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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > climinf3 | Structured version Visualization version GIF version |
Description: A convergent, nonincreasing sequence, converges to the infimum of its range. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
Ref | Expression |
---|---|
climinf3.1 | ⊢ Ⅎ𝑘𝜑 |
climinf3.2 | ⊢ Ⅎ𝑘𝐹 |
climinf3.3 | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
climinf3.4 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
climinf3.5 | ⊢ (𝜑 → 𝐹:𝑍⟶ℝ) |
climinf3.6 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘)) |
climinf3.7 | ⊢ (𝜑 → 𝐹 ∈ dom ⇝ ) |
Ref | Expression |
---|---|
climinf3 | ⊢ (𝜑 → 𝐹 ⇝ inf(ran 𝐹, ℝ*, < )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | climinf3.1 | . 2 ⊢ Ⅎ𝑘𝜑 | |
2 | climinf3.2 | . 2 ⊢ Ⅎ𝑘𝐹 | |
3 | climinf3.4 | . 2 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
4 | climinf3.3 | . 2 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
5 | climinf3.5 | . 2 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ) | |
6 | climinf3.6 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘)) | |
7 | climinf3.7 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ dom ⇝ ) | |
8 | 5 | ffvelrnda 6613 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ) |
9 | 8 | recnd 10392 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) |
10 | 1, 9 | ralrimia 40124 | . . . 4 ⊢ (𝜑 → ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ) |
11 | 2, 3 | climbddf 40708 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ dom ⇝ ∧ ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ) → ∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) ≤ 𝑥) |
12 | 4, 7, 10, 11 | syl3anc 1494 | . . 3 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) ≤ 𝑥) |
13 | renegcl 10672 | . . . . . 6 ⊢ (𝑥 ∈ ℝ → -𝑥 ∈ ℝ) | |
14 | 13 | ad2antlr 718 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) ≤ 𝑥) → -𝑥 ∈ ℝ) |
15 | nfv 2013 | . . . . . . . 8 ⊢ Ⅎ𝑘 𝑥 ∈ ℝ | |
16 | 1, 15 | nfan 2002 | . . . . . . 7 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑥 ∈ ℝ) |
17 | nfra1 3150 | . . . . . . 7 ⊢ Ⅎ𝑘∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) ≤ 𝑥 | |
18 | 16, 17 | nfan 2002 | . . . . . 6 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) ≤ 𝑥) |
19 | simpll 783 | . . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) ≤ 𝑥) ∧ 𝑘 ∈ 𝑍) → (𝜑 ∧ 𝑥 ∈ ℝ)) | |
20 | simpr 479 | . . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) ≤ 𝑥) ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ 𝑍) | |
21 | rspa 3139 | . . . . . . . . 9 ⊢ ((∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) ≤ 𝑥 ∧ 𝑘 ∈ 𝑍) → (abs‘(𝐹‘𝑘)) ≤ 𝑥) | |
22 | 21 | adantll 705 | . . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) ≤ 𝑥) ∧ 𝑘 ∈ 𝑍) → (abs‘(𝐹‘𝑘)) ≤ 𝑥) |
23 | simpr 479 | . . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ 𝑍) ∧ (abs‘(𝐹‘𝑘)) ≤ 𝑥) → (abs‘(𝐹‘𝑘)) ≤ 𝑥) | |
24 | 8 | ad4ant13 757 | . . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ 𝑍) ∧ (abs‘(𝐹‘𝑘)) ≤ 𝑥) → (𝐹‘𝑘) ∈ ℝ) |
25 | simpllr 793 | . . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ 𝑍) ∧ (abs‘(𝐹‘𝑘)) ≤ 𝑥) → 𝑥 ∈ ℝ) | |
26 | 24, 25 | absled 14553 | . . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ 𝑍) ∧ (abs‘(𝐹‘𝑘)) ≤ 𝑥) → ((abs‘(𝐹‘𝑘)) ≤ 𝑥 ↔ (-𝑥 ≤ (𝐹‘𝑘) ∧ (𝐹‘𝑘) ≤ 𝑥))) |
27 | 23, 26 | mpbid 224 | . . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ 𝑍) ∧ (abs‘(𝐹‘𝑘)) ≤ 𝑥) → (-𝑥 ≤ (𝐹‘𝑘) ∧ (𝐹‘𝑘) ≤ 𝑥)) |
28 | 27 | simpld 490 | . . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ 𝑍) ∧ (abs‘(𝐹‘𝑘)) ≤ 𝑥) → -𝑥 ≤ (𝐹‘𝑘)) |
29 | 19, 20, 22, 28 | syl21anc 871 | . . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) ≤ 𝑥) ∧ 𝑘 ∈ 𝑍) → -𝑥 ≤ (𝐹‘𝑘)) |
30 | 29 | ex 403 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) ≤ 𝑥) → (𝑘 ∈ 𝑍 → -𝑥 ≤ (𝐹‘𝑘))) |
31 | 18, 30 | ralrimi 3166 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) ≤ 𝑥) → ∀𝑘 ∈ 𝑍 -𝑥 ≤ (𝐹‘𝑘)) |
32 | breq1 4878 | . . . . . . 7 ⊢ (𝑦 = -𝑥 → (𝑦 ≤ (𝐹‘𝑘) ↔ -𝑥 ≤ (𝐹‘𝑘))) | |
33 | 32 | ralbidv 3195 | . . . . . 6 ⊢ (𝑦 = -𝑥 → (∀𝑘 ∈ 𝑍 𝑦 ≤ (𝐹‘𝑘) ↔ ∀𝑘 ∈ 𝑍 -𝑥 ≤ (𝐹‘𝑘))) |
34 | 33 | rspcev 3526 | . . . . 5 ⊢ ((-𝑥 ∈ ℝ ∧ ∀𝑘 ∈ 𝑍 -𝑥 ≤ (𝐹‘𝑘)) → ∃𝑦 ∈ ℝ ∀𝑘 ∈ 𝑍 𝑦 ≤ (𝐹‘𝑘)) |
35 | 14, 31, 34 | syl2anc 579 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) ≤ 𝑥) → ∃𝑦 ∈ ℝ ∀𝑘 ∈ 𝑍 𝑦 ≤ (𝐹‘𝑘)) |
36 | 35 | rexlimdva2 3243 | . . 3 ⊢ (𝜑 → (∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) ≤ 𝑥 → ∃𝑦 ∈ ℝ ∀𝑘 ∈ 𝑍 𝑦 ≤ (𝐹‘𝑘))) |
37 | 12, 36 | mpd 15 | . 2 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑘 ∈ 𝑍 𝑦 ≤ (𝐹‘𝑘)) |
38 | 1, 2, 3, 4, 5, 6, 37 | climinf2 40728 | 1 ⊢ (𝜑 → 𝐹 ⇝ inf(ran 𝐹, ℝ*, < )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1656 Ⅎwnf 1882 ∈ wcel 2164 Ⅎwnfc 2956 ∀wral 3117 ∃wrex 3118 class class class wbr 4875 dom cdm 5346 ran crn 5347 ⟶wf 6123 ‘cfv 6127 (class class class)co 6910 infcinf 8622 ℂcc 10257 ℝcr 10258 1c1 10260 + caddc 10262 ℝ*cxr 10397 < clt 10398 ≤ cle 10399 -cneg 10593 ℤcz 11711 ℤ≥cuz 11975 abscabs 14358 ⇝ cli 14599 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 ax-pre-sup 10337 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-int 4700 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-om 7332 df-1st 7433 df-2nd 7434 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-1o 7831 df-oadd 7835 df-er 8014 df-en 8229 df-dom 8230 df-sdom 8231 df-fin 8232 df-sup 8623 df-inf 8624 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-div 11017 df-nn 11358 df-2 11421 df-3 11422 df-n0 11626 df-z 11712 df-uz 11976 df-rp 12120 df-fz 12627 df-seq 13103 df-exp 13162 df-cj 14223 df-re 14224 df-im 14225 df-sqrt 14359 df-abs 14360 df-clim 14603 |
This theorem is referenced by: (None) |
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