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Mirrors > Home > MPE Home > Th. List > mopni2 | Structured version Visualization version GIF version |
Description: An open set of a metric space includes a ball around each of its points. (Contributed by NM, 2-May-2007.) (Revised by Mario Carneiro, 12-Nov-2013.) |
Ref | Expression |
---|---|
mopni.1 | ⊢ 𝐽 = (MetOpen‘𝐷) |
Ref | Expression |
---|---|
mopni2 | ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝐽 ∧ 𝑃 ∈ 𝐴) → ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mopni.1 | . . 3 ⊢ 𝐽 = (MetOpen‘𝐷) | |
2 | 1 | mopni 24520 | . 2 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝐽 ∧ 𝑃 ∈ 𝐴) → ∃𝑦 ∈ ran (ball‘𝐷)(𝑃 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴)) |
3 | 1 | mopnss 24471 | . . . . 5 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝐽) → 𝐴 ⊆ 𝑋) |
4 | 3 | sselda 3994 | . . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝐽) ∧ 𝑃 ∈ 𝐴) → 𝑃 ∈ 𝑋) |
5 | blssex 24452 | . . . . 5 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) → (∃𝑦 ∈ ran (ball‘𝐷)(𝑃 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴) ↔ ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ 𝐴)) | |
6 | 5 | adantlr 715 | . . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝐽) ∧ 𝑃 ∈ 𝑋) → (∃𝑦 ∈ ran (ball‘𝐷)(𝑃 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴) ↔ ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ 𝐴)) |
7 | 4, 6 | syldan 591 | . . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝐽) ∧ 𝑃 ∈ 𝐴) → (∃𝑦 ∈ ran (ball‘𝐷)(𝑃 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴) ↔ ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ 𝐴)) |
8 | 7 | 3impa 1109 | . 2 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝐽 ∧ 𝑃 ∈ 𝐴) → (∃𝑦 ∈ ran (ball‘𝐷)(𝑃 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴) ↔ ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ 𝐴)) |
9 | 2, 8 | mpbid 232 | 1 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝐽 ∧ 𝑃 ∈ 𝐴) → ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1536 ∈ wcel 2105 ∃wrex 3067 ⊆ wss 3962 ran crn 5689 ‘cfv 6562 (class class class)co 7430 ℝ+crp 13031 ∞Metcxmet 21366 ballcbl 21368 MetOpencmopn 21371 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-er 8743 df-map 8866 df-en 8984 df-dom 8985 df-sdom 8986 df-sup 9479 df-inf 9480 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-n0 12524 df-z 12611 df-uz 12876 df-q 12988 df-rp 13032 df-xneg 13151 df-xadd 13152 df-xmul 13153 df-topgen 17489 df-psmet 21373 df-xmet 21374 df-bl 21376 df-mopn 21377 df-top 22915 df-topon 22932 df-bases 22968 |
This theorem is referenced by: mopni3 24522 neibl 24529 met1stc 24549 met2ndci 24550 prdsxmslem2 24557 metcnp3 24568 xrsmopn 24847 iccntr 24856 icccmplem3 24859 reconnlem2 24862 opnreen 24866 metdseq0 24889 cnllycmp 25001 nmhmcn 25166 lmmbr 25305 cfilfcls 25321 iscmet3lem2 25339 bcthlem5 25375 opnmbllem 25649 ellimc3 25928 lhop 26069 dvcnvre 26072 xrlimcnp 27025 lgamucov 27095 ubthlem1 30898 cnllysconn 35229 ptrecube 37606 opnmbllem0 37642 heiborlem8 37804 qndenserrnopnlem 46252 opnvonmbllem2 46588 |
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