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Mirrors > Home > MPE Home > Th. List > recos4p | Structured version Visualization version 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 15322 | . 2 ⊢ (𝐴 ∈ ℝ → (cos‘𝐴) = (ℜ‘(exp‘(i · 𝐴)))) | |
2 | recn 10473 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
3 | efi4p.1 | . . . . . 6 ⊢ 𝐹 = (𝑛 ∈ ℕ0 ↦ (((i · 𝐴)↑𝑛) / (!‘𝑛))) | |
4 | 3 | efi4p 15323 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (exp‘(i · 𝐴)) = (((1 − ((𝐴↑2) / 2)) + (i · (𝐴 − ((𝐴↑3) / 6)))) + Σ𝑘 ∈ (ℤ≥‘4)(𝐹‘𝑘))) |
5 | 2, 4 | syl 17 | . . . 4 ⊢ (𝐴 ∈ ℝ → (exp‘(i · 𝐴)) = (((1 − ((𝐴↑2) / 2)) + (i · (𝐴 − ((𝐴↑3) / 6)))) + Σ𝑘 ∈ (ℤ≥‘4)(𝐹‘𝑘))) |
6 | 5 | fveq2d 6542 | . . 3 ⊢ (𝐴 ∈ ℝ → (ℜ‘(exp‘(i · 𝐴))) = (ℜ‘(((1 − ((𝐴↑2) / 2)) + (i · (𝐴 − ((𝐴↑3) / 6)))) + Σ𝑘 ∈ (ℤ≥‘4)(𝐹‘𝑘)))) |
7 | 1re 10487 | . . . . . . 7 ⊢ 1 ∈ ℝ | |
8 | resqcl 13340 | . . . . . . . 8 ⊢ (𝐴 ∈ ℝ → (𝐴↑2) ∈ ℝ) | |
9 | 8 | rehalfcld 11732 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ → ((𝐴↑2) / 2) ∈ ℝ) |
10 | resubcl 10798 | . . . . . . 7 ⊢ ((1 ∈ ℝ ∧ ((𝐴↑2) / 2) ∈ ℝ) → (1 − ((𝐴↑2) / 2)) ∈ ℝ) | |
11 | 7, 9, 10 | sylancr 587 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → (1 − ((𝐴↑2) / 2)) ∈ ℝ) |
12 | 11 | recnd 10515 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (1 − ((𝐴↑2) / 2)) ∈ ℂ) |
13 | ax-icn 10442 | . . . . . 6 ⊢ i ∈ ℂ | |
14 | 3nn0 11763 | . . . . . . . . . 10 ⊢ 3 ∈ ℕ0 | |
15 | reexpcl 13296 | . . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 3 ∈ ℕ0) → (𝐴↑3) ∈ ℝ) | |
16 | 14, 15 | mpan2 687 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → (𝐴↑3) ∈ ℝ) |
17 | 6re 11575 | . . . . . . . . . 10 ⊢ 6 ∈ ℝ | |
18 | 6pos 11595 | . . . . . . . . . . 11 ⊢ 0 < 6 | |
19 | 17, 18 | gt0ne0ii 11024 | . . . . . . . . . 10 ⊢ 6 ≠ 0 |
20 | redivcl 11207 | . . . . . . . . . 10 ⊢ (((𝐴↑3) ∈ ℝ ∧ 6 ∈ ℝ ∧ 6 ≠ 0) → ((𝐴↑3) / 6) ∈ ℝ) | |
21 | 17, 19, 20 | mp3an23 1445 | . . . . . . . . 9 ⊢ ((𝐴↑3) ∈ ℝ → ((𝐴↑3) / 6) ∈ ℝ) |
22 | 16, 21 | syl 17 | . . . . . . . 8 ⊢ (𝐴 ∈ ℝ → ((𝐴↑3) / 6) ∈ ℝ) |
23 | resubcl 10798 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ ((𝐴↑3) / 6) ∈ ℝ) → (𝐴 − ((𝐴↑3) / 6)) ∈ ℝ) | |
24 | 22, 23 | mpdan 683 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ → (𝐴 − ((𝐴↑3) / 6)) ∈ ℝ) |
25 | 24 | recnd 10515 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → (𝐴 − ((𝐴↑3) / 6)) ∈ ℂ) |
26 | mulcl 10467 | . . . . . 6 ⊢ ((i ∈ ℂ ∧ (𝐴 − ((𝐴↑3) / 6)) ∈ ℂ) → (i · (𝐴 − ((𝐴↑3) / 6))) ∈ ℂ) | |
27 | 13, 25, 26 | sylancr 587 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (i · (𝐴 − ((𝐴↑3) / 6))) ∈ ℂ) |
28 | 12, 27 | addcld 10506 | . . . 4 ⊢ (𝐴 ∈ ℝ → ((1 − ((𝐴↑2) / 2)) + (i · (𝐴 − ((𝐴↑3) / 6)))) ∈ ℂ) |
29 | mulcl 10467 | . . . . . 6 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i · 𝐴) ∈ ℂ) | |
30 | 13, 2, 29 | sylancr 587 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (i · 𝐴) ∈ ℂ) |
31 | 4nn0 11764 | . . . . 5 ⊢ 4 ∈ ℕ0 | |
32 | 3 | eftlcl 15293 | . . . . 5 ⊢ (((i · 𝐴) ∈ ℂ ∧ 4 ∈ ℕ0) → Σ𝑘 ∈ (ℤ≥‘4)(𝐹‘𝑘) ∈ ℂ) |
33 | 30, 31, 32 | sylancl 586 | . . . 4 ⊢ (𝐴 ∈ ℝ → Σ𝑘 ∈ (ℤ≥‘4)(𝐹‘𝑘) ∈ ℂ) |
34 | 28, 33 | readdd 14407 | . . 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 14424 | . . . 4 ⊢ (𝐴 ∈ ℝ → (ℜ‘((1 − ((𝐴↑2) / 2)) + (i · (𝐴 − ((𝐴↑3) / 6))))) = (1 − ((𝐴↑2) / 2))) |
36 | 35 | oveq1d 7031 | . . 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 2835 | . 2 ⊢ (𝐴 ∈ ℝ → (ℜ‘(exp‘(i · 𝐴))) = ((1 − ((𝐴↑2) / 2)) + (ℜ‘Σ𝑘 ∈ (ℤ≥‘4)(𝐹‘𝑘)))) |
38 | 1, 37 | eqtrd 2831 | 1 ⊢ (𝐴 ∈ ℝ → (cos‘𝐴) = ((1 − ((𝐴↑2) / 2)) + (ℜ‘Σ𝑘 ∈ (ℤ≥‘4)(𝐹‘𝑘)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1522 ∈ wcel 2081 ≠ wne 2984 ↦ cmpt 5041 ‘cfv 6225 (class class class)co 7016 ℂcc 10381 ℝcr 10382 0cc0 10383 1c1 10384 ici 10385 + caddc 10386 · cmul 10388 − cmin 10717 / cdiv 11145 2c2 11540 3c3 11541 4c4 11542 6c6 11544 ℕ0cn0 11745 ℤ≥cuz 12093 ↑cexp 13279 !cfa 13483 ℜcre 14290 Σcsu 14876 expce 15248 cosccos 15251 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5081 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-inf2 8950 ax-cnex 10439 ax-resscn 10440 ax-1cn 10441 ax-icn 10442 ax-addcl 10443 ax-addrcl 10444 ax-mulcl 10445 ax-mulrcl 10446 ax-mulcom 10447 ax-addass 10448 ax-mulass 10449 ax-distr 10450 ax-i2m1 10451 ax-1ne0 10452 ax-1rid 10453 ax-rnegex 10454 ax-rrecex 10455 ax-cnre 10456 ax-pre-lttri 10457 ax-pre-lttrn 10458 ax-pre-ltadd 10459 ax-pre-mulgt0 10460 ax-pre-sup 10461 ax-addf 10462 ax-mulf 10463 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-fal 1535 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-pss 3876 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-tp 4477 df-op 4479 df-uni 4746 df-int 4783 df-iun 4827 df-br 4963 df-opab 5025 df-mpt 5042 df-tr 5064 df-id 5348 df-eprel 5353 df-po 5362 df-so 5363 df-fr 5402 df-se 5403 df-we 5404 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-pred 6023 df-ord 6069 df-on 6070 df-lim 6071 df-suc 6072 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-isom 6234 df-riota 6977 df-ov 7019 df-oprab 7020 df-mpo 7021 df-om 7437 df-1st 7545 df-2nd 7546 df-wrecs 7798 df-recs 7860 df-rdg 7898 df-1o 7953 df-oadd 7957 df-er 8139 df-pm 8259 df-en 8358 df-dom 8359 df-sdom 8360 df-fin 8361 df-sup 8752 df-inf 8753 df-oi 8820 df-card 9214 df-pnf 10523 df-mnf 10524 df-xr 10525 df-ltxr 10526 df-le 10527 df-sub 10719 df-neg 10720 df-div 11146 df-nn 11487 df-2 11548 df-3 11549 df-4 11550 df-5 11551 df-6 11552 df-n0 11746 df-z 11830 df-uz 12094 df-rp 12240 df-ico 12594 df-fz 12743 df-fzo 12884 df-fl 13012 df-seq 13220 df-exp 13280 df-fac 13484 df-hash 13541 df-shft 14260 df-cj 14292 df-re 14293 df-im 14294 df-sqrt 14428 df-abs 14429 df-limsup 14662 df-clim 14679 df-rlim 14680 df-sum 14877 df-ef 15254 df-cos 15257 |
This theorem is referenced by: cos01bnd 15372 |
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