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Mirrors > Home > ILE Home > Th. List > ef4p | 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 8774 | . 2 ⊢ 4 = (3 + 1) | |
3 | 3nn0 8988 | . 2 ⊢ 3 ∈ ℕ0 | |
4 | id 19 | . 2 ⊢ (𝐴 ∈ ℂ → 𝐴 ∈ ℂ) | |
5 | ax-1cn 7706 | . . . 4 ⊢ 1 ∈ ℂ | |
6 | addcl 7738 | . . . 4 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (1 + 𝐴) ∈ ℂ) | |
7 | 5, 6 | mpan 420 | . . 3 ⊢ (𝐴 ∈ ℂ → (1 + 𝐴) ∈ ℂ) |
8 | sqcl 10347 | . . . 4 ⊢ (𝐴 ∈ ℂ → (𝐴↑2) ∈ ℂ) | |
9 | 8 | halfcld 8957 | . . 3 ⊢ (𝐴 ∈ ℂ → ((𝐴↑2) / 2) ∈ ℂ) |
10 | 7, 9 | addcld 7778 | . 2 ⊢ (𝐴 ∈ ℂ → ((1 + 𝐴) + ((𝐴↑2) / 2)) ∈ ℂ) |
11 | df-3 8773 | . . 3 ⊢ 3 = (2 + 1) | |
12 | 2nn0 8987 | . . 3 ⊢ 2 ∈ ℕ0 | |
13 | df-2 8772 | . . . 4 ⊢ 2 = (1 + 1) | |
14 | 1nn0 8986 | . . . 4 ⊢ 1 ∈ ℕ0 | |
15 | 5 | a1i 9 | . . . 4 ⊢ (𝐴 ∈ ℂ → 1 ∈ ℂ) |
16 | 1e0p1 9216 | . . . . 5 ⊢ 1 = (0 + 1) | |
17 | 0nn0 8985 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
18 | 0cnd 7752 | . . . . 5 ⊢ (𝐴 ∈ ℂ → 0 ∈ ℂ) | |
19 | 1 | efval2 11360 | . . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (exp‘𝐴) = Σ𝑘 ∈ ℕ0 (𝐹‘𝑘)) |
20 | nn0uz 9353 | . . . . . . . . 9 ⊢ ℕ0 = (ℤ≥‘0) | |
21 | 20 | sumeq1i 11125 | . . . . . . . 8 ⊢ Σ𝑘 ∈ ℕ0 (𝐹‘𝑘) = Σ𝑘 ∈ (ℤ≥‘0)(𝐹‘𝑘) |
22 | 19, 21 | syl6req 2187 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → Σ𝑘 ∈ (ℤ≥‘0)(𝐹‘𝑘) = (exp‘𝐴)) |
23 | 22 | oveq2d 5783 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (0 + Σ𝑘 ∈ (ℤ≥‘0)(𝐹‘𝑘)) = (0 + (exp‘𝐴))) |
24 | efcl 11359 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (exp‘𝐴) ∈ ℂ) | |
25 | 24 | addid2d 7905 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (0 + (exp‘𝐴)) = (exp‘𝐴)) |
26 | 23, 25 | eqtr2d 2171 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (exp‘𝐴) = (0 + Σ𝑘 ∈ (ℤ≥‘0)(𝐹‘𝑘))) |
27 | eft0val 11388 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → ((𝐴↑0) / (!‘0)) = 1) | |
28 | 27 | oveq2d 5783 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (0 + ((𝐴↑0) / (!‘0))) = (0 + 1)) |
29 | 0p1e1 8827 | . . . . . 6 ⊢ (0 + 1) = 1 | |
30 | 28, 29 | syl6eq 2186 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (0 + ((𝐴↑0) / (!‘0))) = 1) |
31 | 1, 16, 17, 4, 18, 26, 30 | efsep 11386 | . . . 4 ⊢ (𝐴 ∈ ℂ → (exp‘𝐴) = (1 + Σ𝑘 ∈ (ℤ≥‘1)(𝐹‘𝑘))) |
32 | exp1 10292 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (𝐴↑1) = 𝐴) | |
33 | fac1 10468 | . . . . . . . 8 ⊢ (!‘1) = 1 | |
34 | 33 | a1i 9 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (!‘1) = 1) |
35 | 32, 34 | oveq12d 5785 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → ((𝐴↑1) / (!‘1)) = (𝐴 / 1)) |
36 | div1 8456 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (𝐴 / 1) = 𝐴) | |
37 | 35, 36 | eqtrd 2170 | . . . . 5 ⊢ (𝐴 ∈ ℂ → ((𝐴↑1) / (!‘1)) = 𝐴) |
38 | 37 | oveq2d 5783 | . . . 4 ⊢ (𝐴 ∈ ℂ → (1 + ((𝐴↑1) / (!‘1))) = (1 + 𝐴)) |
39 | 1, 13, 14, 4, 15, 31, 38 | efsep 11386 | . . 3 ⊢ (𝐴 ∈ ℂ → (exp‘𝐴) = ((1 + 𝐴) + Σ𝑘 ∈ (ℤ≥‘2)(𝐹‘𝑘))) |
40 | fac2 10470 | . . . . . 6 ⊢ (!‘2) = 2 | |
41 | 40 | oveq2i 5778 | . . . . 5 ⊢ ((𝐴↑2) / (!‘2)) = ((𝐴↑2) / 2) |
42 | 41 | oveq2i 5778 | . . . 4 ⊢ ((1 + 𝐴) + ((𝐴↑2) / (!‘2))) = ((1 + 𝐴) + ((𝐴↑2) / 2)) |
43 | 42 | a1i 9 | . . 3 ⊢ (𝐴 ∈ ℂ → ((1 + 𝐴) + ((𝐴↑2) / (!‘2))) = ((1 + 𝐴) + ((𝐴↑2) / 2))) |
44 | 1, 11, 12, 4, 7, 39, 43 | efsep 11386 | . 2 ⊢ (𝐴 ∈ ℂ → (exp‘𝐴) = (((1 + 𝐴) + ((𝐴↑2) / 2)) + Σ𝑘 ∈ (ℤ≥‘3)(𝐹‘𝑘))) |
45 | fac3 10471 | . . . . 5 ⊢ (!‘3) = 6 | |
46 | 45 | oveq2i 5778 | . . . 4 ⊢ ((𝐴↑3) / (!‘3)) = ((𝐴↑3) / 6) |
47 | 46 | oveq2i 5778 | . . 3 ⊢ (((1 + 𝐴) + ((𝐴↑2) / 2)) + ((𝐴↑3) / (!‘3))) = (((1 + 𝐴) + ((𝐴↑2) / 2)) + ((𝐴↑3) / 6)) |
48 | 47 | a1i 9 | . 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 11386 | 1 ⊢ (𝐴 ∈ ℂ → (exp‘𝐴) = ((((1 + 𝐴) + ((𝐴↑2) / 2)) + ((𝐴↑3) / 6)) + Σ𝑘 ∈ (ℤ≥‘4)(𝐹‘𝑘))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ∈ wcel 1480 ↦ cmpt 3984 ‘cfv 5118 (class class class)co 5767 ℂcc 7611 0cc0 7613 1c1 7614 + caddc 7616 / cdiv 8425 2c2 8764 3c3 8765 4c4 8766 6c6 8768 ℕ0cn0 8970 ℤ≥cuz 9319 ↑cexp 10285 !cfa 10464 Σcsu 11115 expce 11337 |
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 2119 ax-coll 4038 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-iinf 4497 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-mulrcl 7712 ax-addcom 7713 ax-mulcom 7714 ax-addass 7715 ax-mulass 7716 ax-distr 7717 ax-i2m1 7718 ax-0lt1 7719 ax-1rid 7720 ax-0id 7721 ax-rnegex 7722 ax-precex 7723 ax-cnre 7724 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 ax-pre-lttrn 7727 ax-pre-apti 7728 ax-pre-ltadd 7729 ax-pre-mulgt0 7730 ax-pre-mulext 7731 ax-arch 7732 ax-caucvg 7733 |
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 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-reu 2421 df-rmo 2422 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-if 3470 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-tr 4022 df-id 4210 df-po 4213 df-iso 4214 df-iord 4283 df-on 4285 df-ilim 4286 df-suc 4288 df-iom 4500 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-isom 5127 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-recs 6195 df-irdg 6260 df-frec 6281 df-1o 6306 df-oadd 6310 df-er 6422 df-en 6628 df-dom 6629 df-fin 6630 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-sub 7928 df-neg 7929 df-reap 8330 df-ap 8337 df-div 8426 df-inn 8714 df-2 8772 df-3 8773 df-4 8774 df-5 8775 df-6 8776 df-n0 8971 df-z 9048 df-uz 9320 df-q 9405 df-rp 9435 df-ico 9670 df-fz 9784 df-fzo 9913 df-seqfrec 10212 df-exp 10286 df-fac 10465 df-ihash 10515 df-cj 10607 df-re 10608 df-im 10609 df-rsqrt 10763 df-abs 10764 df-clim 11041 df-sumdc 11116 df-ef 11343 |
This theorem is referenced by: efi4p 11413 |
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