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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 12332 | . 2 ⊢ 4 = (3 + 1) | |
| 3 | 3nn0 12546 | . 2 ⊢ 3 ∈ ℕ0 | |
| 4 | id 22 | . 2 ⊢ (𝐴 ∈ ℂ → 𝐴 ∈ ℂ) | |
| 5 | ax-1cn 11214 | . . . 4 ⊢ 1 ∈ ℂ | |
| 6 | addcl 11238 | . . . 4 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (1 + 𝐴) ∈ ℂ) | |
| 7 | 5, 6 | mpan 690 | . . 3 ⊢ (𝐴 ∈ ℂ → (1 + 𝐴) ∈ ℂ) | 
| 8 | sqcl 14159 | . . . 4 ⊢ (𝐴 ∈ ℂ → (𝐴↑2) ∈ ℂ) | |
| 9 | 8 | halfcld 12513 | . . 3 ⊢ (𝐴 ∈ ℂ → ((𝐴↑2) / 2) ∈ ℂ) | 
| 10 | 7, 9 | addcld 11281 | . 2 ⊢ (𝐴 ∈ ℂ → ((1 + 𝐴) + ((𝐴↑2) / 2)) ∈ ℂ) | 
| 11 | df-3 12331 | . . 3 ⊢ 3 = (2 + 1) | |
| 12 | 2nn0 12545 | . . 3 ⊢ 2 ∈ ℕ0 | |
| 13 | df-2 12330 | . . . 4 ⊢ 2 = (1 + 1) | |
| 14 | 1nn0 12544 | . . . 4 ⊢ 1 ∈ ℕ0 | |
| 15 | 5 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ ℂ → 1 ∈ ℂ) | 
| 16 | 1e0p1 12777 | . . . . 5 ⊢ 1 = (0 + 1) | |
| 17 | 0nn0 12543 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
| 18 | 0cnd 11255 | . . . . 5 ⊢ (𝐴 ∈ ℂ → 0 ∈ ℂ) | |
| 19 | 1 | efval2 16121 | . . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (exp‘𝐴) = Σ𝑘 ∈ ℕ0 (𝐹‘𝑘)) | 
| 20 | nn0uz 12921 | . . . . . . . . 9 ⊢ ℕ0 = (ℤ≥‘0) | |
| 21 | 20 | sumeq1i 15734 | . . . . . . . 8 ⊢ Σ𝑘 ∈ ℕ0 (𝐹‘𝑘) = Σ𝑘 ∈ (ℤ≥‘0)(𝐹‘𝑘) | 
| 22 | 19, 21 | eqtr2di 2793 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → Σ𝑘 ∈ (ℤ≥‘0)(𝐹‘𝑘) = (exp‘𝐴)) | 
| 23 | 22 | oveq2d 7448 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (0 + Σ𝑘 ∈ (ℤ≥‘0)(𝐹‘𝑘)) = (0 + (exp‘𝐴))) | 
| 24 | efcl 16119 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (exp‘𝐴) ∈ ℂ) | |
| 25 | 24 | addlidd 11463 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (0 + (exp‘𝐴)) = (exp‘𝐴)) | 
| 26 | 23, 25 | eqtr2d 2777 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (exp‘𝐴) = (0 + Σ𝑘 ∈ (ℤ≥‘0)(𝐹‘𝑘))) | 
| 27 | eft0val 16149 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → ((𝐴↑0) / (!‘0)) = 1) | |
| 28 | 27 | oveq2d 7448 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (0 + ((𝐴↑0) / (!‘0))) = (0 + 1)) | 
| 29 | 0p1e1 12389 | . . . . . 6 ⊢ (0 + 1) = 1 | |
| 30 | 28, 29 | eqtrdi 2792 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (0 + ((𝐴↑0) / (!‘0))) = 1) | 
| 31 | 1, 16, 17, 4, 18, 26, 30 | efsep 16147 | . . . 4 ⊢ (𝐴 ∈ ℂ → (exp‘𝐴) = (1 + Σ𝑘 ∈ (ℤ≥‘1)(𝐹‘𝑘))) | 
| 32 | exp1 14109 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (𝐴↑1) = 𝐴) | |
| 33 | fac1 14317 | . . . . . . . 8 ⊢ (!‘1) = 1 | |
| 34 | 33 | a1i 11 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (!‘1) = 1) | 
| 35 | 32, 34 | oveq12d 7450 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → ((𝐴↑1) / (!‘1)) = (𝐴 / 1)) | 
| 36 | div1 11958 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (𝐴 / 1) = 𝐴) | |
| 37 | 35, 36 | eqtrd 2776 | . . . . 5 ⊢ (𝐴 ∈ ℂ → ((𝐴↑1) / (!‘1)) = 𝐴) | 
| 38 | 37 | oveq2d 7448 | . . . 4 ⊢ (𝐴 ∈ ℂ → (1 + ((𝐴↑1) / (!‘1))) = (1 + 𝐴)) | 
| 39 | 1, 13, 14, 4, 15, 31, 38 | efsep 16147 | . . 3 ⊢ (𝐴 ∈ ℂ → (exp‘𝐴) = ((1 + 𝐴) + Σ𝑘 ∈ (ℤ≥‘2)(𝐹‘𝑘))) | 
| 40 | fac2 14319 | . . . . . 6 ⊢ (!‘2) = 2 | |
| 41 | 40 | oveq2i 7443 | . . . . 5 ⊢ ((𝐴↑2) / (!‘2)) = ((𝐴↑2) / 2) | 
| 42 | 41 | oveq2i 7443 | . . . 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 16147 | . 2 ⊢ (𝐴 ∈ ℂ → (exp‘𝐴) = (((1 + 𝐴) + ((𝐴↑2) / 2)) + Σ𝑘 ∈ (ℤ≥‘3)(𝐹‘𝑘))) | 
| 45 | fac3 14320 | . . . . 5 ⊢ (!‘3) = 6 | |
| 46 | 45 | oveq2i 7443 | . . . 4 ⊢ ((𝐴↑3) / (!‘3)) = ((𝐴↑3) / 6) | 
| 47 | 46 | oveq2i 7443 | . . 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 16147 | 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 1539 ∈ wcel 2107 ↦ cmpt 5224 ‘cfv 6560 (class class class)co 7432 ℂcc 11154 0cc0 11156 1c1 11157 + caddc 11159 / cdiv 11921 2c2 12322 3c3 12323 4c4 12324 6c6 12326 ℕ0cn0 12528 ℤ≥cuz 12879 ↑cexp 14103 !cfa 14313 Σcsu 15723 expce 16098 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-inf2 9682 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 ax-pre-sup 11234 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-se 5637 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-isom 6569 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-er 8746 df-pm 8870 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-sup 9483 df-inf 9484 df-oi 9551 df-card 9980 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-div 11922 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-n0 12529 df-z 12616 df-uz 12880 df-rp 13036 df-ico 13394 df-fz 13549 df-fzo 13696 df-fl 13833 df-seq 14044 df-exp 14104 df-fac 14314 df-hash 14371 df-shft 15107 df-cj 15139 df-re 15140 df-im 15141 df-sqrt 15275 df-abs 15276 df-limsup 15508 df-clim 15525 df-rlim 15526 df-sum 15724 df-ef 16104 | 
| This theorem is referenced by: efi4p 16174 | 
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