Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > demoivre | GIF version |
Description: De Moivre's Formula. Proof by induction given at http://en.wikipedia.org/wiki/De_Moivre's_formula, but restricted to nonnegative integer powers. See also demoivreALT 11469 for an alternate longer proof not using the exponential function. (Contributed by NM, 24-Jul-2007.) |
Ref | Expression |
---|---|
demoivre | ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ) → (((cos‘𝐴) + (i · (sin‘𝐴)))↑𝑁) = ((cos‘(𝑁 · 𝐴)) + (i · (sin‘(𝑁 · 𝐴))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-icn 7708 | . . . 4 ⊢ i ∈ ℂ | |
2 | mulcl 7740 | . . . 4 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i · 𝐴) ∈ ℂ) | |
3 | 1, 2 | mpan 420 | . . 3 ⊢ (𝐴 ∈ ℂ → (i · 𝐴) ∈ ℂ) |
4 | efexp 11377 | . . 3 ⊢ (((i · 𝐴) ∈ ℂ ∧ 𝑁 ∈ ℤ) → (exp‘(𝑁 · (i · 𝐴))) = ((exp‘(i · 𝐴))↑𝑁)) | |
5 | 3, 4 | sylan 281 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ) → (exp‘(𝑁 · (i · 𝐴))) = ((exp‘(i · 𝐴))↑𝑁)) |
6 | zcn 9052 | . . 3 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ) | |
7 | mul12 7884 | . . . . . . 7 ⊢ ((𝑁 ∈ ℂ ∧ i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (𝑁 · (i · 𝐴)) = (i · (𝑁 · 𝐴))) | |
8 | 1, 7 | mp3an2 1303 | . . . . . 6 ⊢ ((𝑁 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (𝑁 · (i · 𝐴)) = (i · (𝑁 · 𝐴))) |
9 | 8 | fveq2d 5418 | . . . . 5 ⊢ ((𝑁 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (exp‘(𝑁 · (i · 𝐴))) = (exp‘(i · (𝑁 · 𝐴)))) |
10 | mulcl 7740 | . . . . . 6 ⊢ ((𝑁 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (𝑁 · 𝐴) ∈ ℂ) | |
11 | efival 11428 | . . . . . 6 ⊢ ((𝑁 · 𝐴) ∈ ℂ → (exp‘(i · (𝑁 · 𝐴))) = ((cos‘(𝑁 · 𝐴)) + (i · (sin‘(𝑁 · 𝐴))))) | |
12 | 10, 11 | syl 14 | . . . . 5 ⊢ ((𝑁 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (exp‘(i · (𝑁 · 𝐴))) = ((cos‘(𝑁 · 𝐴)) + (i · (sin‘(𝑁 · 𝐴))))) |
13 | 9, 12 | eqtrd 2170 | . . . 4 ⊢ ((𝑁 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (exp‘(𝑁 · (i · 𝐴))) = ((cos‘(𝑁 · 𝐴)) + (i · (sin‘(𝑁 · 𝐴))))) |
14 | 13 | ancoms 266 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℂ) → (exp‘(𝑁 · (i · 𝐴))) = ((cos‘(𝑁 · 𝐴)) + (i · (sin‘(𝑁 · 𝐴))))) |
15 | 6, 14 | sylan2 284 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ) → (exp‘(𝑁 · (i · 𝐴))) = ((cos‘(𝑁 · 𝐴)) + (i · (sin‘(𝑁 · 𝐴))))) |
16 | efival 11428 | . . . 4 ⊢ (𝐴 ∈ ℂ → (exp‘(i · 𝐴)) = ((cos‘𝐴) + (i · (sin‘𝐴)))) | |
17 | 16 | oveq1d 5782 | . . 3 ⊢ (𝐴 ∈ ℂ → ((exp‘(i · 𝐴))↑𝑁) = (((cos‘𝐴) + (i · (sin‘𝐴)))↑𝑁)) |
18 | 17 | adantr 274 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ) → ((exp‘(i · 𝐴))↑𝑁) = (((cos‘𝐴) + (i · (sin‘𝐴)))↑𝑁)) |
19 | 5, 15, 18 | 3eqtr3rd 2179 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ) → (((cos‘𝐴) + (i · (sin‘𝐴)))↑𝑁) = ((cos‘(𝑁 · 𝐴)) + (i · (sin‘(𝑁 · 𝐴))))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1331 ∈ wcel 1480 ‘cfv 5118 (class class class)co 5767 ℂcc 7611 ici 7615 + caddc 7616 · cmul 7618 ℤcz 9047 ↑cexp 10285 expce 11337 sincsin 11339 cosccos 11340 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-coll 4038 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-iinf 4497 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-mulrcl 7712 ax-addcom 7713 ax-mulcom 7714 ax-addass 7715 ax-mulass 7716 ax-distr 7717 ax-i2m1 7718 ax-0lt1 7719 ax-1rid 7720 ax-0id 7721 ax-rnegex 7722 ax-precex 7723 ax-cnre 7724 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 ax-pre-lttrn 7727 ax-pre-apti 7728 ax-pre-ltadd 7729 ax-pre-mulgt0 7730 ax-pre-mulext 7731 ax-arch 7732 ax-caucvg 7733 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-reu 2421 df-rmo 2422 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-if 3470 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-iun 3810 df-disj 3902 df-br 3925 df-opab 3985 df-mpt 3986 df-tr 4022 df-id 4210 df-po 4213 df-iso 4214 df-iord 4283 df-on 4285 df-ilim 4286 df-suc 4288 df-iom 4500 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-isom 5127 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-recs 6195 df-irdg 6260 df-frec 6281 df-1o 6306 df-oadd 6310 df-er 6422 df-en 6628 df-dom 6629 df-fin 6630 df-sup 6864 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-sub 7928 df-neg 7929 df-reap 8330 df-ap 8337 df-div 8426 df-inn 8714 df-2 8772 df-3 8773 df-4 8774 df-n0 8971 df-z 9048 df-uz 9320 df-q 9405 df-rp 9435 df-ico 9670 df-fz 9784 df-fzo 9913 df-seqfrec 10212 df-exp 10286 df-fac 10465 df-bc 10487 df-ihash 10515 df-cj 10607 df-re 10608 df-im 10609 df-rsqrt 10763 df-abs 10764 df-clim 11041 df-sumdc 11116 df-ef 11343 df-sin 11345 df-cos 11346 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |