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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 3951 | . . . 4 ⊢ (𝑥 = (𝑃(ball‘𝐷)𝑅) → (𝑥 ∩ (𝑆 ∖ {𝑃})) = ((𝑃(ball‘𝐷)𝑅) ∩ (𝑆 ∖ {𝑃}))) | |
2 | 1 | neeq1d 2992 | . . 3 ⊢ (𝑥 = (𝑃(ball‘𝐷)𝑅) → ((𝑥 ∩ (𝑆 ∖ {𝑃})) ≠ ∅ ↔ ((𝑃(ball‘𝐷)𝑅) ∩ (𝑆 ∖ {𝑃})) ≠ ∅)) |
3 | simpl3 1232 | . . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ ((limPt‘𝐽)‘𝑆)) ∧ 𝑅 ∈ ℝ+) → 𝑃 ∈ ((limPt‘𝐽)‘𝑆)) | |
4 | simpl1 1228 | . . . . . 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 22467 | . . . . . 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 1230 | . . . . . 6 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ ((limPt‘𝐽)‘𝑆)) ∧ 𝑅 ∈ ℝ+) → 𝑆 ⊆ 𝑋) | |
9 | 5 | mopnuni 22468 | . . . . . . 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 3783 | . . . . 5 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ ((limPt‘𝐽)‘𝑆)) ∧ 𝑅 ∈ ℝ+) → 𝑆 ⊆ ∪ 𝐽) |
12 | eqid 2761 | . . . . . . . 8 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
13 | 12 | lpss 21169 | . . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → ((limPt‘𝐽)‘𝑆) ⊆ ∪ 𝐽) |
14 | 7, 11, 13 | syl2anc 696 | . . . . . 6 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ ((limPt‘𝐽)‘𝑆)) ∧ 𝑅 ∈ ℝ+) → ((limPt‘𝐽)‘𝑆) ⊆ ∪ 𝐽) |
15 | 14, 3 | sseldd 3746 | . . . . 5 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ ((limPt‘𝐽)‘𝑆)) ∧ 𝑅 ∈ ℝ+) → 𝑃 ∈ ∪ 𝐽) |
16 | 12 | islp2 21172 | . . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽 ∧ 𝑃 ∈ ∪ 𝐽) → (𝑃 ∈ ((limPt‘𝐽)‘𝑆) ↔ ∀𝑥 ∈ ((nei‘𝐽)‘{𝑃})(𝑥 ∩ (𝑆 ∖ {𝑃})) ≠ ∅)) |
17 | 7, 11, 15, 16 | syl3anc 1477 | . . . 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 222 | . . 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 479 | . . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ ((limPt‘𝐽)‘𝑆)) ∧ 𝑅 ∈ ℝ+) → 𝑅 ∈ ℝ+) | |
21 | 5 | blnei 22529 | . . . 4 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ+) → (𝑃(ball‘𝐷)𝑅) ∈ ((nei‘𝐽)‘{𝑃})) |
22 | 4, 19, 20, 21 | syl3anc 1477 | . . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ ((limPt‘𝐽)‘𝑆)) ∧ 𝑅 ∈ ℝ+) → (𝑃(ball‘𝐷)𝑅) ∈ ((nei‘𝐽)‘{𝑃})) |
23 | 2, 18, 22 | rspcdva 3456 | . 2 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ ((limPt‘𝐽)‘𝑆)) ∧ 𝑅 ∈ ℝ+) → ((𝑃(ball‘𝐷)𝑅) ∩ (𝑆 ∖ {𝑃})) ≠ ∅) |
24 | elin 3940 | . . . . 5 ⊢ (𝑥 ∈ ((𝑃(ball‘𝐷)𝑅) ∩ (𝑆 ∖ {𝑃})) ↔ (𝑥 ∈ (𝑃(ball‘𝐷)𝑅) ∧ 𝑥 ∈ (𝑆 ∖ {𝑃}))) | |
25 | eldifi 3876 | . . . . . . 7 ⊢ (𝑥 ∈ (𝑆 ∖ {𝑃}) → 𝑥 ∈ 𝑆) | |
26 | 25 | anim2i 594 | . . . . . 6 ⊢ ((𝑥 ∈ (𝑃(ball‘𝐷)𝑅) ∧ 𝑥 ∈ (𝑆 ∖ {𝑃})) → (𝑥 ∈ (𝑃(ball‘𝐷)𝑅) ∧ 𝑥 ∈ 𝑆)) |
27 | 26 | ancomd 466 | . . . . 5 ⊢ ((𝑥 ∈ (𝑃(ball‘𝐷)𝑅) ∧ 𝑥 ∈ (𝑆 ∖ {𝑃})) → (𝑥 ∈ 𝑆 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅))) |
28 | 24, 27 | sylbi 207 | . . . 4 ⊢ (𝑥 ∈ ((𝑃(ball‘𝐷)𝑅) ∩ (𝑆 ∖ {𝑃})) → (𝑥 ∈ 𝑆 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅))) |
29 | 28 | eximi 1911 | . . 3 ⊢ (∃𝑥 𝑥 ∈ ((𝑃(ball‘𝐷)𝑅) ∩ (𝑆 ∖ {𝑃})) → ∃𝑥(𝑥 ∈ 𝑆 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅))) |
30 | n0 4075 | . . 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 281 | . 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 196 ∧ wa 383 ∧ w3a 1072 = wceq 1632 ∃wex 1853 ∈ wcel 2140 ≠ wne 2933 ∀wral 3051 ∃wrex 3052 ∖ cdif 3713 ∩ cin 3715 ⊆ wss 3716 ∅c0 4059 {csn 4322 ∪ cuni 4589 ‘cfv 6050 (class class class)co 6815 ℝ+crp 12046 ∞Metcxmt 19954 ballcbl 19956 MetOpencmopn 19959 Topctop 20921 neicnei 21124 limPtclp 21161 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1989 ax-6 2055 ax-7 2091 ax-8 2142 ax-9 2149 ax-10 2169 ax-11 2184 ax-12 2197 ax-13 2392 ax-ext 2741 ax-rep 4924 ax-sep 4934 ax-nul 4942 ax-pow 4993 ax-pr 5056 ax-un 7116 ax-cnex 10205 ax-resscn 10206 ax-1cn 10207 ax-icn 10208 ax-addcl 10209 ax-addrcl 10210 ax-mulcl 10211 ax-mulrcl 10212 ax-mulcom 10213 ax-addass 10214 ax-mulass 10215 ax-distr 10216 ax-i2m1 10217 ax-1ne0 10218 ax-1rid 10219 ax-rnegex 10220 ax-rrecex 10221 ax-cnre 10222 ax-pre-lttri 10223 ax-pre-lttrn 10224 ax-pre-ltadd 10225 ax-pre-mulgt0 10226 ax-pre-sup 10227 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2048 df-eu 2612 df-mo 2613 df-clab 2748 df-cleq 2754 df-clel 2757 df-nfc 2892 df-ne 2934 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3343 df-sbc 3578 df-csb 3676 df-dif 3719 df-un 3721 df-in 3723 df-ss 3730 df-pss 3732 df-nul 4060 df-if 4232 df-pw 4305 df-sn 4323 df-pr 4325 df-tp 4327 df-op 4329 df-uni 4590 df-int 4629 df-iun 4675 df-iin 4676 df-br 4806 df-opab 4866 df-mpt 4883 df-tr 4906 df-id 5175 df-eprel 5180 df-po 5188 df-so 5189 df-fr 5226 df-we 5228 df-xp 5273 df-rel 5274 df-cnv 5275 df-co 5276 df-dm 5277 df-rn 5278 df-res 5279 df-ima 5280 df-pred 5842 df-ord 5888 df-on 5889 df-lim 5890 df-suc 5891 df-iota 6013 df-fun 6052 df-fn 6053 df-f 6054 df-f1 6055 df-fo 6056 df-f1o 6057 df-fv 6058 df-riota 6776 df-ov 6818 df-oprab 6819 df-mpt2 6820 df-om 7233 df-1st 7335 df-2nd 7336 df-wrecs 7578 df-recs 7639 df-rdg 7677 df-er 7914 df-map 8028 df-en 8125 df-dom 8126 df-sdom 8127 df-sup 8516 df-inf 8517 df-pnf 10289 df-mnf 10290 df-xr 10291 df-ltxr 10292 df-le 10293 df-sub 10481 df-neg 10482 df-div 10898 df-nn 11234 df-2 11292 df-n0 11506 df-z 11591 df-uz 11901 df-q 12003 df-rp 12047 df-xneg 12160 df-xadd 12161 df-xmul 12162 df-topgen 16327 df-psmet 19961 df-xmet 19962 df-bl 19964 df-mopn 19965 df-top 20922 df-topon 20939 df-bases 20973 df-cld 21046 df-ntr 21047 df-cls 21048 df-nei 21125 df-lp 21163 |
This theorem is referenced by: limcrecl 40383 |
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