Theorem xmetresbl 23498

Description: An extended metric restricted to any ball (in particular the infinity ball) is a proper metric. Together with xmetec 23495, 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) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (𝐷 ↾ (𝐵 × 𝐵)) ∈ (Met‘𝐵))

Proof of Theorem xmetresbl

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 1134	. . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → 𝐷 ∈ (∞Met‘𝑋))
2		xmetresbl.1	. . . 4 ⊢ 𝐵 = (𝑃(ball‘𝐷)𝑅)
3		blssm 23479	. . . 4 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑅) ⊆ 𝑋)
4	2, 3	eqsstrid 3965	. . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → 𝐵 ⊆ 𝑋)
5		xmetres2 23422	. . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ⊆ 𝑋) → (𝐷 ↾ (𝐵 × 𝐵)) ∈ (∞Met‘𝐵))
6	1, 4, 5	syl2anc 583	. 2 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (𝐷 ↾ (𝐵 × 𝐵)) ∈ (∞Met‘𝐵))
7		xmetf 23390	. . . . . 6 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
8	1, 7	syl 17	. . . . 5 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
9		xpss12 5595	. . . . . 6 ⊢ ((𝐵 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → (𝐵 × 𝐵) ⊆ (𝑋 × 𝑋))
10	4, 4, 9	syl2anc 583	. . . . 5 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (𝐵 × 𝐵) ⊆ (𝑋 × 𝑋))
11	8, 10	fssresd 6625	. . . 4 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (𝐷 ↾ (𝐵 × 𝐵)):(𝐵 × 𝐵)⟶ℝ*)
12	11	ffnd 6585	. . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (𝐷 ↾ (𝐵 × 𝐵)) Fn (𝐵 × 𝐵))
13		ovres 7416	. . . . . 6 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥(𝐷 ↾ (𝐵 × 𝐵))𝑦) = (𝑥𝐷𝑦))
14	13	adantl 481	. . . . 5 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(𝐷 ↾ (𝐵 × 𝐵))𝑦) = (𝑥𝐷𝑦))
15		simpl1 1189	. . . . . . . . 9 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐷 ∈ (∞Met‘𝑋))
16		eqid 2738	. . . . . . . . . 10 ⊢ (◡𝐷 “ ℝ) = (◡𝐷 “ ℝ)
17	16	xmeter 23494	. . . . . . . . 9 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (◡𝐷 “ ℝ) Er 𝑋)
18	15, 17	syl 17	. . . . . . . 8 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (◡𝐷 “ ℝ) Er 𝑋)
19	16	blssec 23496	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑅) ⊆ [𝑃](◡𝐷 “ ℝ))
20	2, 19	eqsstrid 3965	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → 𝐵 ⊆ [𝑃](◡𝐷 “ ℝ))
21	20	sselda 3917	. . . . . . . . . 10 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ [𝑃](◡𝐷 “ ℝ))
22	21	adantrr 713	. . . . . . . . 9 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ [𝑃](◡𝐷 “ ℝ))
23		simpl2 1190	. . . . . . . . . 10 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑃 ∈ 𝑋)
24		elecg 8499	. . . . . . . . . 10 ⊢ ((𝑥 ∈ [𝑃](◡𝐷 “ ℝ) ∧ 𝑃 ∈ 𝑋) → (𝑥 ∈ [𝑃](◡𝐷 “ ℝ) ↔ 𝑃(◡𝐷 “ ℝ)𝑥))
25	22, 23, 24	syl2anc 583	. . . . . . . . 9 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 ∈ [𝑃](◡𝐷 “ ℝ) ↔ 𝑃(◡𝐷 “ ℝ)𝑥))
26	22, 25	mpbid 231	. . . . . . . 8 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑃(◡𝐷 “ ℝ)𝑥)
27	20	sselda 3917	. . . . . . . . . 10 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ [𝑃](◡𝐷 “ ℝ))
28	27	adantrl 712	. . . . . . . . 9 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ [𝑃](◡𝐷 “ ℝ))
29		elecg 8499	. . . . . . . . . 10 ⊢ ((𝑦 ∈ [𝑃](◡𝐷 “ ℝ) ∧ 𝑃 ∈ 𝑋) → (𝑦 ∈ [𝑃](◡𝐷 “ ℝ) ↔ 𝑃(◡𝐷 “ ℝ)𝑦))
30	28, 23, 29	syl2anc 583	. . . . . . . . 9 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑦 ∈ [𝑃](◡𝐷 “ ℝ) ↔ 𝑃(◡𝐷 “ ℝ)𝑦))
31	28, 30	mpbid 231	. . . . . . . 8 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑃(◡𝐷 “ ℝ)𝑦)
32	18, 26, 31	ertr3d 8474	. . . . . . 7 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑥(◡𝐷 “ ℝ)𝑦)
33	16	xmeterval 23493	. . . . . . . 8 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝑥(◡𝐷 “ ℝ)𝑦 ↔ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ (𝑥𝐷𝑦) ∈ ℝ)))
34	15, 33	syl 17	. . . . . . 7 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(◡𝐷 “ ℝ)𝑦 ↔ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ (𝑥𝐷𝑦) ∈ ℝ)))
35	32, 34	mpbid 231	. . . . . 6 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ (𝑥𝐷𝑦) ∈ ℝ))
36	35	simp3d 1142	. . . . 5 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥𝐷𝑦) ∈ ℝ)
37	14, 36	eqeltrd 2839	. . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(𝐷 ↾ (𝐵 × 𝐵))𝑦) ∈ ℝ)
38	37	ralrimivva 3114	. . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(𝐷 ↾ (𝐵 × 𝐵))𝑦) ∈ ℝ)
39		ffnov 7379	. . 3 ⊢ ((𝐷 ↾ (𝐵 × 𝐵)):(𝐵 × 𝐵)⟶ℝ ↔ ((𝐷 ↾ (𝐵 × 𝐵)) Fn (𝐵 × 𝐵) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(𝐷 ↾ (𝐵 × 𝐵))𝑦) ∈ ℝ))
40	12, 38, 39	sylanbrc 582	. 2 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (𝐷 ↾ (𝐵 × 𝐵)):(𝐵 × 𝐵)⟶ℝ)
41		ismet2 23394	. 2 ⊢ ((𝐷 ↾ (𝐵 × 𝐵)) ∈ (Met‘𝐵) ↔ ((𝐷 ↾ (𝐵 × 𝐵)) ∈ (∞Met‘𝐵) ∧ (𝐷 ↾ (𝐵 × 𝐵)):(𝐵 × 𝐵)⟶ℝ))
42	6, 40, 41	sylanbrc 582	1 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (𝐷 ↾ (𝐵 × 𝐵)) ∈ (Met‘𝐵))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  ∀wral 3063   ⊆ wss 3883   class class class wbr 5070   × cxp 5578  ^◡ccnv 5579   ↾ cres 5582   “ cima 5583   Fn wfn 6413  ⟶wf 6414  ‘cfv 6418  (class class class)co 7255   Er wer 8453  [cec 8454  ℝcr 10801  ℝ^*cxr 10939  ∞Metcxmet 20495  Metcmet 20496  ballcbl 20497

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-po 5494  df-so 5495  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-1st 7804  df-2nd 7805  df-er 8456  df-ec 8458  df-map 8575  df-en 8692  df-dom 8693  df-sdom 8694  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-div 11563  df-2 11966  df-rp 12660  df-xneg 12777  df-xadd 12778  df-xmul 12779  df-psmet 20502  df-xmet 20503  df-met 20504  df-bl 20505

This theorem is referenced by: (None)