦𝐵)) → (𝑦 ∈ [𝑃](𝐷 “ ℝ) ↔ 𝑃(𝐷 “ ℝ)𝑦))
3128, 30mpbid 146 . . . . . . . 8 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑥𝐵𝑦𝐵)) → 𝑃(𝐷 “ ℝ)𝑦)
3218, 26, 31ertr3d 6401 . . . . . . 7 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑥𝐵𝑦𝐵)) → 𝑥(𝐷 “ ℝ)𝑦)
3316xmeterval 12424 . . . . . . . 8 (𝐷 ∈ (∞Met‘𝑋) → (𝑥(𝐷 “ ℝ)𝑦 ↔ (𝑥𝑋𝑦𝑋 ∧ (𝑥𝐷𝑦) ∈ ℝ)))
3415, 33syl 14 . . . . . . 7 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(𝐷 “ ℝ)𝑦 ↔ (𝑥𝑋𝑦𝑋 ∧ (𝑥𝐷𝑦) ∈ ℝ)))
3532, 34mpbid 146 . . . . . 6 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑥𝐵𝑦𝐵)) → (𝑥𝑋𝑦𝑋 ∧ (𝑥𝐷𝑦) ∈ ℝ))
3635simp3d 978 . . . . 5 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑥𝐵𝑦𝐵)) → (𝑥𝐷𝑦) ∈ ℝ)
3714, 36eqeltrd 2191 . . . 4 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(𝐷 ↾ (𝐵 × 𝐵))𝑦) ∈ ℝ)
3837ralrimivva 2488 . . 3 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → ∀𝑥𝐵𝑦𝐵 (𝑥(𝐷 ↾ (𝐵 × 𝐵))𝑦) ∈ ℝ)
39 ffnov 5829 . . 3 ((𝐷 ↾ (𝐵 × 𝐵)):(𝐵 × 𝐵)⟶ℝ ↔ ((𝐷 ↾ (𝐵 × 𝐵)) Fn (𝐵 × 𝐵) ∧ ∀𝑥𝐵𝑦𝐵 (𝑥(𝐷 ↾ (𝐵 × 𝐵))𝑦) ∈ ℝ))
4012, 38, 39sylanbrc 411 . 2 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → (𝐷 ↾ (𝐵 × 𝐵)):(𝐵 × 𝐵)⟶ℝ)
41 ismet2 12343 . 2 ((𝐷 ↾ (𝐵 × 𝐵)) ∈ (Met‘𝐵) ↔ ((𝐷 ↾ (𝐵 × 𝐵)) ∈ (∞Met‘𝐵) ∧ (𝐷 ↾ (𝐵 × 𝐵)):(𝐵 × 𝐵)⟶ℝ))
426, 40, 41sylanbrc 411 1 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → (𝐷 ↾ (𝐵 × 𝐵)) ∈ (Met‘𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  w3a 945   = wceq 1314  wcel 1463  wral 2390  wss 3037   class class class wbr 3895   × cxp 4497  ccnv 4498  cres 4501  cima 4502   Fn wfn 5076  wf 5077  cfv 5081  (class class class)co 5728   Er wer 6380  [cec 6381  cr 7546  *cxr 7723  ∞Metcxmet 11992  Metcmet 11993  ballcbl 11994
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091  ax-un 4315  ax-setind 4412  ax-cnex 7636  ax-resscn 7637  ax-1cn 7638  ax-1re 7639  ax-icn 7640  ax-addcl 7641  ax-addrcl 7642  ax-mulcl 7643  ax-mulrcl 7644  ax-addcom 7645  ax-mulcom 7646  ax-addass 7647  ax-mulass 7648  ax-distr 7649  ax-i2m1 7650  ax-0lt1 7651  ax-1rid 7652  ax-0id 7653  ax-rnegex 7654  ax-precex 7655  ax-cnre 7656  ax-pre-ltirr 7657  ax-pre-ltwlin 7658  ax-pre-lttrn 7659  ax-pre-apti 7660  ax-pre-ltadd 7661  ax-pre-mulgt0 7662
This theorem depends on definitions:  df-bi 116  df-stab 799  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ne 2283  df-nel 2378  df-ral 2395  df-rex 2396  df-reu 2397  df-rab 2399  df-v 2659  df-sbc 2879  df-csb 2972  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050  df-if 3441  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-iun 3781  df-br 3896  df-opab 3950  df-mpt 3951  df-id 4175  df-po 4178  df-iso 4179  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512  df-iota 5046  df-fun 5083  df-fn 5084  df-f 5085  df-fv 5089  df-riota 5684  df-ov 5731  df-oprab 5732  df-mpo 5733  df-1st 5992  df-2nd 5993  df-er 6383  df-ec 6385  df-map 6498  df-pnf 7726  df-mnf 7727  df-xr 7728  df-ltxr 7729  df-le 7730  df-sub 7858  df-neg 7859  df-2 8689  df-xneg 9452  df-xadd 9453  df-psmet 11999  df-xmet 12000  df-met 12001  df-bl 12002
This theorem is referenced by: (None)
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