Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > effsumlt | GIF version |
Description: The partial sums of the series expansion of the exponential function at a positive real number are bounded by the value of the function. (Contributed by Paul Chapman, 21-Aug-2007.) (Revised by Mario Carneiro, 29-Apr-2014.) |
Ref | Expression |
---|---|
effsumlt.1 | ⊢ 𝐹 = (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛))) |
effsumlt.2 | ⊢ (𝜑 → 𝐴 ∈ ℝ+) |
effsumlt.3 | ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
Ref | Expression |
---|---|
effsumlt | ⊢ (𝜑 → (seq0( + , 𝐹)‘𝑁) < (exp‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0uz 9360 | . . . . 5 ⊢ ℕ0 = (ℤ≥‘0) | |
2 | 0zd 9066 | . . . . 5 ⊢ (𝜑 → 0 ∈ ℤ) | |
3 | effsumlt.2 | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℝ+) | |
4 | 3 | rpcnd 9485 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
5 | effsumlt.1 | . . . . . . . 8 ⊢ 𝐹 = (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛))) | |
6 | 5 | eftvalcn 11363 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (𝐹‘𝑘) = ((𝐴↑𝑘) / (!‘𝑘))) |
7 | 4, 6 | sylan 281 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘𝑘) = ((𝐴↑𝑘) / (!‘𝑘))) |
8 | 3 | rpred 9483 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ) |
9 | reeftcl 11361 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝑘 ∈ ℕ0) → ((𝐴↑𝑘) / (!‘𝑘)) ∈ ℝ) | |
10 | 8, 9 | sylan 281 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝐴↑𝑘) / (!‘𝑘)) ∈ ℝ) |
11 | 7, 10 | eqeltrd 2216 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘𝑘) ∈ ℝ) |
12 | 1, 2, 11 | serfre 10248 | . . . 4 ⊢ (𝜑 → seq0( + , 𝐹):ℕ0⟶ℝ) |
13 | effsumlt.3 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
14 | 12, 13 | ffvelrnd 5556 | . . 3 ⊢ (𝜑 → (seq0( + , 𝐹)‘𝑁) ∈ ℝ) |
15 | eqid 2139 | . . . 4 ⊢ (ℤ≥‘(𝑁 + 1)) = (ℤ≥‘(𝑁 + 1)) | |
16 | peano2nn0 9017 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0) | |
17 | 13, 16 | syl 14 | . . . 4 ⊢ (𝜑 → (𝑁 + 1) ∈ ℕ0) |
18 | eqidd 2140 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘𝑘) = (𝐹‘𝑘)) | |
19 | nn0z 9074 | . . . . . . 7 ⊢ (𝑘 ∈ ℕ0 → 𝑘 ∈ ℤ) | |
20 | rpexpcl 10312 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝑘 ∈ ℤ) → (𝐴↑𝑘) ∈ ℝ+) | |
21 | 3, 19, 20 | syl2an 287 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐴↑𝑘) ∈ ℝ+) |
22 | faccl 10481 | . . . . . . . 8 ⊢ (𝑘 ∈ ℕ0 → (!‘𝑘) ∈ ℕ) | |
23 | 22 | adantl 275 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (!‘𝑘) ∈ ℕ) |
24 | 23 | nnrpd 9482 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (!‘𝑘) ∈ ℝ+) |
25 | 21, 24 | rpdivcld 9501 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝐴↑𝑘) / (!‘𝑘)) ∈ ℝ+) |
26 | 7, 25 | eqeltrd 2216 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘𝑘) ∈ ℝ+) |
27 | 5 | efcllem 11365 | . . . . 5 ⊢ (𝐴 ∈ ℂ → seq0( + , 𝐹) ∈ dom ⇝ ) |
28 | 4, 27 | syl 14 | . . . 4 ⊢ (𝜑 → seq0( + , 𝐹) ∈ dom ⇝ ) |
29 | 1, 15, 17, 18, 26, 28 | isumrpcl 11263 | . . 3 ⊢ (𝜑 → Σ𝑘 ∈ (ℤ≥‘(𝑁 + 1))(𝐹‘𝑘) ∈ ℝ+) |
30 | 14, 29 | ltaddrpd 9517 | . 2 ⊢ (𝜑 → (seq0( + , 𝐹)‘𝑁) < ((seq0( + , 𝐹)‘𝑁) + Σ𝑘 ∈ (ℤ≥‘(𝑁 + 1))(𝐹‘𝑘))) |
31 | 5 | efval2 11371 | . . . 4 ⊢ (𝐴 ∈ ℂ → (exp‘𝐴) = Σ𝑘 ∈ ℕ0 (𝐹‘𝑘)) |
32 | 4, 31 | syl 14 | . . 3 ⊢ (𝜑 → (exp‘𝐴) = Σ𝑘 ∈ ℕ0 (𝐹‘𝑘)) |
33 | 11 | recnd 7794 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘𝑘) ∈ ℂ) |
34 | 1, 15, 17, 18, 33, 28 | isumsplit 11260 | . . 3 ⊢ (𝜑 → Σ𝑘 ∈ ℕ0 (𝐹‘𝑘) = (Σ𝑘 ∈ (0...((𝑁 + 1) − 1))(𝐹‘𝑘) + Σ𝑘 ∈ (ℤ≥‘(𝑁 + 1))(𝐹‘𝑘))) |
35 | 13 | nn0cnd 9032 | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℂ) |
36 | ax-1cn 7713 | . . . . . . . 8 ⊢ 1 ∈ ℂ | |
37 | pncan 7968 | . . . . . . . 8 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 + 1) − 1) = 𝑁) | |
38 | 35, 36, 37 | sylancl 409 | . . . . . . 7 ⊢ (𝜑 → ((𝑁 + 1) − 1) = 𝑁) |
39 | 38 | oveq2d 5790 | . . . . . 6 ⊢ (𝜑 → (0...((𝑁 + 1) − 1)) = (0...𝑁)) |
40 | 39 | sumeq1d 11135 | . . . . 5 ⊢ (𝜑 → Σ𝑘 ∈ (0...((𝑁 + 1) − 1))(𝐹‘𝑘) = Σ𝑘 ∈ (0...𝑁)(𝐹‘𝑘)) |
41 | eqidd 2140 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘0)) → (𝐹‘𝑘) = (𝐹‘𝑘)) | |
42 | 13, 1 | eleqtrdi 2232 | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘0)) |
43 | elnn0uz 9363 | . . . . . . 7 ⊢ (𝑘 ∈ ℕ0 ↔ 𝑘 ∈ (ℤ≥‘0)) | |
44 | 43, 33 | sylan2br 286 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘0)) → (𝐹‘𝑘) ∈ ℂ) |
45 | 41, 42, 44 | fsum3ser 11166 | . . . . 5 ⊢ (𝜑 → Σ𝑘 ∈ (0...𝑁)(𝐹‘𝑘) = (seq0( + , 𝐹)‘𝑁)) |
46 | 40, 45 | eqtrd 2172 | . . . 4 ⊢ (𝜑 → Σ𝑘 ∈ (0...((𝑁 + 1) − 1))(𝐹‘𝑘) = (seq0( + , 𝐹)‘𝑁)) |
47 | 46 | oveq1d 5789 | . . 3 ⊢ (𝜑 → (Σ𝑘 ∈ (0...((𝑁 + 1) − 1))(𝐹‘𝑘) + Σ𝑘 ∈ (ℤ≥‘(𝑁 + 1))(𝐹‘𝑘)) = ((seq0( + , 𝐹)‘𝑁) + Σ𝑘 ∈ (ℤ≥‘(𝑁 + 1))(𝐹‘𝑘))) |
48 | 32, 34, 47 | 3eqtrd 2176 | . 2 ⊢ (𝜑 → (exp‘𝐴) = ((seq0( + , 𝐹)‘𝑁) + Σ𝑘 ∈ (ℤ≥‘(𝑁 + 1))(𝐹‘𝑘))) |
49 | 30, 48 | breqtrrd 3956 | 1 ⊢ (𝜑 → (seq0( + , 𝐹)‘𝑁) < (exp‘𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1331 ∈ wcel 1480 class class class wbr 3929 ↦ cmpt 3989 dom cdm 4539 ‘cfv 5123 (class class class)co 5774 ℂcc 7618 ℝcr 7619 0cc0 7620 1c1 7621 + caddc 7623 < clt 7800 − cmin 7933 / cdiv 8432 ℕcn 8720 ℕ0cn0 8977 ℤcz 9054 ℤ≥cuz 9326 ℝ+crp 9441 ...cfz 9790 seqcseq 10218 ↑cexp 10292 !cfa 10471 ⇝ cli 11047 Σcsu 11122 expce 11348 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-precex 7730 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 ax-pre-mulgt0 7737 ax-pre-mulext 7738 ax-arch 7739 ax-caucvg 7740 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-po 4218 df-iso 4219 df-iord 4288 df-on 4290 df-ilim 4291 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-isom 5132 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-irdg 6267 df-frec 6288 df-1o 6313 df-oadd 6317 df-er 6429 df-en 6635 df-dom 6636 df-fin 6637 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-reap 8337 df-ap 8344 df-div 8433 df-inn 8721 df-2 8779 df-3 8780 df-4 8781 df-n0 8978 df-z 9055 df-uz 9327 df-q 9412 df-rp 9442 df-ico 9677 df-fz 9791 df-fzo 9920 df-seqfrec 10219 df-exp 10293 df-fac 10472 df-ihash 10522 df-cj 10614 df-re 10615 df-im 10616 df-rsqrt 10770 df-abs 10771 df-clim 11048 df-sumdc 11123 df-ef 11354 |
This theorem is referenced by: efgt1p2 11401 efgt1p 11402 |
Copyright terms: Public domain | W3C validator |