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Mirrors > Home > ILE Home > Th. List > sinneg | GIF version |
Description: The sine of a negative is the negative of the sine. (Contributed by NM, 30-Apr-2005.) |
Ref | Expression |
---|---|
sinneg | ⊢ (𝐴 ∈ ℂ → (sin‘-𝐴) = -(sin‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | negcl 8171 | . . 3 ⊢ (𝐴 ∈ ℂ → -𝐴 ∈ ℂ) | |
2 | sinval 11724 | . . 3 ⊢ (-𝐴 ∈ ℂ → (sin‘-𝐴) = (((exp‘(i · -𝐴)) − (exp‘(-i · -𝐴))) / (2 · i))) | |
3 | 1, 2 | syl 14 | . 2 ⊢ (𝐴 ∈ ℂ → (sin‘-𝐴) = (((exp‘(i · -𝐴)) − (exp‘(-i · -𝐴))) / (2 · i))) |
4 | sinval 11724 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (sin‘𝐴) = (((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) / (2 · i))) | |
5 | 4 | negeqd 8166 | . . . 4 ⊢ (𝐴 ∈ ℂ → -(sin‘𝐴) = -(((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) / (2 · i))) |
6 | ax-icn 7920 | . . . . . . . 8 ⊢ i ∈ ℂ | |
7 | mulcl 7952 | . . . . . . . 8 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i · 𝐴) ∈ ℂ) | |
8 | 6, 7 | mpan 424 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (i · 𝐴) ∈ ℂ) |
9 | efcl 11686 | . . . . . . 7 ⊢ ((i · 𝐴) ∈ ℂ → (exp‘(i · 𝐴)) ∈ ℂ) | |
10 | 8, 9 | syl 14 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (exp‘(i · 𝐴)) ∈ ℂ) |
11 | negicn 8172 | . . . . . . . 8 ⊢ -i ∈ ℂ | |
12 | mulcl 7952 | . . . . . . . 8 ⊢ ((-i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (-i · 𝐴) ∈ ℂ) | |
13 | 11, 12 | mpan 424 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (-i · 𝐴) ∈ ℂ) |
14 | efcl 11686 | . . . . . . 7 ⊢ ((-i · 𝐴) ∈ ℂ → (exp‘(-i · 𝐴)) ∈ ℂ) | |
15 | 13, 14 | syl 14 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (exp‘(-i · 𝐴)) ∈ ℂ) |
16 | 10, 15 | subcld 8282 | . . . . 5 ⊢ (𝐴 ∈ ℂ → ((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) ∈ ℂ) |
17 | 2mulicn 9155 | . . . . . 6 ⊢ (2 · i) ∈ ℂ | |
18 | 2muliap0 9157 | . . . . . 6 ⊢ (2 · i) # 0 | |
19 | divnegap 8677 | . . . . . 6 ⊢ ((((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) ∈ ℂ ∧ (2 · i) ∈ ℂ ∧ (2 · i) # 0) → -(((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) / (2 · i)) = (-((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) / (2 · i))) | |
20 | 17, 18, 19 | mp3an23 1339 | . . . . 5 ⊢ (((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) ∈ ℂ → -(((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) / (2 · i)) = (-((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) / (2 · i))) |
21 | 16, 20 | syl 14 | . . . 4 ⊢ (𝐴 ∈ ℂ → -(((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) / (2 · i)) = (-((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) / (2 · i))) |
22 | 5, 21 | eqtrd 2220 | . . 3 ⊢ (𝐴 ∈ ℂ → -(sin‘𝐴) = (-((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) / (2 · i))) |
23 | mulneg12 8368 | . . . . . . . . 9 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (-i · 𝐴) = (i · -𝐴)) | |
24 | 6, 23 | mpan 424 | . . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (-i · 𝐴) = (i · -𝐴)) |
25 | 24 | eqcomd 2193 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (i · -𝐴) = (-i · 𝐴)) |
26 | 25 | fveq2d 5531 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (exp‘(i · -𝐴)) = (exp‘(-i · 𝐴))) |
27 | mul2neg 8369 | . . . . . . . 8 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (-i · -𝐴) = (i · 𝐴)) | |
28 | 6, 27 | mpan 424 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (-i · -𝐴) = (i · 𝐴)) |
29 | 28 | fveq2d 5531 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (exp‘(-i · -𝐴)) = (exp‘(i · 𝐴))) |
30 | 26, 29 | oveq12d 5906 | . . . . 5 ⊢ (𝐴 ∈ ℂ → ((exp‘(i · -𝐴)) − (exp‘(-i · -𝐴))) = ((exp‘(-i · 𝐴)) − (exp‘(i · 𝐴)))) |
31 | 10, 15 | negsubdi2d 8298 | . . . . 5 ⊢ (𝐴 ∈ ℂ → -((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) = ((exp‘(-i · 𝐴)) − (exp‘(i · 𝐴)))) |
32 | 30, 31 | eqtr4d 2223 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((exp‘(i · -𝐴)) − (exp‘(-i · -𝐴))) = -((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴)))) |
33 | 32 | oveq1d 5903 | . . 3 ⊢ (𝐴 ∈ ℂ → (((exp‘(i · -𝐴)) − (exp‘(-i · -𝐴))) / (2 · i)) = (-((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) / (2 · i))) |
34 | 22, 33 | eqtr4d 2223 | . 2 ⊢ (𝐴 ∈ ℂ → -(sin‘𝐴) = (((exp‘(i · -𝐴)) − (exp‘(-i · -𝐴))) / (2 · i))) |
35 | 3, 34 | eqtr4d 2223 | 1 ⊢ (𝐴 ∈ ℂ → (sin‘-𝐴) = -(sin‘𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1363 ∈ wcel 2158 class class class wbr 4015 ‘cfv 5228 (class class class)co 5888 ℂcc 7823 0cc0 7825 ici 7827 · cmul 7830 − cmin 8142 -cneg 8143 # cap 8552 / cdiv 8643 2c2 8984 expce 11664 sincsin 11666 |
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 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-coll 4130 ax-sep 4133 ax-nul 4141 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-iinf 4599 ax-cnex 7916 ax-resscn 7917 ax-1cn 7918 ax-1re 7919 ax-icn 7920 ax-addcl 7921 ax-addrcl 7922 ax-mulcl 7923 ax-mulrcl 7924 ax-addcom 7925 ax-mulcom 7926 ax-addass 7927 ax-mulass 7928 ax-distr 7929 ax-i2m1 7930 ax-0lt1 7931 ax-1rid 7932 ax-0id 7933 ax-rnegex 7934 ax-precex 7935 ax-cnre 7936 ax-pre-ltirr 7937 ax-pre-ltwlin 7938 ax-pre-lttrn 7939 ax-pre-apti 7940 ax-pre-ltadd 7941 ax-pre-mulgt0 7942 ax-pre-mulext 7943 ax-arch 7944 ax-caucvg 7945 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 980 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-nel 2453 df-ral 2470 df-rex 2471 df-reu 2472 df-rmo 2473 df-rab 2474 df-v 2751 df-sbc 2975 df-csb 3070 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-nul 3435 df-if 3547 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-int 3857 df-iun 3900 df-br 4016 df-opab 4077 df-mpt 4078 df-tr 4114 df-id 4305 df-po 4308 df-iso 4309 df-iord 4378 df-on 4380 df-ilim 4381 df-suc 4383 df-iom 4602 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-f1 5233 df-fo 5234 df-f1o 5235 df-fv 5236 df-isom 5237 df-riota 5844 df-ov 5891 df-oprab 5892 df-mpo 5893 df-1st 6155 df-2nd 6156 df-recs 6320 df-irdg 6385 df-frec 6406 df-1o 6431 df-oadd 6435 df-er 6549 df-en 6755 df-dom 6756 df-fin 6757 df-pnf 8008 df-mnf 8009 df-xr 8010 df-ltxr 8011 df-le 8012 df-sub 8144 df-neg 8145 df-reap 8546 df-ap 8553 df-div 8644 df-inn 8934 df-2 8992 df-3 8993 df-4 8994 df-n0 9191 df-z 9268 df-uz 9543 df-q 9634 df-rp 9668 df-ico 9908 df-fz 10023 df-fzo 10157 df-seqfrec 10460 df-exp 10534 df-fac 10720 df-ihash 10770 df-cj 10865 df-re 10866 df-im 10867 df-rsqrt 11021 df-abs 11022 df-clim 11301 df-sumdc 11376 df-ef 11670 df-sin 11672 |
This theorem is referenced by: tannegap 11750 sin0 11751 efmival 11755 sinsub 11762 cossub 11763 sincossq 11770 sin2pim 14530 |
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