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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 12283 | . . . . 5 ⊢ ℕ0 = (ℤ≥‘0) | |
2 | 0zd 11996 | . . . . 5 ⊢ (𝜑 → 0 ∈ ℤ) | |
3 | effsumlt.1 | . . . . . . . 8 ⊢ 𝐹 = (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛))) | |
4 | 3 | eftval 15432 | . . . . . . 7 ⊢ (𝑘 ∈ ℕ0 → (𝐹‘𝑘) = ((𝐴↑𝑘) / (!‘𝑘))) |
5 | 4 | adantl 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘𝑘) = ((𝐴↑𝑘) / (!‘𝑘))) |
6 | effsumlt.2 | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℝ+) | |
7 | 6 | rpred 12434 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ) |
8 | reeftcl 15430 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝑘 ∈ ℕ0) → ((𝐴↑𝑘) / (!‘𝑘)) ∈ ℝ) | |
9 | 7, 8 | sylan 582 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝐴↑𝑘) / (!‘𝑘)) ∈ ℝ) |
10 | 5, 9 | eqeltrd 2915 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘𝑘) ∈ ℝ) |
11 | 1, 2, 10 | serfre 13402 | . . . 4 ⊢ (𝜑 → seq0( + , 𝐹):ℕ0⟶ℝ) |
12 | effsumlt.3 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
13 | 11, 12 | ffvelrnd 6854 | . . 3 ⊢ (𝜑 → (seq0( + , 𝐹)‘𝑁) ∈ ℝ) |
14 | eqid 2823 | . . . 4 ⊢ (ℤ≥‘(𝑁 + 1)) = (ℤ≥‘(𝑁 + 1)) | |
15 | peano2nn0 11940 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0) | |
16 | 12, 15 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑁 + 1) ∈ ℕ0) |
17 | eqidd 2824 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘𝑘) = (𝐹‘𝑘)) | |
18 | nn0z 12008 | . . . . . . 7 ⊢ (𝑘 ∈ ℕ0 → 𝑘 ∈ ℤ) | |
19 | rpexpcl 13451 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝑘 ∈ ℤ) → (𝐴↑𝑘) ∈ ℝ+) | |
20 | 6, 18, 19 | syl2an 597 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐴↑𝑘) ∈ ℝ+) |
21 | faccl 13646 | . . . . . . . 8 ⊢ (𝑘 ∈ ℕ0 → (!‘𝑘) ∈ ℕ) | |
22 | 21 | adantl 484 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (!‘𝑘) ∈ ℕ) |
23 | 22 | nnrpd 12432 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (!‘𝑘) ∈ ℝ+) |
24 | 20, 23 | rpdivcld 12451 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝐴↑𝑘) / (!‘𝑘)) ∈ ℝ+) |
25 | 5, 24 | eqeltrd 2915 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘𝑘) ∈ ℝ+) |
26 | 7 | recnd 10671 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
27 | 3 | efcllem 15433 | . . . . 5 ⊢ (𝐴 ∈ ℂ → seq0( + , 𝐹) ∈ dom ⇝ ) |
28 | 26, 27 | syl 17 | . . . 4 ⊢ (𝜑 → seq0( + , 𝐹) ∈ dom ⇝ ) |
29 | 1, 14, 16, 17, 25, 28 | isumrpcl 15200 | . . 3 ⊢ (𝜑 → Σ𝑘 ∈ (ℤ≥‘(𝑁 + 1))(𝐹‘𝑘) ∈ ℝ+) |
30 | 13, 29 | ltaddrpd 12467 | . 2 ⊢ (𝜑 → (seq0( + , 𝐹)‘𝑁) < ((seq0( + , 𝐹)‘𝑁) + Σ𝑘 ∈ (ℤ≥‘(𝑁 + 1))(𝐹‘𝑘))) |
31 | 3 | efval2 15439 | . . . 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 10671 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘𝑘) ∈ ℂ) |
34 | 1, 14, 16, 17, 33, 28 | isumsplit 15197 | . . 3 ⊢ (𝜑 → Σ𝑘 ∈ ℕ0 (𝐹‘𝑘) = (Σ𝑘 ∈ (0...((𝑁 + 1) − 1))(𝐹‘𝑘) + Σ𝑘 ∈ (ℤ≥‘(𝑁 + 1))(𝐹‘𝑘))) |
35 | 12 | nn0cnd 11960 | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℂ) |
36 | ax-1cn 10597 | . . . . . . . 8 ⊢ 1 ∈ ℂ | |
37 | pncan 10894 | . . . . . . . 8 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 + 1) − 1) = 𝑁) | |
38 | 35, 36, 37 | sylancl 588 | . . . . . . 7 ⊢ (𝜑 → ((𝑁 + 1) − 1) = 𝑁) |
39 | 38 | oveq2d 7174 | . . . . . 6 ⊢ (𝜑 → (0...((𝑁 + 1) − 1)) = (0...𝑁)) |
40 | 39 | sumeq1d 15060 | . . . . 5 ⊢ (𝜑 → Σ𝑘 ∈ (0...((𝑁 + 1) − 1))(𝐹‘𝑘) = Σ𝑘 ∈ (0...𝑁)(𝐹‘𝑘)) |
41 | eqidd 2824 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝐹‘𝑘) = (𝐹‘𝑘)) | |
42 | 12, 1 | eleqtrdi 2925 | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘0)) |
43 | elfznn0 13003 | . . . . . . 7 ⊢ (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℕ0) | |
44 | 43, 33 | sylan2 594 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝐹‘𝑘) ∈ ℂ) |
45 | 41, 42, 44 | fsumser 15089 | . . . . 5 ⊢ (𝜑 → Σ𝑘 ∈ (0...𝑁)(𝐹‘𝑘) = (seq0( + , 𝐹)‘𝑁)) |
46 | 40, 45 | eqtrd 2858 | . . . 4 ⊢ (𝜑 → Σ𝑘 ∈ (0...((𝑁 + 1) − 1))(𝐹‘𝑘) = (seq0( + , 𝐹)‘𝑁)) |
47 | 46 | oveq1d 7173 | . . 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 2862 | . 2 ⊢ (𝜑 → (exp‘𝐴) = ((seq0( + , 𝐹)‘𝑁) + Σ𝑘 ∈ (ℤ≥‘(𝑁 + 1))(𝐹‘𝑘))) |
49 | 30, 48 | breqtrrd 5096 | 1 ⊢ (𝜑 → (seq0( + , 𝐹)‘𝑁) < (exp‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 class class class wbr 5068 ↦ cmpt 5148 dom cdm 5557 ‘cfv 6357 (class class class)co 7158 ℂcc 10537 ℝcr 10538 0cc0 10539 1c1 10540 + caddc 10542 < clt 10677 − cmin 10872 / cdiv 11299 ℕcn 11640 ℕ0cn0 11900 ℤcz 11984 ℤ≥cuz 12246 ℝ+crp 12392 ...cfz 12895 seqcseq 13372 ↑cexp 13432 !cfa 13636 ⇝ cli 14843 Σcsu 15044 expce 15417 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 ax-addf 10618 ax-mulf 10619 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-pm 8411 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-sup 8908 df-inf 8909 df-oi 8976 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-ico 12747 df-fz 12896 df-fzo 13037 df-fl 13165 df-seq 13373 df-exp 13433 df-fac 13637 df-hash 13694 df-shft 14428 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-limsup 14830 df-clim 14847 df-rlim 14848 df-sum 15045 df-ef 15423 |
This theorem is referenced by: efgt1p2 15469 efgt1p 15470 |
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