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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 10449 | . . . 4 ⊢ i ∈ ℂ | |
2 | mulneg12 10932 | . . . 4 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (-i · 𝐴) = (i · -𝐴)) | |
3 | 1, 2 | mpan 686 | . . 3 ⊢ (𝐴 ∈ ℂ → (-i · 𝐴) = (i · -𝐴)) |
4 | 3 | fveq2d 6549 | . 2 ⊢ (𝐴 ∈ ℂ → (exp‘(-i · 𝐴)) = (exp‘(i · -𝐴))) |
5 | negcl 10739 | . . . 4 ⊢ (𝐴 ∈ ℂ → -𝐴 ∈ ℂ) | |
6 | efival 15342 | . . . 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 15337 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (cos‘-𝐴) = (cos‘𝐴)) | |
9 | sinneg 15336 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (sin‘-𝐴) = -(sin‘𝐴)) | |
10 | 9 | oveq2d 7039 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (i · (sin‘-𝐴)) = (i · -(sin‘𝐴))) |
11 | sincl 15316 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (sin‘𝐴) ∈ ℂ) | |
12 | mulneg2 10931 | . . . . . . 7 ⊢ ((i ∈ ℂ ∧ (sin‘𝐴) ∈ ℂ) → (i · -(sin‘𝐴)) = -(i · (sin‘𝐴))) | |
13 | 1, 11, 12 | sylancr 587 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (i · -(sin‘𝐴)) = -(i · (sin‘𝐴))) |
14 | 10, 13 | eqtrd 2833 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (i · (sin‘-𝐴)) = -(i · (sin‘𝐴))) |
15 | 8, 14 | oveq12d 7041 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((cos‘-𝐴) + (i · (sin‘-𝐴))) = ((cos‘𝐴) + -(i · (sin‘𝐴)))) |
16 | coscl 15317 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (cos‘𝐴) ∈ ℂ) | |
17 | mulcl 10474 | . . . . . 6 ⊢ ((i ∈ ℂ ∧ (sin‘𝐴) ∈ ℂ) → (i · (sin‘𝐴)) ∈ ℂ) | |
18 | 1, 11, 17 | sylancr 587 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (i · (sin‘𝐴)) ∈ ℂ) |
19 | 16, 18 | negsubd 10857 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((cos‘𝐴) + -(i · (sin‘𝐴))) = ((cos‘𝐴) − (i · (sin‘𝐴)))) |
20 | 15, 19 | eqtrd 2833 | . . 3 ⊢ (𝐴 ∈ ℂ → ((cos‘-𝐴) + (i · (sin‘-𝐴))) = ((cos‘𝐴) − (i · (sin‘𝐴)))) |
21 | 7, 20 | eqtrd 2833 | . 2 ⊢ (𝐴 ∈ ℂ → (exp‘(i · -𝐴)) = ((cos‘𝐴) − (i · (sin‘𝐴)))) |
22 | 4, 21 | eqtrd 2833 | 1 ⊢ (𝐴 ∈ ℂ → (exp‘(-i · 𝐴)) = ((cos‘𝐴) − (i · (sin‘𝐴)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1525 ∈ wcel 2083 ‘cfv 6232 (class class class)co 7023 ℂcc 10388 ici 10392 + caddc 10393 · cmul 10395 − cmin 10723 -cneg 10724 expce 15252 sincsin 15254 cosccos 15255 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-rep 5088 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 ax-un 7326 ax-inf2 8957 ax-cnex 10446 ax-resscn 10447 ax-1cn 10448 ax-icn 10449 ax-addcl 10450 ax-addrcl 10451 ax-mulcl 10452 ax-mulrcl 10453 ax-mulcom 10454 ax-addass 10455 ax-mulass 10456 ax-distr 10457 ax-i2m1 10458 ax-1ne0 10459 ax-1rid 10460 ax-rnegex 10461 ax-rrecex 10462 ax-cnre 10463 ax-pre-lttri 10464 ax-pre-lttrn 10465 ax-pre-ltadd 10466 ax-pre-mulgt0 10467 ax-pre-sup 10468 ax-addf 10469 ax-mulf 10470 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1528 df-fal 1538 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-nel 3093 df-ral 3112 df-rex 3113 df-reu 3114 df-rmo 3115 df-rab 3116 df-v 3442 df-sbc 3712 df-csb 3818 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-pss 3882 df-nul 4218 df-if 4388 df-pw 4461 df-sn 4479 df-pr 4481 df-tp 4483 df-op 4485 df-uni 4752 df-int 4789 df-iun 4833 df-br 4969 df-opab 5031 df-mpt 5048 df-tr 5071 df-id 5355 df-eprel 5360 df-po 5369 df-so 5370 df-fr 5409 df-se 5410 df-we 5411 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-pred 6030 df-ord 6076 df-on 6077 df-lim 6078 df-suc 6079 df-iota 6196 df-fun 6234 df-fn 6235 df-f 6236 df-f1 6237 df-fo 6238 df-f1o 6239 df-fv 6240 df-isom 6241 df-riota 6984 df-ov 7026 df-oprab 7027 df-mpo 7028 df-om 7444 df-1st 7552 df-2nd 7553 df-wrecs 7805 df-recs 7867 df-rdg 7905 df-1o 7960 df-oadd 7964 df-er 8146 df-pm 8266 df-en 8365 df-dom 8366 df-sdom 8367 df-fin 8368 df-sup 8759 df-inf 8760 df-oi 8827 df-card 9221 df-pnf 10530 df-mnf 10531 df-xr 10532 df-ltxr 10533 df-le 10534 df-sub 10725 df-neg 10726 df-div 11152 df-nn 11493 df-2 11554 df-3 11555 df-n0 11752 df-z 11836 df-uz 12098 df-rp 12244 df-ico 12598 df-fz 12747 df-fzo 12888 df-fl 13016 df-seq 13224 df-exp 13284 df-fac 13488 df-hash 13545 df-shft 14264 df-cj 14296 df-re 14297 df-im 14298 df-sqrt 14432 df-abs 14433 df-limsup 14666 df-clim 14683 df-rlim 14684 df-sum 14881 df-ef 15258 df-sin 15260 df-cos 15261 |
This theorem is referenced by: sinadd 15354 cosadd 15355 |
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