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Mirrors > Home > MPE Home > Th. List > lmle | Structured version Visualization version GIF version |
Description: If the distance from each member of a converging sequence to a given point is less than or equal to a given amount, so is the convergence value. (Contributed by NM, 23-Dec-2007.) (Proof shortened by Mario Carneiro, 1-May-2014.) |
Ref | Expression |
---|---|
lmle.1 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
lmle.3 | ⊢ 𝐽 = (MetOpen‘𝐷) |
lmle.4 | ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋)) |
lmle.6 | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
lmle.7 | ⊢ (𝜑 → 𝐹(⇝𝑡‘𝐽)𝑃) |
lmle.8 | ⊢ (𝜑 → 𝑄 ∈ 𝑋) |
lmle.9 | ⊢ (𝜑 → 𝑅 ∈ ℝ*) |
lmle.10 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝑄𝐷(𝐹‘𝑘)) ≤ 𝑅) |
Ref | Expression |
---|---|
lmle | ⊢ (𝜑 → (𝑄𝐷𝑃) ≤ 𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lmle.1 | . . . 4 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
2 | lmle.4 | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋)) | |
3 | lmle.3 | . . . . . 6 ⊢ 𝐽 = (MetOpen‘𝐷) | |
4 | 3 | mopntopon 22656 | . . . . 5 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ (TopOn‘𝑋)) |
5 | 2, 4 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) |
6 | lmle.6 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
7 | lmrel 21446 | . . . . 5 ⊢ Rel (⇝𝑡‘𝐽) | |
8 | lmle.7 | . . . . 5 ⊢ (𝜑 → 𝐹(⇝𝑡‘𝐽)𝑃) | |
9 | releldm 5606 | . . . . 5 ⊢ ((Rel (⇝𝑡‘𝐽) ∧ 𝐹(⇝𝑡‘𝐽)𝑃) → 𝐹 ∈ dom (⇝𝑡‘𝐽)) | |
10 | 7, 8, 9 | sylancr 581 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ dom (⇝𝑡‘𝐽)) |
11 | 1, 5, 6, 10 | lmff 21517 | . . 3 ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶𝑋) |
12 | eqid 2778 | . . . 4 ⊢ (ℤ≥‘𝑗) = (ℤ≥‘𝑗) | |
13 | 5 | adantr 474 | . . . 4 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶𝑋)) → 𝐽 ∈ (TopOn‘𝑋)) |
14 | simprl 761 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶𝑋)) → 𝑗 ∈ 𝑍) | |
15 | 14, 1 | syl6eleq 2869 | . . . . 5 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶𝑋)) → 𝑗 ∈ (ℤ≥‘𝑀)) |
16 | eluzelz 12006 | . . . . 5 ⊢ (𝑗 ∈ (ℤ≥‘𝑀) → 𝑗 ∈ ℤ) | |
17 | 15, 16 | syl 17 | . . . 4 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶𝑋)) → 𝑗 ∈ ℤ) |
18 | 8 | adantr 474 | . . . 4 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶𝑋)) → 𝐹(⇝𝑡‘𝐽)𝑃) |
19 | fvres 6467 | . . . . . . 7 ⊢ (𝑘 ∈ (ℤ≥‘𝑗) → ((𝐹 ↾ (ℤ≥‘𝑗))‘𝑘) = (𝐹‘𝑘)) | |
20 | 19 | adantl 475 | . . . . . 6 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶𝑋)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → ((𝐹 ↾ (ℤ≥‘𝑗))‘𝑘) = (𝐹‘𝑘)) |
21 | simprr 763 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶𝑋)) → (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶𝑋) | |
22 | 21 | ffvelrnda 6625 | . . . . . 6 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶𝑋)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → ((𝐹 ↾ (ℤ≥‘𝑗))‘𝑘) ∈ 𝑋) |
23 | 20, 22 | eqeltrrd 2860 | . . . . 5 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶𝑋)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → (𝐹‘𝑘) ∈ 𝑋) |
24 | 1 | uztrn2 12014 | . . . . . . 7 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → 𝑘 ∈ 𝑍) |
25 | 14, 24 | sylan 575 | . . . . . 6 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶𝑋)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → 𝑘 ∈ 𝑍) |
26 | lmle.10 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝑄𝐷(𝐹‘𝑘)) ≤ 𝑅) | |
27 | 26 | adantlr 705 | . . . . . 6 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶𝑋)) ∧ 𝑘 ∈ 𝑍) → (𝑄𝐷(𝐹‘𝑘)) ≤ 𝑅) |
28 | 25, 27 | syldan 585 | . . . . 5 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶𝑋)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → (𝑄𝐷(𝐹‘𝑘)) ≤ 𝑅) |
29 | oveq2 6932 | . . . . . . 7 ⊢ (𝑥 = (𝐹‘𝑘) → (𝑄𝐷𝑥) = (𝑄𝐷(𝐹‘𝑘))) | |
30 | 29 | breq1d 4898 | . . . . . 6 ⊢ (𝑥 = (𝐹‘𝑘) → ((𝑄𝐷𝑥) ≤ 𝑅 ↔ (𝑄𝐷(𝐹‘𝑘)) ≤ 𝑅)) |
31 | 30 | elrab 3572 | . . . . 5 ⊢ ((𝐹‘𝑘) ∈ {𝑥 ∈ 𝑋 ∣ (𝑄𝐷𝑥) ≤ 𝑅} ↔ ((𝐹‘𝑘) ∈ 𝑋 ∧ (𝑄𝐷(𝐹‘𝑘)) ≤ 𝑅)) |
32 | 23, 28, 31 | sylanbrc 578 | . . . 4 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶𝑋)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → (𝐹‘𝑘) ∈ {𝑥 ∈ 𝑋 ∣ (𝑄𝐷𝑥) ≤ 𝑅}) |
33 | lmle.8 | . . . . . 6 ⊢ (𝜑 → 𝑄 ∈ 𝑋) | |
34 | lmle.9 | . . . . . 6 ⊢ (𝜑 → 𝑅 ∈ ℝ*) | |
35 | eqid 2778 | . . . . . . 7 ⊢ {𝑥 ∈ 𝑋 ∣ (𝑄𝐷𝑥) ≤ 𝑅} = {𝑥 ∈ 𝑋 ∣ (𝑄𝐷𝑥) ≤ 𝑅} | |
36 | 3, 35 | blcld 22722 | . . . . . 6 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑄 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → {𝑥 ∈ 𝑋 ∣ (𝑄𝐷𝑥) ≤ 𝑅} ∈ (Clsd‘𝐽)) |
37 | 2, 33, 34, 36 | syl3anc 1439 | . . . . 5 ⊢ (𝜑 → {𝑥 ∈ 𝑋 ∣ (𝑄𝐷𝑥) ≤ 𝑅} ∈ (Clsd‘𝐽)) |
38 | 37 | adantr 474 | . . . 4 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶𝑋)) → {𝑥 ∈ 𝑋 ∣ (𝑄𝐷𝑥) ≤ 𝑅} ∈ (Clsd‘𝐽)) |
39 | 12, 13, 17, 18, 32, 38 | lmcld 21519 | . . 3 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶𝑋)) → 𝑃 ∈ {𝑥 ∈ 𝑋 ∣ (𝑄𝐷𝑥) ≤ 𝑅}) |
40 | 11, 39 | rexlimddv 3218 | . 2 ⊢ (𝜑 → 𝑃 ∈ {𝑥 ∈ 𝑋 ∣ (𝑄𝐷𝑥) ≤ 𝑅}) |
41 | oveq2 6932 | . . . . 5 ⊢ (𝑥 = 𝑃 → (𝑄𝐷𝑥) = (𝑄𝐷𝑃)) | |
42 | 41 | breq1d 4898 | . . . 4 ⊢ (𝑥 = 𝑃 → ((𝑄𝐷𝑥) ≤ 𝑅 ↔ (𝑄𝐷𝑃) ≤ 𝑅)) |
43 | 42 | elrab 3572 | . . 3 ⊢ (𝑃 ∈ {𝑥 ∈ 𝑋 ∣ (𝑄𝐷𝑥) ≤ 𝑅} ↔ (𝑃 ∈ 𝑋 ∧ (𝑄𝐷𝑃) ≤ 𝑅)) |
44 | 43 | simprbi 492 | . 2 ⊢ (𝑃 ∈ {𝑥 ∈ 𝑋 ∣ (𝑄𝐷𝑥) ≤ 𝑅} → (𝑄𝐷𝑃) ≤ 𝑅) |
45 | 40, 44 | syl 17 | 1 ⊢ (𝜑 → (𝑄𝐷𝑃) ≤ 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2107 {crab 3094 class class class wbr 4888 dom cdm 5357 ↾ cres 5359 Rel wrel 5362 ⟶wf 6133 ‘cfv 6137 (class class class)co 6924 ℝ*cxr 10412 ≤ cle 10414 ℤcz 11732 ℤ≥cuz 11996 ∞Metcxmet 20131 MetOpencmopn 20136 TopOnctopon 21126 Clsdccld 21232 ⇝𝑡clm 21442 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 ax-pre-sup 10352 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-int 4713 df-iun 4757 df-iin 4758 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-1st 7447 df-2nd 7448 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-er 8028 df-map 8144 df-pm 8145 df-en 8244 df-dom 8245 df-sdom 8246 df-sup 8638 df-inf 8639 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-div 11035 df-nn 11379 df-2 11442 df-n0 11647 df-z 11733 df-uz 11997 df-q 12100 df-rp 12142 df-xneg 12261 df-xadd 12262 df-xmul 12263 df-topgen 16494 df-psmet 20138 df-xmet 20139 df-bl 20141 df-mopn 20142 df-top 21110 df-topon 21127 df-bases 21162 df-cld 21235 df-ntr 21236 df-cls 21237 df-lm 21445 |
This theorem is referenced by: nglmle 23512 minvecolem4 28312 |
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