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| Mirrors > Home > MPE Home > Th. List > lpbl | Structured version Visualization version GIF version | ||
| Description: Every ball around a limit point 𝑃 of a subset 𝑆 includes a member of 𝑆 (even if 𝑃 ∉ 𝑆). (Contributed by NM, 9-Nov-2007.) (Revised by Mario Carneiro, 23-Dec-2013.) | 
| Ref | Expression | 
|---|---|
| mopni.1 | ⊢ 𝐽 = (MetOpen‘𝐷) | 
| Ref | Expression | 
|---|---|
| lpbl | ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ ((limPt‘𝐽)‘𝑆)) ∧ 𝑅 ∈ ℝ+) → ∃𝑥 ∈ 𝑆 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ineq1 4212 | . . . 4 ⊢ (𝑥 = (𝑃(ball‘𝐷)𝑅) → (𝑥 ∩ (𝑆 ∖ {𝑃})) = ((𝑃(ball‘𝐷)𝑅) ∩ (𝑆 ∖ {𝑃}))) | |
| 2 | 1 | neeq1d 2999 | . . 3 ⊢ (𝑥 = (𝑃(ball‘𝐷)𝑅) → ((𝑥 ∩ (𝑆 ∖ {𝑃})) ≠ ∅ ↔ ((𝑃(ball‘𝐷)𝑅) ∩ (𝑆 ∖ {𝑃})) ≠ ∅)) | 
| 3 | simpl3 1193 | . . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ ((limPt‘𝐽)‘𝑆)) ∧ 𝑅 ∈ ℝ+) → 𝑃 ∈ ((limPt‘𝐽)‘𝑆)) | |
| 4 | simpl1 1191 | . . . . . 6 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ ((limPt‘𝐽)‘𝑆)) ∧ 𝑅 ∈ ℝ+) → 𝐷 ∈ (∞Met‘𝑋)) | |
| 5 | mopni.1 | . . . . . . 7 ⊢ 𝐽 = (MetOpen‘𝐷) | |
| 6 | 5 | mopntop 24451 | . . . . . 6 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top) | 
| 7 | 4, 6 | syl 17 | . . . . 5 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ ((limPt‘𝐽)‘𝑆)) ∧ 𝑅 ∈ ℝ+) → 𝐽 ∈ Top) | 
| 8 | simpl2 1192 | . . . . . 6 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ ((limPt‘𝐽)‘𝑆)) ∧ 𝑅 ∈ ℝ+) → 𝑆 ⊆ 𝑋) | |
| 9 | 5 | mopnuni 24452 | . . . . . . 7 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = ∪ 𝐽) | 
| 10 | 4, 9 | syl 17 | . . . . . 6 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ ((limPt‘𝐽)‘𝑆)) ∧ 𝑅 ∈ ℝ+) → 𝑋 = ∪ 𝐽) | 
| 11 | 8, 10 | sseqtrd 4019 | . . . . 5 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ ((limPt‘𝐽)‘𝑆)) ∧ 𝑅 ∈ ℝ+) → 𝑆 ⊆ ∪ 𝐽) | 
| 12 | eqid 2736 | . . . . . . . 8 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
| 13 | 12 | lpss 23151 | . . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → ((limPt‘𝐽)‘𝑆) ⊆ ∪ 𝐽) | 
| 14 | 7, 11, 13 | syl2anc 584 | . . . . . 6 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ ((limPt‘𝐽)‘𝑆)) ∧ 𝑅 ∈ ℝ+) → ((limPt‘𝐽)‘𝑆) ⊆ ∪ 𝐽) | 
| 15 | 14, 3 | sseldd 3983 | . . . . 5 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ ((limPt‘𝐽)‘𝑆)) ∧ 𝑅 ∈ ℝ+) → 𝑃 ∈ ∪ 𝐽) | 
| 16 | 12 | islp2 23154 | . . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽 ∧ 𝑃 ∈ ∪ 𝐽) → (𝑃 ∈ ((limPt‘𝐽)‘𝑆) ↔ ∀𝑥 ∈ ((nei‘𝐽)‘{𝑃})(𝑥 ∩ (𝑆 ∖ {𝑃})) ≠ ∅)) | 
| 17 | 7, 11, 15, 16 | syl3anc 1372 | . . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ ((limPt‘𝐽)‘𝑆)) ∧ 𝑅 ∈ ℝ+) → (𝑃 ∈ ((limPt‘𝐽)‘𝑆) ↔ ∀𝑥 ∈ ((nei‘𝐽)‘{𝑃})(𝑥 ∩ (𝑆 ∖ {𝑃})) ≠ ∅)) | 
| 18 | 3, 17 | mpbid 232 | . . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ ((limPt‘𝐽)‘𝑆)) ∧ 𝑅 ∈ ℝ+) → ∀𝑥 ∈ ((nei‘𝐽)‘{𝑃})(𝑥 ∩ (𝑆 ∖ {𝑃})) ≠ ∅) | 
| 19 | 15, 10 | eleqtrrd 2843 | . . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ ((limPt‘𝐽)‘𝑆)) ∧ 𝑅 ∈ ℝ+) → 𝑃 ∈ 𝑋) | 
| 20 | simpr 484 | . . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ ((limPt‘𝐽)‘𝑆)) ∧ 𝑅 ∈ ℝ+) → 𝑅 ∈ ℝ+) | |
| 21 | 5 | blnei 24516 | . . . 4 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ+) → (𝑃(ball‘𝐷)𝑅) ∈ ((nei‘𝐽)‘{𝑃})) | 
| 22 | 4, 19, 20, 21 | syl3anc 1372 | . . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ ((limPt‘𝐽)‘𝑆)) ∧ 𝑅 ∈ ℝ+) → (𝑃(ball‘𝐷)𝑅) ∈ ((nei‘𝐽)‘{𝑃})) | 
| 23 | 2, 18, 22 | rspcdva 3622 | . 2 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ ((limPt‘𝐽)‘𝑆)) ∧ 𝑅 ∈ ℝ+) → ((𝑃(ball‘𝐷)𝑅) ∩ (𝑆 ∖ {𝑃})) ≠ ∅) | 
| 24 | elin 3966 | . . . . 5 ⊢ (𝑥 ∈ ((𝑃(ball‘𝐷)𝑅) ∩ (𝑆 ∖ {𝑃})) ↔ (𝑥 ∈ (𝑃(ball‘𝐷)𝑅) ∧ 𝑥 ∈ (𝑆 ∖ {𝑃}))) | |
| 25 | eldifi 4130 | . . . . . . 7 ⊢ (𝑥 ∈ (𝑆 ∖ {𝑃}) → 𝑥 ∈ 𝑆) | |
| 26 | 25 | anim2i 617 | . . . . . 6 ⊢ ((𝑥 ∈ (𝑃(ball‘𝐷)𝑅) ∧ 𝑥 ∈ (𝑆 ∖ {𝑃})) → (𝑥 ∈ (𝑃(ball‘𝐷)𝑅) ∧ 𝑥 ∈ 𝑆)) | 
| 27 | 26 | ancomd 461 | . . . . 5 ⊢ ((𝑥 ∈ (𝑃(ball‘𝐷)𝑅) ∧ 𝑥 ∈ (𝑆 ∖ {𝑃})) → (𝑥 ∈ 𝑆 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅))) | 
| 28 | 24, 27 | sylbi 217 | . . . 4 ⊢ (𝑥 ∈ ((𝑃(ball‘𝐷)𝑅) ∩ (𝑆 ∖ {𝑃})) → (𝑥 ∈ 𝑆 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅))) | 
| 29 | 28 | eximi 1834 | . . 3 ⊢ (∃𝑥 𝑥 ∈ ((𝑃(ball‘𝐷)𝑅) ∩ (𝑆 ∖ {𝑃})) → ∃𝑥(𝑥 ∈ 𝑆 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅))) | 
| 30 | n0 4352 | . . 3 ⊢ (((𝑃(ball‘𝐷)𝑅) ∩ (𝑆 ∖ {𝑃})) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ ((𝑃(ball‘𝐷)𝑅) ∩ (𝑆 ∖ {𝑃}))) | |
| 31 | df-rex 3070 | . . 3 ⊢ (∃𝑥 ∈ 𝑆 𝑥 ∈ (𝑃(ball‘𝐷)𝑅) ↔ ∃𝑥(𝑥 ∈ 𝑆 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅))) | |
| 32 | 29, 30, 31 | 3imtr4i 292 | . 2 ⊢ (((𝑃(ball‘𝐷)𝑅) ∩ (𝑆 ∖ {𝑃})) ≠ ∅ → ∃𝑥 ∈ 𝑆 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) | 
| 33 | 23, 32 | syl 17 | 1 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ ((limPt‘𝐽)‘𝑆)) ∧ 𝑅 ∈ ℝ+) → ∃𝑥 ∈ 𝑆 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1539 ∃wex 1778 ∈ wcel 2107 ≠ wne 2939 ∀wral 3060 ∃wrex 3069 ∖ cdif 3947 ∩ cin 3949 ⊆ wss 3950 ∅c0 4332 {csn 4625 ∪ cuni 4906 ‘cfv 6560 (class class class)co 7432 ℝ+crp 13035 ∞Metcxmet 21350 ballcbl 21352 MetOpencmopn 21355 Topctop 22900 neicnei 23106 limPtclp 23143 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 ax-pre-sup 11234 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-iin 4993 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-er 8746 df-map 8869 df-en 8987 df-dom 8988 df-sdom 8989 df-sup 9483 df-inf 9484 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-div 11922 df-nn 12268 df-2 12330 df-n0 12529 df-z 12616 df-uz 12880 df-q 12992 df-rp 13036 df-xneg 13155 df-xadd 13156 df-xmul 13157 df-topgen 17489 df-psmet 21357 df-xmet 21358 df-bl 21360 df-mopn 21361 df-top 22901 df-topon 22918 df-bases 22954 df-cld 23028 df-ntr 23029 df-cls 23030 df-nei 23107 df-lp 23145 | 
| This theorem is referenced by: limcrecl 45649 | 
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