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Mirrors > Home > ILE Home > Th. List > climge0 | GIF version |
Description: A nonnegative sequence converges to a nonnegative number. (Contributed by NM, 11-Sep-2005.) |
Ref | Expression |
---|---|
climrecl.1 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
climrecl.2 | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
climrecl.3 | ⊢ (𝜑 → 𝐹 ⇝ 𝐴) |
climrecl.4 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ) |
climge0.5 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 0 ≤ (𝐹‘𝑘)) |
Ref | Expression |
---|---|
climge0 | ⊢ (𝜑 → 0 ≤ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | climrecl.1 | . . . . . 6 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
2 | climrecl.2 | . . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
3 | 2 | adantr 274 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐴 < 0) → 𝑀 ∈ ℤ) |
4 | climrecl.3 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ⇝ 𝐴) | |
5 | climrecl.4 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ) | |
6 | 1, 2, 4, 5 | climrecl 11061 | . . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℝ) |
7 | 6 | adantr 274 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 < 0) → 𝐴 ∈ ℝ) |
8 | 7 | renegcld 8110 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 < 0) → -𝐴 ∈ ℝ) |
9 | 6 | lt0neg1d 8245 | . . . . . . . 8 ⊢ (𝜑 → (𝐴 < 0 ↔ 0 < -𝐴)) |
10 | 9 | biimpa 294 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 < 0) → 0 < -𝐴) |
11 | 8, 10 | elrpd 9449 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐴 < 0) → -𝐴 ∈ ℝ+) |
12 | eqidd 2118 | . . . . . 6 ⊢ (((𝜑 ∧ 𝐴 < 0) ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = (𝐹‘𝑘)) | |
13 | 4 | adantr 274 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐴 < 0) → 𝐹 ⇝ 𝐴) |
14 | 1, 3, 11, 12, 13 | climi2 11025 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 < 0) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < -𝐴) |
15 | 1 | r19.2uz 10733 | . . . . 5 ⊢ (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < -𝐴 → ∃𝑘 ∈ 𝑍 (abs‘((𝐹‘𝑘) − 𝐴)) < -𝐴) |
16 | 14, 15 | syl 14 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 < 0) → ∃𝑘 ∈ 𝑍 (abs‘((𝐹‘𝑘) − 𝐴)) < -𝐴) |
17 | simprr 506 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 < 0) ∧ (𝑘 ∈ 𝑍 ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < -𝐴)) → (abs‘((𝐹‘𝑘) − 𝐴)) < -𝐴) | |
18 | 5 | ad2ant2r 500 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 < 0) ∧ (𝑘 ∈ 𝑍 ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < -𝐴)) → (𝐹‘𝑘) ∈ ℝ) |
19 | 7 | adantr 274 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 < 0) ∧ (𝑘 ∈ 𝑍 ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < -𝐴)) → 𝐴 ∈ ℝ) |
20 | 8 | adantr 274 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 < 0) ∧ (𝑘 ∈ 𝑍 ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < -𝐴)) → -𝐴 ∈ ℝ) |
21 | 18, 19, 20 | absdifltd 10918 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 < 0) ∧ (𝑘 ∈ 𝑍 ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < -𝐴)) → ((abs‘((𝐹‘𝑘) − 𝐴)) < -𝐴 ↔ ((𝐴 − -𝐴) < (𝐹‘𝑘) ∧ (𝐹‘𝑘) < (𝐴 + -𝐴)))) |
22 | 17, 21 | mpbid 146 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 < 0) ∧ (𝑘 ∈ 𝑍 ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < -𝐴)) → ((𝐴 − -𝐴) < (𝐹‘𝑘) ∧ (𝐹‘𝑘) < (𝐴 + -𝐴))) |
23 | 22 | simprd 113 | . . . . . 6 ⊢ (((𝜑 ∧ 𝐴 < 0) ∧ (𝑘 ∈ 𝑍 ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < -𝐴)) → (𝐹‘𝑘) < (𝐴 + -𝐴)) |
24 | 19 | recnd 7762 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 < 0) ∧ (𝑘 ∈ 𝑍 ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < -𝐴)) → 𝐴 ∈ ℂ) |
25 | 24 | negidd 8031 | . . . . . 6 ⊢ (((𝜑 ∧ 𝐴 < 0) ∧ (𝑘 ∈ 𝑍 ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < -𝐴)) → (𝐴 + -𝐴) = 0) |
26 | 23, 25 | breqtrd 3924 | . . . . 5 ⊢ (((𝜑 ∧ 𝐴 < 0) ∧ (𝑘 ∈ 𝑍 ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < -𝐴)) → (𝐹‘𝑘) < 0) |
27 | climge0.5 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 0 ≤ (𝐹‘𝑘)) | |
28 | 27 | ad2ant2r 500 | . . . . . 6 ⊢ (((𝜑 ∧ 𝐴 < 0) ∧ (𝑘 ∈ 𝑍 ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < -𝐴)) → 0 ≤ (𝐹‘𝑘)) |
29 | 0red 7735 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 < 0) ∧ (𝑘 ∈ 𝑍 ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < -𝐴)) → 0 ∈ ℝ) | |
30 | 29, 18 | lenltd 7848 | . . . . . 6 ⊢ (((𝜑 ∧ 𝐴 < 0) ∧ (𝑘 ∈ 𝑍 ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < -𝐴)) → (0 ≤ (𝐹‘𝑘) ↔ ¬ (𝐹‘𝑘) < 0)) |
31 | 28, 30 | mpbid 146 | . . . . 5 ⊢ (((𝜑 ∧ 𝐴 < 0) ∧ (𝑘 ∈ 𝑍 ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < -𝐴)) → ¬ (𝐹‘𝑘) < 0) |
32 | 26, 31 | pm2.21fal 1336 | . . . 4 ⊢ (((𝜑 ∧ 𝐴 < 0) ∧ (𝑘 ∈ 𝑍 ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < -𝐴)) → ⊥) |
33 | 16, 32 | rexlimddv 2531 | . . 3 ⊢ ((𝜑 ∧ 𝐴 < 0) → ⊥) |
34 | 33 | inegd 1335 | . 2 ⊢ (𝜑 → ¬ 𝐴 < 0) |
35 | 0re 7734 | . . 3 ⊢ 0 ∈ ℝ | |
36 | lenlt 7808 | . . 3 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (0 ≤ 𝐴 ↔ ¬ 𝐴 < 0)) | |
37 | 35, 6, 36 | sylancr 410 | . 2 ⊢ (𝜑 → (0 ≤ 𝐴 ↔ ¬ 𝐴 < 0)) |
38 | 34, 37 | mpbird 166 | 1 ⊢ (𝜑 → 0 ≤ 𝐴) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1316 ⊥wfal 1321 ∈ wcel 1465 ∀wral 2393 ∃wrex 2394 class class class wbr 3899 ‘cfv 5093 (class class class)co 5742 ℝcr 7587 0cc0 7588 + caddc 7591 < clt 7768 ≤ cle 7769 − cmin 7901 -cneg 7902 ℤcz 9022 ℤ≥cuz 9294 abscabs 10737 ⇝ cli 11015 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-coll 4013 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-iinf 4472 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-mulrcl 7687 ax-addcom 7688 ax-mulcom 7689 ax-addass 7690 ax-mulass 7691 ax-distr 7692 ax-i2m1 7693 ax-0lt1 7694 ax-1rid 7695 ax-0id 7696 ax-rnegex 7697 ax-precex 7698 ax-cnre 7699 ax-pre-ltirr 7700 ax-pre-ltwlin 7701 ax-pre-lttrn 7702 ax-pre-apti 7703 ax-pre-ltadd 7704 ax-pre-mulgt0 7705 ax-pre-mulext 7706 ax-arch 7707 ax-caucvg 7708 |
This theorem depends on definitions: df-bi 116 df-dc 805 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-reu 2400 df-rmo 2401 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-if 3445 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-tr 3997 df-id 4185 df-po 4188 df-iso 4189 df-iord 4258 df-on 4260 df-ilim 4261 df-suc 4263 df-iom 4475 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-1st 6006 df-2nd 6007 df-recs 6170 df-frec 6256 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 df-sub 7903 df-neg 7904 df-reap 8305 df-ap 8312 df-div 8401 df-inn 8689 df-2 8747 df-3 8748 df-4 8749 df-n0 8946 df-z 9023 df-uz 9295 df-rp 9410 df-seqfrec 10187 df-exp 10261 df-cj 10582 df-re 10583 df-im 10584 df-rsqrt 10738 df-abs 10739 df-clim 11016 |
This theorem is referenced by: climle 11071 |
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