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Mirrors > Home > ILE Home > Th. List > blval | GIF version |
Description: The ball around a point 𝑃 is the set of all points whose distance from 𝑃 is less than the ball's radius 𝑅. (Contributed by NM, 31-Aug-2006.) (Revised by Mario Carneiro, 11-Nov-2013.) |
Ref | Expression |
---|---|
blval | ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑅) = {𝑥 ∈ 𝑋 ∣ (𝑃𝐷𝑥) < 𝑅}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | blfval 12786 | . . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (ball‘𝐷) = (𝑦 ∈ 𝑋, 𝑟 ∈ ℝ* ↦ {𝑥 ∈ 𝑋 ∣ (𝑦𝐷𝑥) < 𝑟})) | |
2 | 1 | 3ad2ant1 1003 | . 2 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (ball‘𝐷) = (𝑦 ∈ 𝑋, 𝑟 ∈ ℝ* ↦ {𝑥 ∈ 𝑋 ∣ (𝑦𝐷𝑥) < 𝑟})) |
3 | simprl 521 | . . . . 5 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) ∧ (𝑦 = 𝑃 ∧ 𝑟 = 𝑅)) → 𝑦 = 𝑃) | |
4 | 3 | oveq1d 5839 | . . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) ∧ (𝑦 = 𝑃 ∧ 𝑟 = 𝑅)) → (𝑦𝐷𝑥) = (𝑃𝐷𝑥)) |
5 | simprr 522 | . . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) ∧ (𝑦 = 𝑃 ∧ 𝑟 = 𝑅)) → 𝑟 = 𝑅) | |
6 | 4, 5 | breq12d 3978 | . . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) ∧ (𝑦 = 𝑃 ∧ 𝑟 = 𝑅)) → ((𝑦𝐷𝑥) < 𝑟 ↔ (𝑃𝐷𝑥) < 𝑅)) |
7 | 6 | rabbidv 2701 | . 2 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) ∧ (𝑦 = 𝑃 ∧ 𝑟 = 𝑅)) → {𝑥 ∈ 𝑋 ∣ (𝑦𝐷𝑥) < 𝑟} = {𝑥 ∈ 𝑋 ∣ (𝑃𝐷𝑥) < 𝑅}) |
8 | simp2 983 | . 2 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → 𝑃 ∈ 𝑋) | |
9 | simp3 984 | . 2 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → 𝑅 ∈ ℝ*) | |
10 | xmetrel 12743 | . . . . 5 ⊢ Rel ∞Met | |
11 | relelfvdm 5500 | . . . . 5 ⊢ ((Rel ∞Met ∧ 𝐷 ∈ (∞Met‘𝑋)) → 𝑋 ∈ dom ∞Met) | |
12 | 10, 11 | mpan 421 | . . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 ∈ dom ∞Met) |
13 | 12 | 3ad2ant1 1003 | . . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → 𝑋 ∈ dom ∞Met) |
14 | rabexg 4107 | . . 3 ⊢ (𝑋 ∈ dom ∞Met → {𝑥 ∈ 𝑋 ∣ (𝑃𝐷𝑥) < 𝑅} ∈ V) | |
15 | 13, 14 | syl 14 | . 2 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → {𝑥 ∈ 𝑋 ∣ (𝑃𝐷𝑥) < 𝑅} ∈ V) |
16 | 2, 7, 8, 9, 15 | ovmpod 5948 | 1 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑅) = {𝑥 ∈ 𝑋 ∣ (𝑃𝐷𝑥) < 𝑅}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∧ w3a 963 = wceq 1335 ∈ wcel 2128 {crab 2439 Vcvv 2712 class class class wbr 3965 dom cdm 4586 Rel wrel 4591 ‘cfv 5170 (class class class)co 5824 ∈ cmpo 5826 ℝ*cxr 7911 < clt 7912 ∞Metcxmet 12380 ballcbl 12382 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4496 ax-cnex 7823 ax-resscn 7824 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-iun 3851 df-br 3966 df-opab 4026 df-mpt 4027 df-id 4253 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-rn 4597 df-res 4598 df-ima 4599 df-iota 5135 df-fun 5172 df-fn 5173 df-f 5174 df-fv 5178 df-ov 5827 df-oprab 5828 df-mpo 5829 df-1st 6088 df-2nd 6089 df-map 6595 df-pnf 7914 df-mnf 7915 df-xr 7916 df-psmet 12387 df-xmet 12388 df-bl 12390 |
This theorem is referenced by: elbl 12791 metss2lem 12897 bdbl 12903 xmetxpbl 12908 |
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