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Mirrors > Home > ILE Home > Th. List > recos4p | GIF version |
Description: Separate out the first four terms of the infinite series expansion of the cosine of a real number. (Contributed by Paul Chapman, 19-Jan-2008.) (Revised by Mario Carneiro, 30-Apr-2014.) |
Ref | Expression |
---|---|
efi4p.1 | ⊢ 𝐹 = (𝑛 ∈ ℕ0 ↦ (((i · 𝐴)↑𝑛) / (!‘𝑛))) |
Ref | Expression |
---|---|
recos4p | ⊢ (𝐴 ∈ ℝ → (cos‘𝐴) = ((1 − ((𝐴↑2) / 2)) + (ℜ‘Σ𝑘 ∈ (ℤ≥‘4)(𝐹‘𝑘)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | recosval 11719 | . 2 ⊢ (𝐴 ∈ ℝ → (cos‘𝐴) = (ℜ‘(exp‘(i · 𝐴)))) | |
2 | recn 7943 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
3 | efi4p.1 | . . . . . 6 ⊢ 𝐹 = (𝑛 ∈ ℕ0 ↦ (((i · 𝐴)↑𝑛) / (!‘𝑛))) | |
4 | 3 | efi4p 11720 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (exp‘(i · 𝐴)) = (((1 − ((𝐴↑2) / 2)) + (i · (𝐴 − ((𝐴↑3) / 6)))) + Σ𝑘 ∈ (ℤ≥‘4)(𝐹‘𝑘))) |
5 | 2, 4 | syl 14 | . . . 4 ⊢ (𝐴 ∈ ℝ → (exp‘(i · 𝐴)) = (((1 − ((𝐴↑2) / 2)) + (i · (𝐴 − ((𝐴↑3) / 6)))) + Σ𝑘 ∈ (ℤ≥‘4)(𝐹‘𝑘))) |
6 | 5 | fveq2d 5519 | . . 3 ⊢ (𝐴 ∈ ℝ → (ℜ‘(exp‘(i · 𝐴))) = (ℜ‘(((1 − ((𝐴↑2) / 2)) + (i · (𝐴 − ((𝐴↑3) / 6)))) + Σ𝑘 ∈ (ℤ≥‘4)(𝐹‘𝑘)))) |
7 | 1re 7955 | . . . . . . 7 ⊢ 1 ∈ ℝ | |
8 | resqcl 10584 | . . . . . . . 8 ⊢ (𝐴 ∈ ℝ → (𝐴↑2) ∈ ℝ) | |
9 | 8 | rehalfcld 9163 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ → ((𝐴↑2) / 2) ∈ ℝ) |
10 | resubcl 8219 | . . . . . . 7 ⊢ ((1 ∈ ℝ ∧ ((𝐴↑2) / 2) ∈ ℝ) → (1 − ((𝐴↑2) / 2)) ∈ ℝ) | |
11 | 7, 9, 10 | sylancr 414 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → (1 − ((𝐴↑2) / 2)) ∈ ℝ) |
12 | 11 | recnd 7984 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (1 − ((𝐴↑2) / 2)) ∈ ℂ) |
13 | ax-icn 7905 | . . . . . 6 ⊢ i ∈ ℂ | |
14 | 3nn0 9192 | . . . . . . . . . 10 ⊢ 3 ∈ ℕ0 | |
15 | reexpcl 10534 | . . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 3 ∈ ℕ0) → (𝐴↑3) ∈ ℝ) | |
16 | 14, 15 | mpan2 425 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → (𝐴↑3) ∈ ℝ) |
17 | 6re 8998 | . . . . . . . . . 10 ⊢ 6 ∈ ℝ | |
18 | 6pos 9018 | . . . . . . . . . . 11 ⊢ 0 < 6 | |
19 | 17, 18 | gt0ap0ii 8583 | . . . . . . . . . 10 ⊢ 6 # 0 |
20 | redivclap 8686 | . . . . . . . . . 10 ⊢ (((𝐴↑3) ∈ ℝ ∧ 6 ∈ ℝ ∧ 6 # 0) → ((𝐴↑3) / 6) ∈ ℝ) | |
21 | 17, 19, 20 | mp3an23 1329 | . . . . . . . . 9 ⊢ ((𝐴↑3) ∈ ℝ → ((𝐴↑3) / 6) ∈ ℝ) |
22 | 16, 21 | syl 14 | . . . . . . . 8 ⊢ (𝐴 ∈ ℝ → ((𝐴↑3) / 6) ∈ ℝ) |
23 | resubcl 8219 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ ((𝐴↑3) / 6) ∈ ℝ) → (𝐴 − ((𝐴↑3) / 6)) ∈ ℝ) | |
24 | 22, 23 | mpdan 421 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ → (𝐴 − ((𝐴↑3) / 6)) ∈ ℝ) |
25 | 24 | recnd 7984 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → (𝐴 − ((𝐴↑3) / 6)) ∈ ℂ) |
26 | mulcl 7937 | . . . . . 6 ⊢ ((i ∈ ℂ ∧ (𝐴 − ((𝐴↑3) / 6)) ∈ ℂ) → (i · (𝐴 − ((𝐴↑3) / 6))) ∈ ℂ) | |
27 | 13, 25, 26 | sylancr 414 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (i · (𝐴 − ((𝐴↑3) / 6))) ∈ ℂ) |
28 | 12, 27 | addcld 7975 | . . . 4 ⊢ (𝐴 ∈ ℝ → ((1 − ((𝐴↑2) / 2)) + (i · (𝐴 − ((𝐴↑3) / 6)))) ∈ ℂ) |
29 | mulcl 7937 | . . . . . 6 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i · 𝐴) ∈ ℂ) | |
30 | 13, 2, 29 | sylancr 414 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (i · 𝐴) ∈ ℂ) |
31 | 4nn0 9193 | . . . . 5 ⊢ 4 ∈ ℕ0 | |
32 | 3 | eftlcl 11691 | . . . . 5 ⊢ (((i · 𝐴) ∈ ℂ ∧ 4 ∈ ℕ0) → Σ𝑘 ∈ (ℤ≥‘4)(𝐹‘𝑘) ∈ ℂ) |
33 | 30, 31, 32 | sylancl 413 | . . . 4 ⊢ (𝐴 ∈ ℝ → Σ𝑘 ∈ (ℤ≥‘4)(𝐹‘𝑘) ∈ ℂ) |
34 | 28, 33 | readdd 10963 | . . 3 ⊢ (𝐴 ∈ ℝ → (ℜ‘(((1 − ((𝐴↑2) / 2)) + (i · (𝐴 − ((𝐴↑3) / 6)))) + Σ𝑘 ∈ (ℤ≥‘4)(𝐹‘𝑘))) = ((ℜ‘((1 − ((𝐴↑2) / 2)) + (i · (𝐴 − ((𝐴↑3) / 6))))) + (ℜ‘Σ𝑘 ∈ (ℤ≥‘4)(𝐹‘𝑘)))) |
35 | 11, 24 | crred 10980 | . . . 4 ⊢ (𝐴 ∈ ℝ → (ℜ‘((1 − ((𝐴↑2) / 2)) + (i · (𝐴 − ((𝐴↑3) / 6))))) = (1 − ((𝐴↑2) / 2))) |
36 | 35 | oveq1d 5889 | . . 3 ⊢ (𝐴 ∈ ℝ → ((ℜ‘((1 − ((𝐴↑2) / 2)) + (i · (𝐴 − ((𝐴↑3) / 6))))) + (ℜ‘Σ𝑘 ∈ (ℤ≥‘4)(𝐹‘𝑘))) = ((1 − ((𝐴↑2) / 2)) + (ℜ‘Σ𝑘 ∈ (ℤ≥‘4)(𝐹‘𝑘)))) |
37 | 6, 34, 36 | 3eqtrd 2214 | . 2 ⊢ (𝐴 ∈ ℝ → (ℜ‘(exp‘(i · 𝐴))) = ((1 − ((𝐴↑2) / 2)) + (ℜ‘Σ𝑘 ∈ (ℤ≥‘4)(𝐹‘𝑘)))) |
38 | 1, 37 | eqtrd 2210 | 1 ⊢ (𝐴 ∈ ℝ → (cos‘𝐴) = ((1 − ((𝐴↑2) / 2)) + (ℜ‘Σ𝑘 ∈ (ℤ≥‘4)(𝐹‘𝑘)))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1353 ∈ wcel 2148 class class class wbr 4003 ↦ cmpt 4064 ‘cfv 5216 (class class class)co 5874 ℂcc 7808 ℝcr 7809 0cc0 7810 1c1 7811 ici 7812 + caddc 7813 · cmul 7815 − cmin 8126 # cap 8536 / cdiv 8627 2c2 8968 3c3 8969 4c4 8970 6c6 8972 ℕ0cn0 9174 ℤ≥cuz 9526 ↑cexp 10516 !cfa 10700 ℜcre 10844 Σcsu 11356 expce 11645 cosccos 11648 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4118 ax-sep 4121 ax-nul 4129 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-iinf 4587 ax-cnex 7901 ax-resscn 7902 ax-1cn 7903 ax-1re 7904 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-mulrcl 7909 ax-addcom 7910 ax-mulcom 7911 ax-addass 7912 ax-mulass 7913 ax-distr 7914 ax-i2m1 7915 ax-0lt1 7916 ax-1rid 7917 ax-0id 7918 ax-rnegex 7919 ax-precex 7920 ax-cnre 7921 ax-pre-ltirr 7922 ax-pre-ltwlin 7923 ax-pre-lttrn 7924 ax-pre-apti 7925 ax-pre-ltadd 7926 ax-pre-mulgt0 7927 ax-pre-mulext 7928 ax-arch 7929 ax-caucvg 7930 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-if 3535 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-iun 3888 df-br 4004 df-opab 4065 df-mpt 4066 df-tr 4102 df-id 4293 df-po 4296 df-iso 4297 df-iord 4366 df-on 4368 df-ilim 4369 df-suc 4371 df-iom 4590 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-f1 5221 df-fo 5222 df-f1o 5223 df-fv 5224 df-isom 5225 df-riota 5830 df-ov 5877 df-oprab 5878 df-mpo 5879 df-1st 6140 df-2nd 6141 df-recs 6305 df-irdg 6370 df-frec 6391 df-1o 6416 df-oadd 6420 df-er 6534 df-en 6740 df-dom 6741 df-fin 6742 df-pnf 7992 df-mnf 7993 df-xr 7994 df-ltxr 7995 df-le 7996 df-sub 8128 df-neg 8129 df-reap 8530 df-ap 8537 df-div 8628 df-inn 8918 df-2 8976 df-3 8977 df-4 8978 df-5 8979 df-6 8980 df-n0 9175 df-z 9252 df-uz 9527 df-q 9618 df-rp 9652 df-ico 9892 df-fz 10007 df-fzo 10140 df-seqfrec 10443 df-exp 10517 df-fac 10701 df-ihash 10751 df-cj 10846 df-re 10847 df-im 10848 df-rsqrt 11002 df-abs 11003 df-clim 11282 df-sumdc 11357 df-ef 11651 df-cos 11654 |
This theorem is referenced by: cos01bnd 11761 |
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