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Mirrors > Home > ILE Home > Th. List > sinval | GIF version |
Description: Value of the sine function. (Contributed by NM, 14-Mar-2005.) (Revised by Mario Carneiro, 10-Nov-2013.) |
Ref | Expression |
---|---|
sinval | ⊢ (𝐴 ∈ ℂ → (sin‘𝐴) = (((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) / (2 · i))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-icn 7924 | . . . . . . 7 ⊢ i ∈ ℂ | |
2 | 1 | a1i 9 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → i ∈ ℂ) |
3 | id 19 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → 𝐴 ∈ ℂ) | |
4 | 2, 3 | mulcld 7996 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (i · 𝐴) ∈ ℂ) |
5 | efcl 11690 | . . . . 5 ⊢ ((i · 𝐴) ∈ ℂ → (exp‘(i · 𝐴)) ∈ ℂ) | |
6 | 4, 5 | syl 14 | . . . 4 ⊢ (𝐴 ∈ ℂ → (exp‘(i · 𝐴)) ∈ ℂ) |
7 | negicn 8176 | . . . . . . 7 ⊢ -i ∈ ℂ | |
8 | 7 | a1i 9 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → -i ∈ ℂ) |
9 | 8, 3 | mulcld 7996 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (-i · 𝐴) ∈ ℂ) |
10 | efcl 11690 | . . . . 5 ⊢ ((-i · 𝐴) ∈ ℂ → (exp‘(-i · 𝐴)) ∈ ℂ) | |
11 | 9, 10 | syl 14 | . . . 4 ⊢ (𝐴 ∈ ℂ → (exp‘(-i · 𝐴)) ∈ ℂ) |
12 | 6, 11 | subcld 8286 | . . 3 ⊢ (𝐴 ∈ ℂ → ((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) ∈ ℂ) |
13 | 2mulicn 9159 | . . . 4 ⊢ (2 · i) ∈ ℂ | |
14 | 13 | a1i 9 | . . 3 ⊢ (𝐴 ∈ ℂ → (2 · i) ∈ ℂ) |
15 | 2muliap0 9161 | . . . 4 ⊢ (2 · i) # 0 | |
16 | 15 | a1i 9 | . . 3 ⊢ (𝐴 ∈ ℂ → (2 · i) # 0) |
17 | 12, 14, 16 | divclapd 8765 | . 2 ⊢ (𝐴 ∈ ℂ → (((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) / (2 · i)) ∈ ℂ) |
18 | oveq2 5899 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (i · 𝑥) = (i · 𝐴)) | |
19 | 18 | fveq2d 5534 | . . . . 5 ⊢ (𝑥 = 𝐴 → (exp‘(i · 𝑥)) = (exp‘(i · 𝐴))) |
20 | oveq2 5899 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (-i · 𝑥) = (-i · 𝐴)) | |
21 | 20 | fveq2d 5534 | . . . . 5 ⊢ (𝑥 = 𝐴 → (exp‘(-i · 𝑥)) = (exp‘(-i · 𝐴))) |
22 | 19, 21 | oveq12d 5909 | . . . 4 ⊢ (𝑥 = 𝐴 → ((exp‘(i · 𝑥)) − (exp‘(-i · 𝑥))) = ((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴)))) |
23 | 22 | oveq1d 5906 | . . 3 ⊢ (𝑥 = 𝐴 → (((exp‘(i · 𝑥)) − (exp‘(-i · 𝑥))) / (2 · i)) = (((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) / (2 · i))) |
24 | df-sin 11676 | . . 3 ⊢ sin = (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) − (exp‘(-i · 𝑥))) / (2 · i))) | |
25 | 23, 24 | fvmptg 5608 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ (((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) / (2 · i)) ∈ ℂ) → (sin‘𝐴) = (((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) / (2 · i))) |
26 | 17, 25 | mpdan 421 | 1 ⊢ (𝐴 ∈ ℂ → (sin‘𝐴) = (((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) / (2 · i))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1364 ∈ wcel 2160 class class class wbr 4018 ‘cfv 5231 (class class class)co 5891 ℂcc 7827 0cc0 7829 ici 7831 · cmul 7834 − cmin 8146 -cneg 8147 # cap 8556 / cdiv 8647 2c2 8988 expce 11668 sincsin 11670 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-nul 4144 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-iinf 4602 ax-cnex 7920 ax-resscn 7921 ax-1cn 7922 ax-1re 7923 ax-icn 7924 ax-addcl 7925 ax-addrcl 7926 ax-mulcl 7927 ax-mulrcl 7928 ax-addcom 7929 ax-mulcom 7930 ax-addass 7931 ax-mulass 7932 ax-distr 7933 ax-i2m1 7934 ax-0lt1 7935 ax-1rid 7936 ax-0id 7937 ax-rnegex 7938 ax-precex 7939 ax-cnre 7940 ax-pre-ltirr 7941 ax-pre-ltwlin 7942 ax-pre-lttrn 7943 ax-pre-apti 7944 ax-pre-ltadd 7945 ax-pre-mulgt0 7946 ax-pre-mulext 7947 ax-arch 7948 ax-caucvg 7949 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-if 3550 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-tr 4117 df-id 4308 df-po 4311 df-iso 4312 df-iord 4381 df-on 4383 df-ilim 4384 df-suc 4386 df-iom 4605 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-ima 4654 df-iota 5193 df-fun 5233 df-fn 5234 df-f 5235 df-f1 5236 df-fo 5237 df-f1o 5238 df-fv 5239 df-isom 5240 df-riota 5847 df-ov 5894 df-oprab 5895 df-mpo 5896 df-1st 6159 df-2nd 6160 df-recs 6324 df-irdg 6389 df-frec 6410 df-1o 6435 df-oadd 6439 df-er 6553 df-en 6759 df-dom 6760 df-fin 6761 df-pnf 8012 df-mnf 8013 df-xr 8014 df-ltxr 8015 df-le 8016 df-sub 8148 df-neg 8149 df-reap 8550 df-ap 8557 df-div 8648 df-inn 8938 df-2 8996 df-3 8997 df-4 8998 df-n0 9195 df-z 9272 df-uz 9547 df-q 9638 df-rp 9672 df-ico 9912 df-fz 10027 df-fzo 10161 df-seqfrec 10464 df-exp 10538 df-fac 10724 df-ihash 10774 df-cj 10869 df-re 10870 df-im 10871 df-rsqrt 11025 df-abs 11026 df-clim 11305 df-sumdc 11380 df-ef 11674 df-sin 11676 |
This theorem is referenced by: tanval2ap 11739 resinval 11741 sinneg 11752 efival 11758 sinadd 11762 sinper 14627 |
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