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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 11251 | . . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℝ) |
7 | 6 | adantr 274 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 < 0) → 𝐴 ∈ ℝ) |
8 | 7 | renegcld 8269 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 < 0) → -𝐴 ∈ ℝ) |
9 | 6 | lt0neg1d 8404 | . . . . . . . 8 ⊢ (𝜑 → (𝐴 < 0 ↔ 0 < -𝐴)) |
10 | 9 | biimpa 294 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 < 0) → 0 < -𝐴) |
11 | 8, 10 | elrpd 9620 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐴 < 0) → -𝐴 ∈ ℝ+) |
12 | eqidd 2165 | . . . . . 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 11215 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 < 0) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < -𝐴) |
15 | 1 | r19.2uz 10921 | . . . . 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 522 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 < 0) ∧ (𝑘 ∈ 𝑍 ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < -𝐴)) → (abs‘((𝐹‘𝑘) − 𝐴)) < -𝐴) | |
18 | 5 | ad2ant2r 501 | . . . . . . . . 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 11106 | . . . . . . . 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 7918 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 < 0) ∧ (𝑘 ∈ 𝑍 ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < -𝐴)) → 𝐴 ∈ ℂ) |
25 | 24 | negidd 8190 | . . . . . 6 ⊢ (((𝜑 ∧ 𝐴 < 0) ∧ (𝑘 ∈ 𝑍 ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < -𝐴)) → (𝐴 + -𝐴) = 0) |
26 | 23, 25 | breqtrd 4002 | . . . . 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 501 | . . . . . 6 ⊢ (((𝜑 ∧ 𝐴 < 0) ∧ (𝑘 ∈ 𝑍 ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < -𝐴)) → 0 ≤ (𝐹‘𝑘)) |
29 | 0red 7891 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 < 0) ∧ (𝑘 ∈ 𝑍 ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < -𝐴)) → 0 ∈ ℝ) | |
30 | 29, 18 | lenltd 8007 | . . . . . 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 1362 | . . . 4 ⊢ (((𝜑 ∧ 𝐴 < 0) ∧ (𝑘 ∈ 𝑍 ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < -𝐴)) → ⊥) |
33 | 16, 32 | rexlimddv 2586 | . . 3 ⊢ ((𝜑 ∧ 𝐴 < 0) → ⊥) |
34 | 33 | inegd 1361 | . 2 ⊢ (𝜑 → ¬ 𝐴 < 0) |
35 | 0re 7890 | . . 3 ⊢ 0 ∈ ℝ | |
36 | lenlt 7965 | . . 3 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (0 ≤ 𝐴 ↔ ¬ 𝐴 < 0)) | |
37 | 35, 6, 36 | sylancr 411 | . 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 1342 ⊥wfal 1347 ∈ wcel 2135 ∀wral 2442 ∃wrex 2443 class class class wbr 3976 ‘cfv 5182 (class class class)co 5836 ℝcr 7743 0cc0 7744 + caddc 7747 < clt 7924 ≤ cle 7925 − cmin 8060 -cneg 8061 ℤcz 9182 ℤ≥cuz 9457 abscabs 10925 ⇝ cli 11205 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-mulrcl 7843 ax-addcom 7844 ax-mulcom 7845 ax-addass 7846 ax-mulass 7847 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-1rid 7851 ax-0id 7852 ax-rnegex 7853 ax-precex 7854 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-apti 7859 ax-pre-ltadd 7860 ax-pre-mulgt0 7861 ax-pre-mulext 7862 ax-arch 7863 ax-caucvg 7864 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rmo 2450 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-if 3516 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-id 4265 df-po 4268 df-iso 4269 df-iord 4338 df-on 4340 df-ilim 4341 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-recs 6264 df-frec 6350 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-reap 8464 df-ap 8471 df-div 8560 df-inn 8849 df-2 8907 df-3 8908 df-4 8909 df-n0 9106 df-z 9183 df-uz 9458 df-rp 9581 df-seqfrec 10371 df-exp 10445 df-cj 10770 df-re 10771 df-im 10772 df-rsqrt 10926 df-abs 10927 df-clim 11206 |
This theorem is referenced by: climle 11261 |
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