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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 12329 | . 2 ⊢ 4 = (3 + 1) | |
3 | 3nn0 12542 | . 2 ⊢ 3 ∈ ℕ0 | |
4 | id 22 | . 2 ⊢ (𝐴 ∈ ℂ → 𝐴 ∈ ℂ) | |
5 | ax-1cn 11216 | . . . 4 ⊢ 1 ∈ ℂ | |
6 | addcl 11240 | . . . 4 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (1 + 𝐴) ∈ ℂ) | |
7 | 5, 6 | mpan 688 | . . 3 ⊢ (𝐴 ∈ ℂ → (1 + 𝐴) ∈ ℂ) |
8 | sqcl 14137 | . . . 4 ⊢ (𝐴 ∈ ℂ → (𝐴↑2) ∈ ℂ) | |
9 | 8 | halfcld 12509 | . . 3 ⊢ (𝐴 ∈ ℂ → ((𝐴↑2) / 2) ∈ ℂ) |
10 | 7, 9 | addcld 11283 | . 2 ⊢ (𝐴 ∈ ℂ → ((1 + 𝐴) + ((𝐴↑2) / 2)) ∈ ℂ) |
11 | df-3 12328 | . . 3 ⊢ 3 = (2 + 1) | |
12 | 2nn0 12541 | . . 3 ⊢ 2 ∈ ℕ0 | |
13 | df-2 12327 | . . . 4 ⊢ 2 = (1 + 1) | |
14 | 1nn0 12540 | . . . 4 ⊢ 1 ∈ ℕ0 | |
15 | 5 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ ℂ → 1 ∈ ℂ) |
16 | 1e0p1 12771 | . . . . 5 ⊢ 1 = (0 + 1) | |
17 | 0nn0 12539 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
18 | 0cnd 11257 | . . . . 5 ⊢ (𝐴 ∈ ℂ → 0 ∈ ℂ) | |
19 | 1 | efval2 16086 | . . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (exp‘𝐴) = Σ𝑘 ∈ ℕ0 (𝐹‘𝑘)) |
20 | nn0uz 12916 | . . . . . . . . 9 ⊢ ℕ0 = (ℤ≥‘0) | |
21 | 20 | sumeq1i 15702 | . . . . . . . 8 ⊢ Σ𝑘 ∈ ℕ0 (𝐹‘𝑘) = Σ𝑘 ∈ (ℤ≥‘0)(𝐹‘𝑘) |
22 | 19, 21 | eqtr2di 2783 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → Σ𝑘 ∈ (ℤ≥‘0)(𝐹‘𝑘) = (exp‘𝐴)) |
23 | 22 | oveq2d 7440 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (0 + Σ𝑘 ∈ (ℤ≥‘0)(𝐹‘𝑘)) = (0 + (exp‘𝐴))) |
24 | efcl 16084 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (exp‘𝐴) ∈ ℂ) | |
25 | 24 | addlidd 11465 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (0 + (exp‘𝐴)) = (exp‘𝐴)) |
26 | 23, 25 | eqtr2d 2767 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (exp‘𝐴) = (0 + Σ𝑘 ∈ (ℤ≥‘0)(𝐹‘𝑘))) |
27 | eft0val 16114 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → ((𝐴↑0) / (!‘0)) = 1) | |
28 | 27 | oveq2d 7440 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (0 + ((𝐴↑0) / (!‘0))) = (0 + 1)) |
29 | 0p1e1 12386 | . . . . . 6 ⊢ (0 + 1) = 1 | |
30 | 28, 29 | eqtrdi 2782 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (0 + ((𝐴↑0) / (!‘0))) = 1) |
31 | 1, 16, 17, 4, 18, 26, 30 | efsep 16112 | . . . 4 ⊢ (𝐴 ∈ ℂ → (exp‘𝐴) = (1 + Σ𝑘 ∈ (ℤ≥‘1)(𝐹‘𝑘))) |
32 | exp1 14087 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (𝐴↑1) = 𝐴) | |
33 | fac1 14294 | . . . . . . . 8 ⊢ (!‘1) = 1 | |
34 | 33 | a1i 11 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (!‘1) = 1) |
35 | 32, 34 | oveq12d 7442 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → ((𝐴↑1) / (!‘1)) = (𝐴 / 1)) |
36 | div1 11954 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (𝐴 / 1) = 𝐴) | |
37 | 35, 36 | eqtrd 2766 | . . . . 5 ⊢ (𝐴 ∈ ℂ → ((𝐴↑1) / (!‘1)) = 𝐴) |
38 | 37 | oveq2d 7440 | . . . 4 ⊢ (𝐴 ∈ ℂ → (1 + ((𝐴↑1) / (!‘1))) = (1 + 𝐴)) |
39 | 1, 13, 14, 4, 15, 31, 38 | efsep 16112 | . . 3 ⊢ (𝐴 ∈ ℂ → (exp‘𝐴) = ((1 + 𝐴) + Σ𝑘 ∈ (ℤ≥‘2)(𝐹‘𝑘))) |
40 | fac2 14296 | . . . . . 6 ⊢ (!‘2) = 2 | |
41 | 40 | oveq2i 7435 | . . . . 5 ⊢ ((𝐴↑2) / (!‘2)) = ((𝐴↑2) / 2) |
42 | 41 | oveq2i 7435 | . . . 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 16112 | . 2 ⊢ (𝐴 ∈ ℂ → (exp‘𝐴) = (((1 + 𝐴) + ((𝐴↑2) / 2)) + Σ𝑘 ∈ (ℤ≥‘3)(𝐹‘𝑘))) |
45 | fac3 14297 | . . . . 5 ⊢ (!‘3) = 6 | |
46 | 45 | oveq2i 7435 | . . . 4 ⊢ ((𝐴↑3) / (!‘3)) = ((𝐴↑3) / 6) |
47 | 46 | oveq2i 7435 | . . 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 16112 | 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 1534 ∈ wcel 2099 ↦ cmpt 5236 ‘cfv 6554 (class class class)co 7424 ℂcc 11156 0cc0 11158 1c1 11159 + caddc 11161 / cdiv 11921 2c2 12319 3c3 12320 4c4 12321 6c6 12323 ℕ0cn0 12524 ℤ≥cuz 12874 ↑cexp 14081 !cfa 14290 Σcsu 15690 expce 16063 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-inf2 9684 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 ax-pre-sup 11236 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-int 4955 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-isom 6563 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-1st 8003 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-1o 8496 df-er 8734 df-pm 8858 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-sup 9485 df-inf 9486 df-oi 9553 df-card 9982 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-div 11922 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-n0 12525 df-z 12611 df-uz 12875 df-rp 13029 df-ico 13384 df-fz 13539 df-fzo 13682 df-fl 13812 df-seq 14022 df-exp 14082 df-fac 14291 df-hash 14348 df-shft 15072 df-cj 15104 df-re 15105 df-im 15106 df-sqrt 15240 df-abs 15241 df-limsup 15473 df-clim 15490 df-rlim 15491 df-sum 15691 df-ef 16069 |
This theorem is referenced by: efi4p 16139 |
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