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Theorem xmetresbl 22463
Description: An extended metric restricted to any ball (in particular the infinity ball) is a proper metric. Together with xmetec 22460, 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 1131 . . 3 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → 𝐷 ∈ (∞Met‘𝑋))
2 xmetresbl.1 . . . 4 𝐵 = (𝑃(ball‘𝐷)𝑅)
3 blssm 22444 . . . 4 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑅) ⊆ 𝑋)
42, 3syl5eqss 3790 . . 3 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → 𝐵𝑋)
5 xmetres2 22387 . . 3 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵𝑋) → (𝐷 ↾ (𝐵 × 𝐵)) ∈ (∞Met‘𝐵))
61, 4, 5syl2anc 696 . 2 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → (𝐷 ↾ (𝐵 × 𝐵)) ∈ (∞Met‘𝐵))
7 xmetf 22355 . . . . . 6 (𝐷 ∈ (∞Met‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
81, 7syl 17 . . . . 5 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
9 xpss12 5281 . . . . . 6 ((𝐵𝑋𝐵𝑋) → (𝐵 × 𝐵) ⊆ (𝑋 × 𝑋))
104, 4, 9syl2anc 696 . . . . 5 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → (𝐵 × 𝐵) ⊆ (𝑋 × 𝑋))
118, 10fssresd 6232 . . . 4 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → (𝐷 ↾ (𝐵 × 𝐵)):(𝐵 × 𝐵)⟶ℝ*)
12 ffn 6206 . . . 4 ((𝐷 ↾ (𝐵 × 𝐵)):(𝐵 × 𝐵)⟶ℝ* → (𝐷 ↾ (𝐵 × 𝐵)) Fn (𝐵 × 𝐵))
1311, 12syl 17 . . 3 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → (𝐷 ↾ (𝐵 × 𝐵)) Fn (𝐵 × 𝐵))
14 ovres 6966 . . . . . 6 ((𝑥𝐵𝑦𝐵) → (𝑥(𝐷 ↾ (𝐵 × 𝐵))𝑦) = (𝑥𝐷𝑦))
1514adantl 473 . . . . 5 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(𝐷 ↾ (𝐵 × 𝐵))𝑦) = (𝑥𝐷𝑦))
16 simpl1 1228 . . . . . . . . 9 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑥𝐵𝑦𝐵)) → 𝐷 ∈ (∞Met‘𝑋))
17 eqid 2760 . . . . . . . . . 10 (𝐷 “ ℝ) = (𝐷 “ ℝ)
1817xmeter 22459 . . . . . . . . 9 (𝐷 ∈ (∞Met‘𝑋) → (𝐷 “ ℝ) Er 𝑋)
1916, 18syl 17 . . . . . . . 8 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑥𝐵𝑦𝐵)) → (𝐷 “ ℝ) Er 𝑋)
2017blssec 22461 . . . . . . . . . . . 12 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑅) ⊆ [𝑃](𝐷 “ ℝ))
212, 20syl5eqss 3790 . . . . . . . . . . 11 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → 𝐵 ⊆ [𝑃](𝐷 “ ℝ))
2221sselda 3744 . . . . . . . . . 10 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ 𝑥𝐵) → 𝑥 ∈ [𝑃](𝐷 “ ℝ))
2322adantrr 755 . . . . . . . . 9 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑥𝐵𝑦𝐵)) → 𝑥 ∈ [𝑃](𝐷 “ ℝ))
24 simpl2 1230 . . . . . . . . . 10 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑥𝐵𝑦𝐵)) → 𝑃𝑋)
25 elecg 7954 . . . . . . . . . 10 ((𝑥 ∈ [𝑃](𝐷 “ ℝ) ∧ 𝑃𝑋) → (𝑥 ∈ [𝑃](𝐷 “ ℝ) ↔ 𝑃(𝐷 “ ℝ)𝑥))
2623, 24, 25syl2anc 696 . . . . . . . . 9 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑥𝐵𝑦𝐵)) → (𝑥 ∈ [𝑃](𝐷 “ ℝ) ↔ 𝑃(𝐷 “ ℝ)𝑥))
2723, 26mpbid 222 . . . . . . . 8 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑥𝐵𝑦𝐵)) → 𝑃(𝐷 “ ℝ)𝑥)
2821sselda 3744 . . . . . . . . . 10 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ 𝑦𝐵) → 𝑦 ∈ [𝑃](𝐷 “ ℝ))
2928adantrl 754 . . . . . . . . 9 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑥𝐵𝑦𝐵)) → 𝑦 ∈ [𝑃](𝐷 “ ℝ))
30 elecg 7954 . . . . . . . . . 10 ((𝑦 ∈ [𝑃](𝐷 “ ℝ) ∧ 𝑃𝑋) → (𝑦 ∈ [𝑃](𝐷 “ ℝ) ↔ 𝑃(𝐷 “ ℝ)𝑦))
3129, 24, 30syl2anc 696 . . . . . . . . 9 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑥𝐵𝑦𝐵)) → (𝑦 ∈ [𝑃](𝐷 “ ℝ) ↔ 𝑃(𝐷 “ ℝ)𝑦))
3229, 31mpbid 222 . . . . . . . 8 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑥𝐵𝑦𝐵)) → 𝑃(𝐷 “ ℝ)𝑦)
3319, 27, 32ertr3d 7931 . . . . . . 7 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑥𝐵𝑦𝐵)) → 𝑥(𝐷 “ ℝ)𝑦)
3417xmeterval 22458 . . . . . . . 8 (𝐷 ∈ (∞Met‘𝑋) → (𝑥(𝐷 “ ℝ)𝑦 ↔ (𝑥𝑋𝑦𝑋 ∧ (𝑥𝐷𝑦) ∈ ℝ)))
3516, 34syl 17 . . . . . . 7 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(𝐷 “ ℝ)𝑦 ↔ (𝑥𝑋𝑦𝑋 ∧ (𝑥𝐷𝑦) ∈ ℝ)))
3633, 35mpbid 222 . . . . . 6 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑥𝐵𝑦𝐵)) → (𝑥𝑋𝑦𝑋 ∧ (𝑥𝐷𝑦) ∈ ℝ))
3736simp3d 1139 . . . . 5 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑥𝐵𝑦𝐵)) → (𝑥𝐷𝑦) ∈ ℝ)
3815, 37eqeltrd 2839 . . . 4 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(𝐷 ↾ (𝐵 × 𝐵))𝑦) ∈ ℝ)
3938ralrimivva 3109 . . 3 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → ∀𝑥𝐵𝑦𝐵 (𝑥(𝐷 ↾ (𝐵 × 𝐵))𝑦) ∈ ℝ)
40 ffnov 6930 . . 3 ((𝐷 ↾ (𝐵 × 𝐵)):(𝐵 × 𝐵)⟶ℝ ↔ ((𝐷 ↾ (𝐵 × 𝐵)) Fn (𝐵 × 𝐵) ∧ ∀𝑥𝐵𝑦𝐵 (𝑥(𝐷 ↾ (𝐵 × 𝐵))𝑦) ∈ ℝ))
4113, 39, 40sylanbrc 701 . 2 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → (𝐷 ↾ (𝐵 × 𝐵)):(𝐵 × 𝐵)⟶ℝ)
42 ismet2 22359 . 2 ((𝐷 ↾ (𝐵 × 𝐵)) ∈ (Met‘𝐵) ↔ ((𝐷 ↾ (𝐵 × 𝐵)) ∈ (∞Met‘𝐵) ∧ (𝐷 ↾ (𝐵 × 𝐵)):(𝐵 × 𝐵)⟶ℝ))
436, 41, 42sylanbrc 701 1 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → (𝐷 ↾ (𝐵 × 𝐵)) ∈ (Met‘𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1632  wcel 2139  wral 3050  wss 3715   class class class wbr 4804   × cxp 5264  ccnv 5265  cres 5268  cima 5269   Fn wfn 6044  wf 6045  cfv 6049  (class class class)co 6814   Er wer 7910  [cec 7911  cr 10147  *cxr 10285  ∞Metcxmt 19953  Metcme 19954  ballcbl 19955
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-cnex 10204  ax-resscn 10205  ax-1cn 10206  ax-icn 10207  ax-addcl 10208  ax-addrcl 10209  ax-mulcl 10210  ax-mulrcl 10211  ax-mulcom 10212  ax-addass 10213  ax-mulass 10214  ax-distr 10215  ax-i2m1 10216  ax-1ne0 10217  ax-1rid 10218  ax-rnegex 10219  ax-rrecex 10220  ax-cnre 10221  ax-pre-lttri 10222  ax-pre-lttrn 10223  ax-pre-ltadd 10224  ax-pre-mulgt0 10225
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-po 5187  df-so 5188  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-1st 7334  df-2nd 7335  df-er 7913  df-ec 7915  df-map 8027  df-en 8124  df-dom 8125  df-sdom 8126  df-pnf 10288  df-mnf 10289  df-xr 10290  df-ltxr 10291  df-le 10292  df-sub 10480  df-neg 10481  df-div 10897  df-2 11291  df-rp 12046  df-xneg 12159  df-xadd 12160  df-xmul 12161  df-psmet 19960  df-xmet 19961  df-met 19962  df-bl 19963
This theorem is referenced by: (None)
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