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Mirrors > Home > MPE Home > Th. List > elbl | Structured version Visualization version GIF version |
Description: Membership in a ball. (Contributed by NM, 2-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.) |
Ref | Expression |
---|---|
elbl | ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (𝐴 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝐴 ∈ 𝑋 ∧ (𝑃𝐷𝐴) < 𝑅))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | blval 22191 | . . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑅) = {𝑥 ∈ 𝑋 ∣ (𝑃𝐷𝑥) < 𝑅}) | |
2 | 1 | eleq2d 2687 | . 2 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (𝐴 ∈ (𝑃(ball‘𝐷)𝑅) ↔ 𝐴 ∈ {𝑥 ∈ 𝑋 ∣ (𝑃𝐷𝑥) < 𝑅})) |
3 | oveq2 6658 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑃𝐷𝑥) = (𝑃𝐷𝐴)) | |
4 | 3 | breq1d 4663 | . . 3 ⊢ (𝑥 = 𝐴 → ((𝑃𝐷𝑥) < 𝑅 ↔ (𝑃𝐷𝐴) < 𝑅)) |
5 | 4 | elrab 3363 | . 2 ⊢ (𝐴 ∈ {𝑥 ∈ 𝑋 ∣ (𝑃𝐷𝑥) < 𝑅} ↔ (𝐴 ∈ 𝑋 ∧ (𝑃𝐷𝐴) < 𝑅)) |
6 | 2, 5 | syl6bb 276 | 1 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (𝐴 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝐴 ∈ 𝑋 ∧ (𝑃𝐷𝐴) < 𝑅))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 {crab 2916 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 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: elbl2 22195 xblpnf 22201 bldisj 22203 blgt0 22204 xblss2 22207 blhalf 22210 xblcntr 22216 xbln0 22219 blin 22226 blss 22230 blres 22236 imasf1obl 22293 prdsbl 22296 blcls 22311 metcnp 22346 dscopn 22378 cnbl0 22577 bl2ioo 22595 blcvx 22601 xrsmopn 22615 recld2 22617 cnheibor 22754 nmhmcn 22920 lmmbr2 23057 iscau2 23075 dvlip2 23758 psercn 24180 abelth 24195 logtayl 24406 logtayl2 24408 poimirlem29 33438 heicant 33444 iooabslt 39721 limcrecl 39861 islpcn 39871 qndenserrnbllem 40514 |
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