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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 4106 | . . . 4 ⊢ (𝑥 = (𝑃(ball‘𝐷)𝑅) → (𝑥 ∩ (𝑆 ∖ {𝑃})) = ((𝑃(ball‘𝐷)𝑅) ∩ (𝑆 ∖ {𝑃}))) | |
2 | 1 | neeq1d 2991 | . . 3 ⊢ (𝑥 = (𝑃(ball‘𝐷)𝑅) → ((𝑥 ∩ (𝑆 ∖ {𝑃})) ≠ ∅ ↔ ((𝑃(ball‘𝐷)𝑅) ∩ (𝑆 ∖ {𝑃})) ≠ ∅)) |
3 | simpl3 1195 | . . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ ((limPt‘𝐽)‘𝑆)) ∧ 𝑅 ∈ ℝ+) → 𝑃 ∈ ((limPt‘𝐽)‘𝑆)) | |
4 | simpl1 1193 | . . . . . 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 23292 | . . . . . 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 1194 | . . . . . 6 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ ((limPt‘𝐽)‘𝑆)) ∧ 𝑅 ∈ ℝ+) → 𝑆 ⊆ 𝑋) | |
9 | 5 | mopnuni 23293 | . . . . . . 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 3927 | . . . . 5 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ ((limPt‘𝐽)‘𝑆)) ∧ 𝑅 ∈ ℝ+) → 𝑆 ⊆ ∪ 𝐽) |
12 | eqid 2736 | . . . . . . . 8 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
13 | 12 | lpss 21993 | . . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → ((limPt‘𝐽)‘𝑆) ⊆ ∪ 𝐽) |
14 | 7, 11, 13 | syl2anc 587 | . . . . . 6 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ ((limPt‘𝐽)‘𝑆)) ∧ 𝑅 ∈ ℝ+) → ((limPt‘𝐽)‘𝑆) ⊆ ∪ 𝐽) |
15 | 14, 3 | sseldd 3888 | . . . . 5 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ ((limPt‘𝐽)‘𝑆)) ∧ 𝑅 ∈ ℝ+) → 𝑃 ∈ ∪ 𝐽) |
16 | 12 | islp2 21996 | . . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽 ∧ 𝑃 ∈ ∪ 𝐽) → (𝑃 ∈ ((limPt‘𝐽)‘𝑆) ↔ ∀𝑥 ∈ ((nei‘𝐽)‘{𝑃})(𝑥 ∩ (𝑆 ∖ {𝑃})) ≠ ∅)) |
17 | 7, 11, 15, 16 | syl3anc 1373 | . . . 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 235 | . . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ ((limPt‘𝐽)‘𝑆)) ∧ 𝑅 ∈ ℝ+) → ∀𝑥 ∈ ((nei‘𝐽)‘{𝑃})(𝑥 ∩ (𝑆 ∖ {𝑃})) ≠ ∅) |
19 | 15, 10 | eleqtrrd 2834 | . . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ ((limPt‘𝐽)‘𝑆)) ∧ 𝑅 ∈ ℝ+) → 𝑃 ∈ 𝑋) |
20 | simpr 488 | . . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ ((limPt‘𝐽)‘𝑆)) ∧ 𝑅 ∈ ℝ+) → 𝑅 ∈ ℝ+) | |
21 | 5 | blnei 23354 | . . . 4 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ+) → (𝑃(ball‘𝐷)𝑅) ∈ ((nei‘𝐽)‘{𝑃})) |
22 | 4, 19, 20, 21 | syl3anc 1373 | . . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ ((limPt‘𝐽)‘𝑆)) ∧ 𝑅 ∈ ℝ+) → (𝑃(ball‘𝐷)𝑅) ∈ ((nei‘𝐽)‘{𝑃})) |
23 | 2, 18, 22 | rspcdva 3529 | . 2 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ ((limPt‘𝐽)‘𝑆)) ∧ 𝑅 ∈ ℝ+) → ((𝑃(ball‘𝐷)𝑅) ∩ (𝑆 ∖ {𝑃})) ≠ ∅) |
24 | elin 3869 | . . . . 5 ⊢ (𝑥 ∈ ((𝑃(ball‘𝐷)𝑅) ∩ (𝑆 ∖ {𝑃})) ↔ (𝑥 ∈ (𝑃(ball‘𝐷)𝑅) ∧ 𝑥 ∈ (𝑆 ∖ {𝑃}))) | |
25 | eldifi 4027 | . . . . . . 7 ⊢ (𝑥 ∈ (𝑆 ∖ {𝑃}) → 𝑥 ∈ 𝑆) | |
26 | 25 | anim2i 620 | . . . . . 6 ⊢ ((𝑥 ∈ (𝑃(ball‘𝐷)𝑅) ∧ 𝑥 ∈ (𝑆 ∖ {𝑃})) → (𝑥 ∈ (𝑃(ball‘𝐷)𝑅) ∧ 𝑥 ∈ 𝑆)) |
27 | 26 | ancomd 465 | . . . . 5 ⊢ ((𝑥 ∈ (𝑃(ball‘𝐷)𝑅) ∧ 𝑥 ∈ (𝑆 ∖ {𝑃})) → (𝑥 ∈ 𝑆 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅))) |
28 | 24, 27 | sylbi 220 | . . . 4 ⊢ (𝑥 ∈ ((𝑃(ball‘𝐷)𝑅) ∩ (𝑆 ∖ {𝑃})) → (𝑥 ∈ 𝑆 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅))) |
29 | 28 | eximi 1842 | . . 3 ⊢ (∃𝑥 𝑥 ∈ ((𝑃(ball‘𝐷)𝑅) ∩ (𝑆 ∖ {𝑃})) → ∃𝑥(𝑥 ∈ 𝑆 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅))) |
30 | n0 4247 | . . 3 ⊢ (((𝑃(ball‘𝐷)𝑅) ∩ (𝑆 ∖ {𝑃})) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ ((𝑃(ball‘𝐷)𝑅) ∩ (𝑆 ∖ {𝑃}))) | |
31 | df-rex 3057 | . . 3 ⊢ (∃𝑥 ∈ 𝑆 𝑥 ∈ (𝑃(ball‘𝐷)𝑅) ↔ ∃𝑥(𝑥 ∈ 𝑆 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅))) | |
32 | 29, 30, 31 | 3imtr4i 295 | . 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 209 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∃wex 1787 ∈ wcel 2112 ≠ wne 2932 ∀wral 3051 ∃wrex 3052 ∖ cdif 3850 ∩ cin 3852 ⊆ wss 3853 ∅c0 4223 {csn 4527 ∪ cuni 4805 ‘cfv 6358 (class class class)co 7191 ℝ+crp 12551 ∞Metcxmet 20302 ballcbl 20304 MetOpencmopn 20307 Topctop 21744 neicnei 21948 limPtclp 21985 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 ax-pre-sup 10772 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-iin 4893 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-er 8369 df-map 8488 df-en 8605 df-dom 8606 df-sdom 8607 df-sup 9036 df-inf 9037 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-div 11455 df-nn 11796 df-2 11858 df-n0 12056 df-z 12142 df-uz 12404 df-q 12510 df-rp 12552 df-xneg 12669 df-xadd 12670 df-xmul 12671 df-topgen 16902 df-psmet 20309 df-xmet 20310 df-bl 20312 df-mopn 20313 df-top 21745 df-topon 21762 df-bases 21797 df-cld 21870 df-ntr 21871 df-cls 21872 df-nei 21949 df-lp 21987 |
This theorem is referenced by: limcrecl 42788 |
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