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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 276 | . . . . . 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 11313 | . . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℝ) |
7 | 6 | adantr 276 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 < 0) → 𝐴 ∈ ℝ) |
8 | 7 | renegcld 8324 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 < 0) → -𝐴 ∈ ℝ) |
9 | 6 | lt0neg1d 8459 | . . . . . . . 8 ⊢ (𝜑 → (𝐴 < 0 ↔ 0 < -𝐴)) |
10 | 9 | biimpa 296 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 < 0) → 0 < -𝐴) |
11 | 8, 10 | elrpd 9677 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐴 < 0) → -𝐴 ∈ ℝ+) |
12 | eqidd 2178 | . . . . . 6 ⊢ (((𝜑 ∧ 𝐴 < 0) ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = (𝐹‘𝑘)) | |
13 | 4 | adantr 276 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐴 < 0) → 𝐹 ⇝ 𝐴) |
14 | 1, 3, 11, 12, 13 | climi2 11277 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 < 0) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < -𝐴) |
15 | 1 | r19.2uz 10983 | . . . . 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 531 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 < 0) ∧ (𝑘 ∈ 𝑍 ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < -𝐴)) → (abs‘((𝐹‘𝑘) − 𝐴)) < -𝐴) | |
18 | 5 | ad2ant2r 509 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 < 0) ∧ (𝑘 ∈ 𝑍 ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < -𝐴)) → (𝐹‘𝑘) ∈ ℝ) |
19 | 7 | adantr 276 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 < 0) ∧ (𝑘 ∈ 𝑍 ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < -𝐴)) → 𝐴 ∈ ℝ) |
20 | 8 | adantr 276 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 < 0) ∧ (𝑘 ∈ 𝑍 ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < -𝐴)) → -𝐴 ∈ ℝ) |
21 | 18, 19, 20 | absdifltd 11168 | . . . . . . . 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 147 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 < 0) ∧ (𝑘 ∈ 𝑍 ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < -𝐴)) → ((𝐴 − -𝐴) < (𝐹‘𝑘) ∧ (𝐹‘𝑘) < (𝐴 + -𝐴))) |
23 | 22 | simprd 114 | . . . . . 6 ⊢ (((𝜑 ∧ 𝐴 < 0) ∧ (𝑘 ∈ 𝑍 ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < -𝐴)) → (𝐹‘𝑘) < (𝐴 + -𝐴)) |
24 | 19 | recnd 7973 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 < 0) ∧ (𝑘 ∈ 𝑍 ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < -𝐴)) → 𝐴 ∈ ℂ) |
25 | 24 | negidd 8245 | . . . . . 6 ⊢ (((𝜑 ∧ 𝐴 < 0) ∧ (𝑘 ∈ 𝑍 ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < -𝐴)) → (𝐴 + -𝐴) = 0) |
26 | 23, 25 | breqtrd 4026 | . . . . 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 509 | . . . . . 6 ⊢ (((𝜑 ∧ 𝐴 < 0) ∧ (𝑘 ∈ 𝑍 ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < -𝐴)) → 0 ≤ (𝐹‘𝑘)) |
29 | 0red 7946 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 < 0) ∧ (𝑘 ∈ 𝑍 ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < -𝐴)) → 0 ∈ ℝ) | |
30 | 29, 18 | lenltd 8062 | . . . . . 6 ⊢ (((𝜑 ∧ 𝐴 < 0) ∧ (𝑘 ∈ 𝑍 ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < -𝐴)) → (0 ≤ (𝐹‘𝑘) ↔ ¬ (𝐹‘𝑘) < 0)) |
31 | 28, 30 | mpbid 147 | . . . . 5 ⊢ (((𝜑 ∧ 𝐴 < 0) ∧ (𝑘 ∈ 𝑍 ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < -𝐴)) → ¬ (𝐹‘𝑘) < 0) |
32 | 26, 31 | pm2.21fal 1373 | . . . 4 ⊢ (((𝜑 ∧ 𝐴 < 0) ∧ (𝑘 ∈ 𝑍 ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < -𝐴)) → ⊥) |
33 | 16, 32 | rexlimddv 2599 | . . 3 ⊢ ((𝜑 ∧ 𝐴 < 0) → ⊥) |
34 | 33 | inegd 1372 | . 2 ⊢ (𝜑 → ¬ 𝐴 < 0) |
35 | 0re 7945 | . . 3 ⊢ 0 ∈ ℝ | |
36 | lenlt 8020 | . . 3 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (0 ≤ 𝐴 ↔ ¬ 𝐴 < 0)) | |
37 | 35, 6, 36 | sylancr 414 | . 2 ⊢ (𝜑 → (0 ≤ 𝐴 ↔ ¬ 𝐴 < 0)) |
38 | 34, 37 | mpbird 167 | 1 ⊢ (𝜑 → 0 ≤ 𝐴) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1353 ⊥wfal 1358 ∈ wcel 2148 ∀wral 2455 ∃wrex 2456 class class class wbr 4000 ‘cfv 5212 (class class class)co 5869 ℝcr 7798 0cc0 7799 + caddc 7802 < clt 7979 ≤ cle 7980 − cmin 8115 -cneg 8116 ℤcz 9239 ℤ≥cuz 9514 abscabs 10987 ⇝ cli 11267 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4115 ax-sep 4118 ax-nul 4126 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533 ax-iinf 4584 ax-cnex 7890 ax-resscn 7891 ax-1cn 7892 ax-1re 7893 ax-icn 7894 ax-addcl 7895 ax-addrcl 7896 ax-mulcl 7897 ax-mulrcl 7898 ax-addcom 7899 ax-mulcom 7900 ax-addass 7901 ax-mulass 7902 ax-distr 7903 ax-i2m1 7904 ax-0lt1 7905 ax-1rid 7906 ax-0id 7907 ax-rnegex 7908 ax-precex 7909 ax-cnre 7910 ax-pre-ltirr 7911 ax-pre-ltwlin 7912 ax-pre-lttrn 7913 ax-pre-apti 7914 ax-pre-ltadd 7915 ax-pre-mulgt0 7916 ax-pre-mulext 7917 ax-arch 7918 ax-caucvg 7919 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-if 3535 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-tr 4099 df-id 4290 df-po 4293 df-iso 4294 df-iord 4363 df-on 4365 df-ilim 4366 df-suc 4368 df-iom 4587 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-res 4635 df-ima 4636 df-iota 5174 df-fun 5214 df-fn 5215 df-f 5216 df-f1 5217 df-fo 5218 df-f1o 5219 df-fv 5220 df-riota 5825 df-ov 5872 df-oprab 5873 df-mpo 5874 df-1st 6135 df-2nd 6136 df-recs 6300 df-frec 6386 df-pnf 7981 df-mnf 7982 df-xr 7983 df-ltxr 7984 df-le 7985 df-sub 8117 df-neg 8118 df-reap 8519 df-ap 8526 df-div 8616 df-inn 8906 df-2 8964 df-3 8965 df-4 8966 df-n0 9163 df-z 9240 df-uz 9515 df-rp 9638 df-seqfrec 10429 df-exp 10503 df-cj 10832 df-re 10833 df-im 10834 df-rsqrt 10988 df-abs 10989 df-clim 11268 |
This theorem is referenced by: climle 11323 |
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