Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > blval | Structured version Visualization version 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 22189 | . . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (ball‘𝐷) = (𝑦 ∈ 𝑋, 𝑟 ∈ ℝ* ↦ {𝑥 ∈ 𝑋 ∣ (𝑦𝐷𝑥) < 𝑟})) | |
2 | 1 | 3ad2ant1 1082 | . 2 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (ball‘𝐷) = (𝑦 ∈ 𝑋, 𝑟 ∈ ℝ* ↦ {𝑥 ∈ 𝑋 ∣ (𝑦𝐷𝑥) < 𝑟})) |
3 | simprl 794 | . . . . 5 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) ∧ (𝑦 = 𝑃 ∧ 𝑟 = 𝑅)) → 𝑦 = 𝑃) | |
4 | 3 | oveq1d 6665 | . . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) ∧ (𝑦 = 𝑃 ∧ 𝑟 = 𝑅)) → (𝑦𝐷𝑥) = (𝑃𝐷𝑥)) |
5 | simprr 796 | . . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) ∧ (𝑦 = 𝑃 ∧ 𝑟 = 𝑅)) → 𝑟 = 𝑅) | |
6 | 4, 5 | breq12d 4666 | . . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) ∧ (𝑦 = 𝑃 ∧ 𝑟 = 𝑅)) → ((𝑦𝐷𝑥) < 𝑟 ↔ (𝑃𝐷𝑥) < 𝑅)) |
7 | 6 | rabbidv 3189 | . 2 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) ∧ (𝑦 = 𝑃 ∧ 𝑟 = 𝑅)) → {𝑥 ∈ 𝑋 ∣ (𝑦𝐷𝑥) < 𝑟} = {𝑥 ∈ 𝑋 ∣ (𝑃𝐷𝑥) < 𝑅}) |
8 | simp2 1062 | . 2 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → 𝑃 ∈ 𝑋) | |
9 | simp3 1063 | . 2 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → 𝑅 ∈ ℝ*) | |
10 | elfvdm 6220 | . . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 ∈ dom ∞Met) | |
11 | 10 | 3ad2ant1 1082 | . . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → 𝑋 ∈ dom ∞Met) |
12 | rabexg 4812 | . . 3 ⊢ (𝑋 ∈ dom ∞Met → {𝑥 ∈ 𝑋 ∣ (𝑃𝐷𝑥) < 𝑅} ∈ V) | |
13 | 11, 12 | syl 17 | . 2 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → {𝑥 ∈ 𝑋 ∣ (𝑃𝐷𝑥) < 𝑅} ∈ V) |
14 | 2, 7, 8, 9, 13 | ovmpt2d 6788 | 1 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑅) = {𝑥 ∈ 𝑋 ∣ (𝑃𝐷𝑥) < 𝑅}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 {crab 2916 Vcvv 3200 class class class wbr 4653 dom cdm 5114 ‘cfv 5888 (class class class)co 6650 ↦ cmpt2 6652 ℝ*cxr 10073 < clt 10074 ∞Metcxmt 19731 ballcbl 19733 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-map 7859 df-xr 10078 df-psmet 19738 df-xmet 19739 df-bl 19741 |
This theorem is referenced by: elbl 22193 metss2lem 22316 stdbdbl 22322 nmhmcn 22920 lgamucov 24764 isbnd3 33583 |
Copyright terms: Public domain | W3C validator |