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Mirrors > Home > MPE Home > Th. List > ef4p | Structured version Visualization version GIF version |
Description: Separate out the first four terms of the infinite series expansion of the exponential function. (Contributed by Paul Chapman, 19-Jan-2008.) (Revised by Mario Carneiro, 29-Apr-2014.) |
Ref | Expression |
---|---|
ef4p.1 | ⊢ 𝐹 = (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛))) |
Ref | Expression |
---|---|
ef4p | ⊢ (𝐴 ∈ ℂ → (exp‘𝐴) = ((((1 + 𝐴) + ((𝐴↑2) / 2)) + ((𝐴↑3) / 6)) + Σ𝑘 ∈ (ℤ≥‘4)(𝐹‘𝑘))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ef4p.1 | . 2 ⊢ 𝐹 = (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛))) | |
2 | df-4 11690 | . 2 ⊢ 4 = (3 + 1) | |
3 | 3nn0 11903 | . 2 ⊢ 3 ∈ ℕ0 | |
4 | id 22 | . 2 ⊢ (𝐴 ∈ ℂ → 𝐴 ∈ ℂ) | |
5 | ax-1cn 10584 | . . . 4 ⊢ 1 ∈ ℂ | |
6 | addcl 10608 | . . . 4 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (1 + 𝐴) ∈ ℂ) | |
7 | 5, 6 | mpan 689 | . . 3 ⊢ (𝐴 ∈ ℂ → (1 + 𝐴) ∈ ℂ) |
8 | sqcl 13480 | . . . 4 ⊢ (𝐴 ∈ ℂ → (𝐴↑2) ∈ ℂ) | |
9 | 8 | halfcld 11870 | . . 3 ⊢ (𝐴 ∈ ℂ → ((𝐴↑2) / 2) ∈ ℂ) |
10 | 7, 9 | addcld 10649 | . 2 ⊢ (𝐴 ∈ ℂ → ((1 + 𝐴) + ((𝐴↑2) / 2)) ∈ ℂ) |
11 | df-3 11689 | . . 3 ⊢ 3 = (2 + 1) | |
12 | 2nn0 11902 | . . 3 ⊢ 2 ∈ ℕ0 | |
13 | df-2 11688 | . . . 4 ⊢ 2 = (1 + 1) | |
14 | 1nn0 11901 | . . . 4 ⊢ 1 ∈ ℕ0 | |
15 | 5 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ ℂ → 1 ∈ ℂ) |
16 | 1e0p1 12128 | . . . . 5 ⊢ 1 = (0 + 1) | |
17 | 0nn0 11900 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
18 | 0cnd 10623 | . . . . 5 ⊢ (𝐴 ∈ ℂ → 0 ∈ ℂ) | |
19 | 1 | efval2 15429 | . . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (exp‘𝐴) = Σ𝑘 ∈ ℕ0 (𝐹‘𝑘)) |
20 | nn0uz 12268 | . . . . . . . . 9 ⊢ ℕ0 = (ℤ≥‘0) | |
21 | 20 | sumeq1i 15047 | . . . . . . . 8 ⊢ Σ𝑘 ∈ ℕ0 (𝐹‘𝑘) = Σ𝑘 ∈ (ℤ≥‘0)(𝐹‘𝑘) |
22 | 19, 21 | eqtr2di 2850 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → Σ𝑘 ∈ (ℤ≥‘0)(𝐹‘𝑘) = (exp‘𝐴)) |
23 | 22 | oveq2d 7151 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (0 + Σ𝑘 ∈ (ℤ≥‘0)(𝐹‘𝑘)) = (0 + (exp‘𝐴))) |
24 | efcl 15428 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (exp‘𝐴) ∈ ℂ) | |
25 | 24 | addid2d 10830 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (0 + (exp‘𝐴)) = (exp‘𝐴)) |
26 | 23, 25 | eqtr2d 2834 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (exp‘𝐴) = (0 + Σ𝑘 ∈ (ℤ≥‘0)(𝐹‘𝑘))) |
27 | eft0val 15457 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → ((𝐴↑0) / (!‘0)) = 1) | |
28 | 27 | oveq2d 7151 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (0 + ((𝐴↑0) / (!‘0))) = (0 + 1)) |
29 | 0p1e1 11747 | . . . . . 6 ⊢ (0 + 1) = 1 | |
30 | 28, 29 | eqtrdi 2849 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (0 + ((𝐴↑0) / (!‘0))) = 1) |
31 | 1, 16, 17, 4, 18, 26, 30 | efsep 15455 | . . . 4 ⊢ (𝐴 ∈ ℂ → (exp‘𝐴) = (1 + Σ𝑘 ∈ (ℤ≥‘1)(𝐹‘𝑘))) |
32 | exp1 13431 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (𝐴↑1) = 𝐴) | |
33 | fac1 13633 | . . . . . . . 8 ⊢ (!‘1) = 1 | |
34 | 33 | a1i 11 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (!‘1) = 1) |
35 | 32, 34 | oveq12d 7153 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → ((𝐴↑1) / (!‘1)) = (𝐴 / 1)) |
36 | div1 11318 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (𝐴 / 1) = 𝐴) | |
37 | 35, 36 | eqtrd 2833 | . . . . 5 ⊢ (𝐴 ∈ ℂ → ((𝐴↑1) / (!‘1)) = 𝐴) |
38 | 37 | oveq2d 7151 | . . . 4 ⊢ (𝐴 ∈ ℂ → (1 + ((𝐴↑1) / (!‘1))) = (1 + 𝐴)) |
39 | 1, 13, 14, 4, 15, 31, 38 | efsep 15455 | . . 3 ⊢ (𝐴 ∈ ℂ → (exp‘𝐴) = ((1 + 𝐴) + Σ𝑘 ∈ (ℤ≥‘2)(𝐹‘𝑘))) |
40 | fac2 13635 | . . . . . 6 ⊢ (!‘2) = 2 | |
41 | 40 | oveq2i 7146 | . . . . 5 ⊢ ((𝐴↑2) / (!‘2)) = ((𝐴↑2) / 2) |
42 | 41 | oveq2i 7146 | . . . 4 ⊢ ((1 + 𝐴) + ((𝐴↑2) / (!‘2))) = ((1 + 𝐴) + ((𝐴↑2) / 2)) |
43 | 42 | a1i 11 | . . 3 ⊢ (𝐴 ∈ ℂ → ((1 + 𝐴) + ((𝐴↑2) / (!‘2))) = ((1 + 𝐴) + ((𝐴↑2) / 2))) |
44 | 1, 11, 12, 4, 7, 39, 43 | efsep 15455 | . 2 ⊢ (𝐴 ∈ ℂ → (exp‘𝐴) = (((1 + 𝐴) + ((𝐴↑2) / 2)) + Σ𝑘 ∈ (ℤ≥‘3)(𝐹‘𝑘))) |
45 | fac3 13636 | . . . . 5 ⊢ (!‘3) = 6 | |
46 | 45 | oveq2i 7146 | . . . 4 ⊢ ((𝐴↑3) / (!‘3)) = ((𝐴↑3) / 6) |
47 | 46 | oveq2i 7146 | . . 3 ⊢ (((1 + 𝐴) + ((𝐴↑2) / 2)) + ((𝐴↑3) / (!‘3))) = (((1 + 𝐴) + ((𝐴↑2) / 2)) + ((𝐴↑3) / 6)) |
48 | 47 | a1i 11 | . 2 ⊢ (𝐴 ∈ ℂ → (((1 + 𝐴) + ((𝐴↑2) / 2)) + ((𝐴↑3) / (!‘3))) = (((1 + 𝐴) + ((𝐴↑2) / 2)) + ((𝐴↑3) / 6))) |
49 | 1, 2, 3, 4, 10, 44, 48 | efsep 15455 | 1 ⊢ (𝐴 ∈ ℂ → (exp‘𝐴) = ((((1 + 𝐴) + ((𝐴↑2) / 2)) + ((𝐴↑3) / 6)) + Σ𝑘 ∈ (ℤ≥‘4)(𝐹‘𝑘))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ↦ cmpt 5110 ‘cfv 6324 (class class class)co 7135 ℂcc 10524 0cc0 10526 1c1 10527 + caddc 10529 / cdiv 11286 2c2 11680 3c3 11681 4c4 11682 6c6 11684 ℕ0cn0 11885 ℤ≥cuz 12231 ↑cexp 13425 !cfa 13629 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-4 11690 df-5 11691 df-6 11692 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: efi4p 15482 |
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