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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 of 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 11598 | . . . . 5 ⊢ ℕ0 = (ℤ≥‘0) | |
2 | 0zd 11266 | . . . . 5 ⊢ (𝜑 → 0 ∈ ℤ) | |
3 | effsumlt.1 | . . . . . . . 8 ⊢ 𝐹 = (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛))) | |
4 | 3 | eftval 14646 | . . . . . . 7 ⊢ (𝑘 ∈ ℕ0 → (𝐹‘𝑘) = ((𝐴↑𝑘) / (!‘𝑘))) |
5 | 4 | adantl 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘𝑘) = ((𝐴↑𝑘) / (!‘𝑘))) |
6 | effsumlt.2 | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℝ+) | |
7 | 6 | rpred 11748 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ) |
8 | reeftcl 14644 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝑘 ∈ ℕ0) → ((𝐴↑𝑘) / (!‘𝑘)) ∈ ℝ) | |
9 | 7, 8 | sylan 487 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝐴↑𝑘) / (!‘𝑘)) ∈ ℝ) |
10 | 5, 9 | eqeltrd 2688 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘𝑘) ∈ ℝ) |
11 | 1, 2, 10 | serfre 12692 | . . . 4 ⊢ (𝜑 → seq0( + , 𝐹):ℕ0⟶ℝ) |
12 | effsumlt.3 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
13 | 11, 12 | ffvelrnd 6268 | . . 3 ⊢ (𝜑 → (seq0( + , 𝐹)‘𝑁) ∈ ℝ) |
14 | eqid 2610 | . . . 4 ⊢ (ℤ≥‘(𝑁 + 1)) = (ℤ≥‘(𝑁 + 1)) | |
15 | peano2nn0 11210 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0) | |
16 | 12, 15 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑁 + 1) ∈ ℕ0) |
17 | eqidd 2611 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘𝑘) = (𝐹‘𝑘)) | |
18 | nn0z 11277 | . . . . . . 7 ⊢ (𝑘 ∈ ℕ0 → 𝑘 ∈ ℤ) | |
19 | rpexpcl 12741 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝑘 ∈ ℤ) → (𝐴↑𝑘) ∈ ℝ+) | |
20 | 6, 18, 19 | syl2an 493 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐴↑𝑘) ∈ ℝ+) |
21 | faccl 12932 | . . . . . . . 8 ⊢ (𝑘 ∈ ℕ0 → (!‘𝑘) ∈ ℕ) | |
22 | 21 | adantl 481 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (!‘𝑘) ∈ ℕ) |
23 | 22 | nnrpd 11746 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (!‘𝑘) ∈ ℝ+) |
24 | 20, 23 | rpdivcld 11765 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝐴↑𝑘) / (!‘𝑘)) ∈ ℝ+) |
25 | 5, 24 | eqeltrd 2688 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘𝑘) ∈ ℝ+) |
26 | 7 | recnd 9947 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
27 | 3 | efcllem 14647 | . . . . 5 ⊢ (𝐴 ∈ ℂ → seq0( + , 𝐹) ∈ dom ⇝ ) |
28 | 26, 27 | syl 17 | . . . 4 ⊢ (𝜑 → seq0( + , 𝐹) ∈ dom ⇝ ) |
29 | 1, 14, 16, 17, 25, 28 | isumrpcl 14414 | . . 3 ⊢ (𝜑 → Σ𝑘 ∈ (ℤ≥‘(𝑁 + 1))(𝐹‘𝑘) ∈ ℝ+) |
30 | 13, 29 | ltaddrpd 11781 | . 2 ⊢ (𝜑 → (seq0( + , 𝐹)‘𝑁) < ((seq0( + , 𝐹)‘𝑁) + Σ𝑘 ∈ (ℤ≥‘(𝑁 + 1))(𝐹‘𝑘))) |
31 | 3 | efval2 14653 | . . . 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 9947 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘𝑘) ∈ ℂ) |
34 | 1, 14, 16, 17, 33, 28 | isumsplit 14411 | . . 3 ⊢ (𝜑 → Σ𝑘 ∈ ℕ0 (𝐹‘𝑘) = (Σ𝑘 ∈ (0...((𝑁 + 1) − 1))(𝐹‘𝑘) + Σ𝑘 ∈ (ℤ≥‘(𝑁 + 1))(𝐹‘𝑘))) |
35 | 12 | nn0cnd 11230 | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℂ) |
36 | ax-1cn 9873 | . . . . . . . 8 ⊢ 1 ∈ ℂ | |
37 | pncan 10166 | . . . . . . . 8 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 + 1) − 1) = 𝑁) | |
38 | 35, 36, 37 | sylancl 693 | . . . . . . 7 ⊢ (𝜑 → ((𝑁 + 1) − 1) = 𝑁) |
39 | 38 | oveq2d 6565 | . . . . . 6 ⊢ (𝜑 → (0...((𝑁 + 1) − 1)) = (0...𝑁)) |
40 | 39 | sumeq1d 14279 | . . . . 5 ⊢ (𝜑 → Σ𝑘 ∈ (0...((𝑁 + 1) − 1))(𝐹‘𝑘) = Σ𝑘 ∈ (0...𝑁)(𝐹‘𝑘)) |
41 | eqidd 2611 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝐹‘𝑘) = (𝐹‘𝑘)) | |
42 | 12, 1 | syl6eleq 2698 | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘0)) |
43 | elfznn0 12302 | . . . . . . 7 ⊢ (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℕ0) | |
44 | 43, 33 | sylan2 490 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝐹‘𝑘) ∈ ℂ) |
45 | 41, 42, 44 | fsumser 14308 | . . . . 5 ⊢ (𝜑 → Σ𝑘 ∈ (0...𝑁)(𝐹‘𝑘) = (seq0( + , 𝐹)‘𝑁)) |
46 | 40, 45 | eqtrd 2644 | . . . 4 ⊢ (𝜑 → Σ𝑘 ∈ (0...((𝑁 + 1) − 1))(𝐹‘𝑘) = (seq0( + , 𝐹)‘𝑁)) |
47 | 46 | oveq1d 6564 | . . 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 2648 | . 2 ⊢ (𝜑 → (exp‘𝐴) = ((seq0( + , 𝐹)‘𝑁) + Σ𝑘 ∈ (ℤ≥‘(𝑁 + 1))(𝐹‘𝑘))) |
49 | 30, 48 | breqtrrd 4611 | 1 ⊢ (𝜑 → (seq0( + , 𝐹)‘𝑁) < (exp‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 class class class wbr 4583 ↦ cmpt 4643 dom cdm 5038 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 ℝcr 9814 0cc0 9815 1c1 9816 + caddc 9818 < clt 9953 − cmin 10145 / cdiv 10563 ℕcn 10897 ℕ0cn0 11169 ℤcz 11254 ℤ≥cuz 11563 ℝ+crp 11708 ...cfz 12197 seqcseq 12663 ↑cexp 12722 !cfa 12922 ⇝ cli 14063 Σcsu 14264 expce 14631 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-ico 12052 df-fz 12198 df-fzo 12335 df-fl 12455 df-seq 12664 df-exp 12723 df-fac 12923 df-hash 12980 df-shft 13655 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-limsup 14050 df-clim 14067 df-rlim 14068 df-sum 14265 df-ef 14637 |
This theorem is referenced by: efgt1p2 14683 efgt1p 14684 |
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