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Mirrors > Home > MPE Home > Th. List > efmival | Structured version Visualization version GIF version |
Description: The exponential function in terms of sine and cosine. (Contributed by NM, 14-Jan-2006.) |
Ref | Expression |
---|---|
efmival | ⊢ (𝐴 ∈ ℂ → (exp‘(-i · 𝐴)) = ((cos‘𝐴) − (i · (sin‘𝐴)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-icn 9995 | . . . 4 ⊢ i ∈ ℂ | |
2 | mulneg12 10468 | . . . 4 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (-i · 𝐴) = (i · -𝐴)) | |
3 | 1, 2 | mpan 706 | . . 3 ⊢ (𝐴 ∈ ℂ → (-i · 𝐴) = (i · -𝐴)) |
4 | 3 | fveq2d 6195 | . 2 ⊢ (𝐴 ∈ ℂ → (exp‘(-i · 𝐴)) = (exp‘(i · -𝐴))) |
5 | negcl 10281 | . . . 4 ⊢ (𝐴 ∈ ℂ → -𝐴 ∈ ℂ) | |
6 | efival 14882 | . . . 4 ⊢ (-𝐴 ∈ ℂ → (exp‘(i · -𝐴)) = ((cos‘-𝐴) + (i · (sin‘-𝐴)))) | |
7 | 5, 6 | syl 17 | . . 3 ⊢ (𝐴 ∈ ℂ → (exp‘(i · -𝐴)) = ((cos‘-𝐴) + (i · (sin‘-𝐴)))) |
8 | cosneg 14877 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (cos‘-𝐴) = (cos‘𝐴)) | |
9 | sinneg 14876 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (sin‘-𝐴) = -(sin‘𝐴)) | |
10 | 9 | oveq2d 6666 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (i · (sin‘-𝐴)) = (i · -(sin‘𝐴))) |
11 | sincl 14856 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (sin‘𝐴) ∈ ℂ) | |
12 | mulneg2 10467 | . . . . . . 7 ⊢ ((i ∈ ℂ ∧ (sin‘𝐴) ∈ ℂ) → (i · -(sin‘𝐴)) = -(i · (sin‘𝐴))) | |
13 | 1, 11, 12 | sylancr 695 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (i · -(sin‘𝐴)) = -(i · (sin‘𝐴))) |
14 | 10, 13 | eqtrd 2656 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (i · (sin‘-𝐴)) = -(i · (sin‘𝐴))) |
15 | 8, 14 | oveq12d 6668 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((cos‘-𝐴) + (i · (sin‘-𝐴))) = ((cos‘𝐴) + -(i · (sin‘𝐴)))) |
16 | coscl 14857 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (cos‘𝐴) ∈ ℂ) | |
17 | mulcl 10020 | . . . . . 6 ⊢ ((i ∈ ℂ ∧ (sin‘𝐴) ∈ ℂ) → (i · (sin‘𝐴)) ∈ ℂ) | |
18 | 1, 11, 17 | sylancr 695 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (i · (sin‘𝐴)) ∈ ℂ) |
19 | 16, 18 | negsubd 10398 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((cos‘𝐴) + -(i · (sin‘𝐴))) = ((cos‘𝐴) − (i · (sin‘𝐴)))) |
20 | 15, 19 | eqtrd 2656 | . . 3 ⊢ (𝐴 ∈ ℂ → ((cos‘-𝐴) + (i · (sin‘-𝐴))) = ((cos‘𝐴) − (i · (sin‘𝐴)))) |
21 | 7, 20 | eqtrd 2656 | . 2 ⊢ (𝐴 ∈ ℂ → (exp‘(i · -𝐴)) = ((cos‘𝐴) − (i · (sin‘𝐴)))) |
22 | 4, 21 | eqtrd 2656 | 1 ⊢ (𝐴 ∈ ℂ → (exp‘(-i · 𝐴)) = ((cos‘𝐴) − (i · (sin‘𝐴)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 ici 9938 + caddc 9939 · cmul 9941 − cmin 10266 -cneg 10267 expce 14792 sincsin 14794 cosccos 14795 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-ico 12181 df-fz 12327 df-fzo 12466 df-fl 12593 df-seq 12802 df-exp 12861 df-fac 13061 df-hash 13118 df-shft 13807 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-limsup 14202 df-clim 14219 df-rlim 14220 df-sum 14417 df-ef 14798 df-sin 14800 df-cos 14801 |
This theorem is referenced by: sinadd 14894 cosadd 14895 |
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