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Mirrors > Home > MPE Home > Th. List > efsep | Structured version Visualization version GIF version |
Description: Separate out the next term of the power series expansion of the exponential function. The last hypothesis allows the separated terms to be rearranged as desired. (Contributed by Paul Chapman, 23-Nov-2007.) (Revised by Mario Carneiro, 29-Apr-2014.) |
Ref | Expression |
---|---|
efsep.1 | ⊢ 𝐹 = (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛))) |
efsep.2 | ⊢ 𝑁 = (𝑀 + 1) |
efsep.3 | ⊢ 𝑀 ∈ ℕ0 |
efsep.4 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
efsep.5 | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
efsep.6 | ⊢ (𝜑 → (exp‘𝐴) = (𝐵 + Σ𝑘 ∈ (ℤ≥‘𝑀)(𝐹‘𝑘))) |
efsep.7 | ⊢ (𝜑 → (𝐵 + ((𝐴↑𝑀) / (!‘𝑀))) = 𝐷) |
Ref | Expression |
---|---|
efsep | ⊢ (𝜑 → (exp‘𝐴) = (𝐷 + Σ𝑘 ∈ (ℤ≥‘𝑁)(𝐹‘𝑘))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | efsep.6 | . 2 ⊢ (𝜑 → (exp‘𝐴) = (𝐵 + Σ𝑘 ∈ (ℤ≥‘𝑀)(𝐹‘𝑘))) | |
2 | eqid 2778 | . . . . . 6 ⊢ (ℤ≥‘𝑀) = (ℤ≥‘𝑀) | |
3 | efsep.3 | . . . . . . . 8 ⊢ 𝑀 ∈ ℕ0 | |
4 | 3 | nn0zi 11754 | . . . . . . 7 ⊢ 𝑀 ∈ ℤ |
5 | 4 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ) |
6 | eqidd 2779 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) = (𝐹‘𝑘)) | |
7 | eluznn0 12064 | . . . . . . . 8 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → 𝑘 ∈ ℕ0) | |
8 | 3, 7 | mpan 680 | . . . . . . 7 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → 𝑘 ∈ ℕ0) |
9 | efsep.1 | . . . . . . . . . 10 ⊢ 𝐹 = (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛))) | |
10 | 9 | eftval 15209 | . . . . . . . . 9 ⊢ (𝑘 ∈ ℕ0 → (𝐹‘𝑘) = ((𝐴↑𝑘) / (!‘𝑘))) |
11 | 10 | adantl 475 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘𝑘) = ((𝐴↑𝑘) / (!‘𝑘))) |
12 | efsep.4 | . . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
13 | eftcl 15206 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → ((𝐴↑𝑘) / (!‘𝑘)) ∈ ℂ) | |
14 | 12, 13 | sylan 575 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝐴↑𝑘) / (!‘𝑘)) ∈ ℂ) |
15 | 11, 14 | eqeltrd 2859 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘𝑘) ∈ ℂ) |
16 | 8, 15 | sylan2 586 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) ∈ ℂ) |
17 | 9 | eftlcvg 15238 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑀 ∈ ℕ0) → seq𝑀( + , 𝐹) ∈ dom ⇝ ) |
18 | 12, 3, 17 | sylancl 580 | . . . . . 6 ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ ) |
19 | 2, 5, 6, 16, 18 | isum1p 14977 | . . . . 5 ⊢ (𝜑 → Σ𝑘 ∈ (ℤ≥‘𝑀)(𝐹‘𝑘) = ((𝐹‘𝑀) + Σ𝑘 ∈ (ℤ≥‘(𝑀 + 1))(𝐹‘𝑘))) |
20 | 9 | eftval 15209 | . . . . . . 7 ⊢ (𝑀 ∈ ℕ0 → (𝐹‘𝑀) = ((𝐴↑𝑀) / (!‘𝑀))) |
21 | 3, 20 | ax-mp 5 | . . . . . 6 ⊢ (𝐹‘𝑀) = ((𝐴↑𝑀) / (!‘𝑀)) |
22 | efsep.2 | . . . . . . . . 9 ⊢ 𝑁 = (𝑀 + 1) | |
23 | 22 | eqcomi 2787 | . . . . . . . 8 ⊢ (𝑀 + 1) = 𝑁 |
24 | 23 | fveq2i 6449 | . . . . . . 7 ⊢ (ℤ≥‘(𝑀 + 1)) = (ℤ≥‘𝑁) |
25 | 24 | sumeq1i 14836 | . . . . . 6 ⊢ Σ𝑘 ∈ (ℤ≥‘(𝑀 + 1))(𝐹‘𝑘) = Σ𝑘 ∈ (ℤ≥‘𝑁)(𝐹‘𝑘) |
26 | 21, 25 | oveq12i 6934 | . . . . 5 ⊢ ((𝐹‘𝑀) + Σ𝑘 ∈ (ℤ≥‘(𝑀 + 1))(𝐹‘𝑘)) = (((𝐴↑𝑀) / (!‘𝑀)) + Σ𝑘 ∈ (ℤ≥‘𝑁)(𝐹‘𝑘)) |
27 | 19, 26 | syl6eq 2830 | . . . 4 ⊢ (𝜑 → Σ𝑘 ∈ (ℤ≥‘𝑀)(𝐹‘𝑘) = (((𝐴↑𝑀) / (!‘𝑀)) + Σ𝑘 ∈ (ℤ≥‘𝑁)(𝐹‘𝑘))) |
28 | 27 | oveq2d 6938 | . . 3 ⊢ (𝜑 → (𝐵 + Σ𝑘 ∈ (ℤ≥‘𝑀)(𝐹‘𝑘)) = (𝐵 + (((𝐴↑𝑀) / (!‘𝑀)) + Σ𝑘 ∈ (ℤ≥‘𝑁)(𝐹‘𝑘)))) |
29 | efsep.5 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
30 | eftcl 15206 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑀 ∈ ℕ0) → ((𝐴↑𝑀) / (!‘𝑀)) ∈ ℂ) | |
31 | 12, 3, 30 | sylancl 580 | . . . 4 ⊢ (𝜑 → ((𝐴↑𝑀) / (!‘𝑀)) ∈ ℂ) |
32 | peano2nn0 11684 | . . . . . . 7 ⊢ (𝑀 ∈ ℕ0 → (𝑀 + 1) ∈ ℕ0) | |
33 | 3, 32 | ax-mp 5 | . . . . . 6 ⊢ (𝑀 + 1) ∈ ℕ0 |
34 | 22, 33 | eqeltri 2855 | . . . . 5 ⊢ 𝑁 ∈ ℕ0 |
35 | 9 | eftlcl 15239 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → Σ𝑘 ∈ (ℤ≥‘𝑁)(𝐹‘𝑘) ∈ ℂ) |
36 | 12, 34, 35 | sylancl 580 | . . . 4 ⊢ (𝜑 → Σ𝑘 ∈ (ℤ≥‘𝑁)(𝐹‘𝑘) ∈ ℂ) |
37 | 29, 31, 36 | addassd 10399 | . . 3 ⊢ (𝜑 → ((𝐵 + ((𝐴↑𝑀) / (!‘𝑀))) + Σ𝑘 ∈ (ℤ≥‘𝑁)(𝐹‘𝑘)) = (𝐵 + (((𝐴↑𝑀) / (!‘𝑀)) + Σ𝑘 ∈ (ℤ≥‘𝑁)(𝐹‘𝑘)))) |
38 | 28, 37 | eqtr4d 2817 | . 2 ⊢ (𝜑 → (𝐵 + Σ𝑘 ∈ (ℤ≥‘𝑀)(𝐹‘𝑘)) = ((𝐵 + ((𝐴↑𝑀) / (!‘𝑀))) + Σ𝑘 ∈ (ℤ≥‘𝑁)(𝐹‘𝑘))) |
39 | efsep.7 | . . 3 ⊢ (𝜑 → (𝐵 + ((𝐴↑𝑀) / (!‘𝑀))) = 𝐷) | |
40 | 39 | oveq1d 6937 | . 2 ⊢ (𝜑 → ((𝐵 + ((𝐴↑𝑀) / (!‘𝑀))) + Σ𝑘 ∈ (ℤ≥‘𝑁)(𝐹‘𝑘)) = (𝐷 + Σ𝑘 ∈ (ℤ≥‘𝑁)(𝐹‘𝑘))) |
41 | 1, 38, 40 | 3eqtrd 2818 | 1 ⊢ (𝜑 → (exp‘𝐴) = (𝐷 + Σ𝑘 ∈ (ℤ≥‘𝑁)(𝐹‘𝑘))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ↦ cmpt 4965 dom cdm 5355 ‘cfv 6135 (class class class)co 6922 ℂcc 10270 1c1 10273 + caddc 10275 / cdiv 11032 ℕ0cn0 11642 ℤcz 11728 ℤ≥cuz 11992 seqcseq 13119 ↑cexp 13178 !cfa 13378 ⇝ cli 14623 Σcsu 14824 expce 15194 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-inf2 8835 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-pre-sup 10350 ax-addf 10351 ax-mulf 10352 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-int 4711 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-se 5315 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-isom 6144 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-oadd 7847 df-er 8026 df-pm 8143 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-sup 8636 df-inf 8637 df-oi 8704 df-card 9098 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-div 11033 df-nn 11375 df-2 11438 df-3 11439 df-n0 11643 df-z 11729 df-uz 11993 df-rp 12138 df-ico 12493 df-fz 12644 df-fzo 12785 df-fl 12912 df-seq 13120 df-exp 13179 df-fac 13379 df-hash 13436 df-shft 14214 df-cj 14246 df-re 14247 df-im 14248 df-sqrt 14382 df-abs 14383 df-limsup 14610 df-clim 14627 df-rlim 14628 df-sum 14825 |
This theorem is referenced by: ef4p 15245 dveflem 24179 |
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