![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > resin4p | Structured version Visualization version GIF version |
Description: Separate out the first four terms of the infinite series expansion of the sine 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 |
---|---|
resin4p | ⊢ (𝐴 ∈ ℝ → (sin‘𝐴) = ((𝐴 − ((𝐴↑3) / 6)) + (ℑ‘Σ𝑘 ∈ (ℤ≥‘4)(𝐹‘𝑘)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resinval 16168 | . 2 ⊢ (𝐴 ∈ ℝ → (sin‘𝐴) = (ℑ‘(exp‘(i · 𝐴)))) | |
2 | recn 11243 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
3 | efi4p.1 | . . . . . 6 ⊢ 𝐹 = (𝑛 ∈ ℕ0 ↦ (((i · 𝐴)↑𝑛) / (!‘𝑛))) | |
4 | 3 | efi4p 16170 | . . . . 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 6911 | . . 3 ⊢ (𝐴 ∈ ℝ → (ℑ‘(exp‘(i · 𝐴))) = (ℑ‘(((1 − ((𝐴↑2) / 2)) + (i · (𝐴 − ((𝐴↑3) / 6)))) + Σ𝑘 ∈ (ℤ≥‘4)(𝐹‘𝑘)))) |
7 | 1re 11259 | . . . . . . 7 ⊢ 1 ∈ ℝ | |
8 | resqcl 14161 | . . . . . . . 8 ⊢ (𝐴 ∈ ℝ → (𝐴↑2) ∈ ℝ) | |
9 | 8 | rehalfcld 12511 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ → ((𝐴↑2) / 2) ∈ ℝ) |
10 | resubcl 11571 | . . . . . . 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 11287 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (1 − ((𝐴↑2) / 2)) ∈ ℂ) |
13 | ax-icn 11212 | . . . . . 6 ⊢ i ∈ ℂ | |
14 | 3nn0 12542 | . . . . . . . . . 10 ⊢ 3 ∈ ℕ0 | |
15 | reexpcl 14116 | . . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 3 ∈ ℕ0) → (𝐴↑3) ∈ ℝ) | |
16 | 14, 15 | mpan2 691 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → (𝐴↑3) ∈ ℝ) |
17 | 6re 12354 | . . . . . . . . . 10 ⊢ 6 ∈ ℝ | |
18 | 6pos 12374 | . . . . . . . . . . 11 ⊢ 0 < 6 | |
19 | 17, 18 | gt0ne0ii 11797 | . . . . . . . . . 10 ⊢ 6 ≠ 0 |
20 | redivcl 11984 | . . . . . . . . . 10 ⊢ (((𝐴↑3) ∈ ℝ ∧ 6 ∈ ℝ ∧ 6 ≠ 0) → ((𝐴↑3) / 6) ∈ ℝ) | |
21 | 17, 19, 20 | mp3an23 1452 | . . . . . . . . 9 ⊢ ((𝐴↑3) ∈ ℝ → ((𝐴↑3) / 6) ∈ ℝ) |
22 | 16, 21 | syl 17 | . . . . . . . 8 ⊢ (𝐴 ∈ ℝ → ((𝐴↑3) / 6) ∈ ℝ) |
23 | resubcl 11571 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ ((𝐴↑3) / 6) ∈ ℝ) → (𝐴 − ((𝐴↑3) / 6)) ∈ ℝ) | |
24 | 22, 23 | mpdan 687 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ → (𝐴 − ((𝐴↑3) / 6)) ∈ ℝ) |
25 | 24 | recnd 11287 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → (𝐴 − ((𝐴↑3) / 6)) ∈ ℂ) |
26 | mulcl 11237 | . . . . . 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 11278 | . . . 4 ⊢ (𝐴 ∈ ℝ → ((1 − ((𝐴↑2) / 2)) + (i · (𝐴 − ((𝐴↑3) / 6)))) ∈ ℂ) |
29 | mulcl 11237 | . . . . . 6 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i · 𝐴) ∈ ℂ) | |
30 | 13, 2, 29 | sylancr 587 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (i · 𝐴) ∈ ℂ) |
31 | 4nn0 12543 | . . . . 5 ⊢ 4 ∈ ℕ0 | |
32 | 3 | eftlcl 16140 | . . . . 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 | imaddd 15251 | . . 3 ⊢ (𝐴 ∈ ℝ → (ℑ‘(((1 − ((𝐴↑2) / 2)) + (i · (𝐴 − ((𝐴↑3) / 6)))) + Σ𝑘 ∈ (ℤ≥‘4)(𝐹‘𝑘))) = ((ℑ‘((1 − ((𝐴↑2) / 2)) + (i · (𝐴 − ((𝐴↑3) / 6))))) + (ℑ‘Σ𝑘 ∈ (ℤ≥‘4)(𝐹‘𝑘)))) |
35 | 11, 24 | crimd 15268 | . . . 4 ⊢ (𝐴 ∈ ℝ → (ℑ‘((1 − ((𝐴↑2) / 2)) + (i · (𝐴 − ((𝐴↑3) / 6))))) = (𝐴 − ((𝐴↑3) / 6))) |
36 | 35 | oveq1d 7446 | . . 3 ⊢ (𝐴 ∈ ℝ → ((ℑ‘((1 − ((𝐴↑2) / 2)) + (i · (𝐴 − ((𝐴↑3) / 6))))) + (ℑ‘Σ𝑘 ∈ (ℤ≥‘4)(𝐹‘𝑘))) = ((𝐴 − ((𝐴↑3) / 6)) + (ℑ‘Σ𝑘 ∈ (ℤ≥‘4)(𝐹‘𝑘)))) |
37 | 6, 34, 36 | 3eqtrd 2779 | . 2 ⊢ (𝐴 ∈ ℝ → (ℑ‘(exp‘(i · 𝐴))) = ((𝐴 − ((𝐴↑3) / 6)) + (ℑ‘Σ𝑘 ∈ (ℤ≥‘4)(𝐹‘𝑘)))) |
38 | 1, 37 | eqtrd 2775 | 1 ⊢ (𝐴 ∈ ℝ → (sin‘𝐴) = ((𝐴 − ((𝐴↑3) / 6)) + (ℑ‘Σ𝑘 ∈ (ℤ≥‘4)(𝐹‘𝑘)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ↦ cmpt 5231 ‘cfv 6563 (class class class)co 7431 ℂcc 11151 ℝcr 11152 0cc0 11153 1c1 11154 ici 11155 + caddc 11156 · cmul 11158 − cmin 11490 / cdiv 11918 2c2 12319 3c3 12320 4c4 12321 6c6 12323 ℕ0cn0 12524 ℤ≥cuz 12876 ↑cexp 14099 !cfa 14309 ℑcim 15134 Σcsu 15719 expce 16094 sincsin 16096 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-pm 8868 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-inf 9481 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-n0 12525 df-z 12612 df-uz 12877 df-rp 13033 df-ico 13390 df-fz 13545 df-fzo 13692 df-fl 13829 df-seq 14040 df-exp 14100 df-fac 14310 df-hash 14367 df-shft 15103 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-limsup 15504 df-clim 15521 df-rlim 15522 df-sum 15720 df-ef 16100 df-sin 16102 |
This theorem is referenced by: sin01bnd 16218 |
Copyright terms: Public domain | W3C validator |