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Mirrors > Home > MPE Home > Th. List > effsumlt | Structured version Visualization version 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 12268 | . . . . 5 ⊢ ℕ0 = (ℤ≥‘0) | |
2 | 0zd 11981 | . . . . 5 ⊢ (𝜑 → 0 ∈ ℤ) | |
3 | effsumlt.1 | . . . . . . . 8 ⊢ 𝐹 = (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛))) | |
4 | 3 | eftval 15422 | . . . . . . 7 ⊢ (𝑘 ∈ ℕ0 → (𝐹‘𝑘) = ((𝐴↑𝑘) / (!‘𝑘))) |
5 | 4 | adantl 485 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘𝑘) = ((𝐴↑𝑘) / (!‘𝑘))) |
6 | effsumlt.2 | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℝ+) | |
7 | 6 | rpred 12419 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ) |
8 | reeftcl 15420 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝑘 ∈ ℕ0) → ((𝐴↑𝑘) / (!‘𝑘)) ∈ ℝ) | |
9 | 7, 8 | sylan 583 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝐴↑𝑘) / (!‘𝑘)) ∈ ℝ) |
10 | 5, 9 | eqeltrd 2890 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘𝑘) ∈ ℝ) |
11 | 1, 2, 10 | serfre 13395 | . . . 4 ⊢ (𝜑 → seq0( + , 𝐹):ℕ0⟶ℝ) |
12 | effsumlt.3 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
13 | 11, 12 | ffvelrnd 6829 | . . 3 ⊢ (𝜑 → (seq0( + , 𝐹)‘𝑁) ∈ ℝ) |
14 | eqid 2798 | . . . 4 ⊢ (ℤ≥‘(𝑁 + 1)) = (ℤ≥‘(𝑁 + 1)) | |
15 | peano2nn0 11925 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0) | |
16 | 12, 15 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑁 + 1) ∈ ℕ0) |
17 | eqidd 2799 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘𝑘) = (𝐹‘𝑘)) | |
18 | nn0z 11993 | . . . . . . 7 ⊢ (𝑘 ∈ ℕ0 → 𝑘 ∈ ℤ) | |
19 | rpexpcl 13444 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝑘 ∈ ℤ) → (𝐴↑𝑘) ∈ ℝ+) | |
20 | 6, 18, 19 | syl2an 598 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐴↑𝑘) ∈ ℝ+) |
21 | faccl 13639 | . . . . . . . 8 ⊢ (𝑘 ∈ ℕ0 → (!‘𝑘) ∈ ℕ) | |
22 | 21 | adantl 485 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (!‘𝑘) ∈ ℕ) |
23 | 22 | nnrpd 12417 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (!‘𝑘) ∈ ℝ+) |
24 | 20, 23 | rpdivcld 12436 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝐴↑𝑘) / (!‘𝑘)) ∈ ℝ+) |
25 | 5, 24 | eqeltrd 2890 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘𝑘) ∈ ℝ+) |
26 | 7 | recnd 10658 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
27 | 3 | efcllem 15423 | . . . . 5 ⊢ (𝐴 ∈ ℂ → seq0( + , 𝐹) ∈ dom ⇝ ) |
28 | 26, 27 | syl 17 | . . . 4 ⊢ (𝜑 → seq0( + , 𝐹) ∈ dom ⇝ ) |
29 | 1, 14, 16, 17, 25, 28 | isumrpcl 15190 | . . 3 ⊢ (𝜑 → Σ𝑘 ∈ (ℤ≥‘(𝑁 + 1))(𝐹‘𝑘) ∈ ℝ+) |
30 | 13, 29 | ltaddrpd 12452 | . 2 ⊢ (𝜑 → (seq0( + , 𝐹)‘𝑁) < ((seq0( + , 𝐹)‘𝑁) + Σ𝑘 ∈ (ℤ≥‘(𝑁 + 1))(𝐹‘𝑘))) |
31 | 3 | efval2 15429 | . . . 4 ⊢ (𝐴 ∈ ℂ → (exp‘𝐴) = Σ𝑘 ∈ ℕ0 (𝐹‘𝑘)) |
32 | 26, 31 | syl 17 | . . 3 ⊢ (𝜑 → (exp‘𝐴) = Σ𝑘 ∈ ℕ0 (𝐹‘𝑘)) |
33 | 10 | recnd 10658 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘𝑘) ∈ ℂ) |
34 | 1, 14, 16, 17, 33, 28 | isumsplit 15187 | . . 3 ⊢ (𝜑 → Σ𝑘 ∈ ℕ0 (𝐹‘𝑘) = (Σ𝑘 ∈ (0...((𝑁 + 1) − 1))(𝐹‘𝑘) + Σ𝑘 ∈ (ℤ≥‘(𝑁 + 1))(𝐹‘𝑘))) |
35 | 12 | nn0cnd 11945 | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℂ) |
36 | ax-1cn 10584 | . . . . . . . 8 ⊢ 1 ∈ ℂ | |
37 | pncan 10881 | . . . . . . . 8 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 + 1) − 1) = 𝑁) | |
38 | 35, 36, 37 | sylancl 589 | . . . . . . 7 ⊢ (𝜑 → ((𝑁 + 1) − 1) = 𝑁) |
39 | 38 | oveq2d 7151 | . . . . . 6 ⊢ (𝜑 → (0...((𝑁 + 1) − 1)) = (0...𝑁)) |
40 | 39 | sumeq1d 15050 | . . . . 5 ⊢ (𝜑 → Σ𝑘 ∈ (0...((𝑁 + 1) − 1))(𝐹‘𝑘) = Σ𝑘 ∈ (0...𝑁)(𝐹‘𝑘)) |
41 | eqidd 2799 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝐹‘𝑘) = (𝐹‘𝑘)) | |
42 | 12, 1 | eleqtrdi 2900 | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘0)) |
43 | elfznn0 12995 | . . . . . . 7 ⊢ (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℕ0) | |
44 | 43, 33 | sylan2 595 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝐹‘𝑘) ∈ ℂ) |
45 | 41, 42, 44 | fsumser 15079 | . . . . 5 ⊢ (𝜑 → Σ𝑘 ∈ (0...𝑁)(𝐹‘𝑘) = (seq0( + , 𝐹)‘𝑁)) |
46 | 40, 45 | eqtrd 2833 | . . . 4 ⊢ (𝜑 → Σ𝑘 ∈ (0...((𝑁 + 1) − 1))(𝐹‘𝑘) = (seq0( + , 𝐹)‘𝑁)) |
47 | 46 | oveq1d 7150 | . . 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 2837 | . 2 ⊢ (𝜑 → (exp‘𝐴) = ((seq0( + , 𝐹)‘𝑁) + Σ𝑘 ∈ (ℤ≥‘(𝑁 + 1))(𝐹‘𝑘))) |
49 | 30, 48 | breqtrrd 5058 | 1 ⊢ (𝜑 → (seq0( + , 𝐹)‘𝑁) < (exp‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 class class class wbr 5030 ↦ cmpt 5110 dom cdm 5519 ‘cfv 6324 (class class class)co 7135 ℂcc 10524 ℝcr 10525 0cc0 10526 1c1 10527 + caddc 10529 < clt 10664 − cmin 10859 / cdiv 11286 ℕcn 11625 ℕ0cn0 11885 ℤcz 11969 ℤ≥cuz 12231 ℝ+crp 12377 ...cfz 12885 seqcseq 13364 ↑cexp 13425 !cfa 13629 ⇝ cli 14833 Σcsu 15034 expce 15407 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 ax-mulf 10606 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-pm 8392 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-inf 8891 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-ico 12732 df-fz 12886 df-fzo 13029 df-fl 13157 df-seq 13365 df-exp 13426 df-fac 13630 df-hash 13687 df-shft 14418 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-limsup 14820 df-clim 14837 df-rlim 14838 df-sum 15035 df-ef 15413 |
This theorem is referenced by: efgt1p2 15459 efgt1p 15460 |
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