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Mirrors > Home > MPE Home > Th. List > sinneg | Structured version Visualization version 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 10722 | . . 3 ⊢ (𝐴 ∈ ℂ → -𝐴 ∈ ℂ) | |
2 | sinval 15296 | . . 3 ⊢ (-𝐴 ∈ ℂ → (sin‘-𝐴) = (((exp‘(i · -𝐴)) − (exp‘(-i · -𝐴))) / (2 · i))) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝐴 ∈ ℂ → (sin‘-𝐴) = (((exp‘(i · -𝐴)) − (exp‘(-i · -𝐴))) / (2 · i))) |
4 | sinval 15296 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (sin‘𝐴) = (((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) / (2 · i))) | |
5 | 4 | negeqd 10716 | . . . 4 ⊢ (𝐴 ∈ ℂ → -(sin‘𝐴) = -(((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) / (2 · i))) |
6 | ax-icn 10431 | . . . . . . . 8 ⊢ i ∈ ℂ | |
7 | mulcl 10456 | . . . . . . . 8 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i · 𝐴) ∈ ℂ) | |
8 | 6, 7 | mpan 686 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (i · 𝐴) ∈ ℂ) |
9 | efcl 15257 | . . . . . . 7 ⊢ ((i · 𝐴) ∈ ℂ → (exp‘(i · 𝐴)) ∈ ℂ) | |
10 | 8, 9 | syl 17 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (exp‘(i · 𝐴)) ∈ ℂ) |
11 | negicn 10723 | . . . . . . . 8 ⊢ -i ∈ ℂ | |
12 | mulcl 10456 | . . . . . . . 8 ⊢ ((-i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (-i · 𝐴) ∈ ℂ) | |
13 | 11, 12 | mpan 686 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (-i · 𝐴) ∈ ℂ) |
14 | efcl 15257 | . . . . . . 7 ⊢ ((-i · 𝐴) ∈ ℂ → (exp‘(-i · 𝐴)) ∈ ℂ) | |
15 | 13, 14 | syl 17 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (exp‘(-i · 𝐴)) ∈ ℂ) |
16 | 10, 15 | subcld 10834 | . . . . 5 ⊢ (𝐴 ∈ ℂ → ((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) ∈ ℂ) |
17 | 2mulicn 11697 | . . . . . 6 ⊢ (2 · i) ∈ ℂ | |
18 | 2muline0 11698 | . . . . . 6 ⊢ (2 · i) ≠ 0 | |
19 | divneg 11169 | . . . . . 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 1443 | . . . . 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 17 | . . . 4 ⊢ (𝐴 ∈ ℂ → -(((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) / (2 · i)) = (-((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) / (2 · i))) |
22 | 5, 21 | eqtrd 2829 | . . 3 ⊢ (𝐴 ∈ ℂ → -(sin‘𝐴) = (-((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) / (2 · i))) |
23 | mulneg12 10915 | . . . . . . . . 9 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (-i · 𝐴) = (i · -𝐴)) | |
24 | 6, 23 | mpan 686 | . . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (-i · 𝐴) = (i · -𝐴)) |
25 | 24 | eqcomd 2799 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (i · -𝐴) = (-i · 𝐴)) |
26 | 25 | fveq2d 6534 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (exp‘(i · -𝐴)) = (exp‘(-i · 𝐴))) |
27 | mul2neg 10916 | . . . . . . . 8 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (-i · -𝐴) = (i · 𝐴)) | |
28 | 6, 27 | mpan 686 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (-i · -𝐴) = (i · 𝐴)) |
29 | 28 | fveq2d 6534 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (exp‘(-i · -𝐴)) = (exp‘(i · 𝐴))) |
30 | 26, 29 | oveq12d 7025 | . . . . 5 ⊢ (𝐴 ∈ ℂ → ((exp‘(i · -𝐴)) − (exp‘(-i · -𝐴))) = ((exp‘(-i · 𝐴)) − (exp‘(i · 𝐴)))) |
31 | 10, 15 | negsubdi2d 10850 | . . . . 5 ⊢ (𝐴 ∈ ℂ → -((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) = ((exp‘(-i · 𝐴)) − (exp‘(i · 𝐴)))) |
32 | 30, 31 | eqtr4d 2832 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((exp‘(i · -𝐴)) − (exp‘(-i · -𝐴))) = -((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴)))) |
33 | 32 | oveq1d 7022 | . . 3 ⊢ (𝐴 ∈ ℂ → (((exp‘(i · -𝐴)) − (exp‘(-i · -𝐴))) / (2 · i)) = (-((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) / (2 · i))) |
34 | 22, 33 | eqtr4d 2832 | . 2 ⊢ (𝐴 ∈ ℂ → -(sin‘𝐴) = (((exp‘(i · -𝐴)) − (exp‘(-i · -𝐴))) / (2 · i))) |
35 | 3, 34 | eqtr4d 2832 | 1 ⊢ (𝐴 ∈ ℂ → (sin‘-𝐴) = -(sin‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1520 ∈ wcel 2079 ≠ wne 2982 ‘cfv 6217 (class class class)co 7007 ℂcc 10370 0cc0 10372 ici 10374 · cmul 10377 − cmin 10706 -cneg 10707 / cdiv 11134 2c2 11529 expce 15236 sincsin 15238 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1775 ax-4 1789 ax-5 1886 ax-6 1945 ax-7 1990 ax-8 2081 ax-9 2089 ax-10 2110 ax-11 2124 ax-12 2139 ax-13 2342 ax-ext 2767 ax-rep 5075 ax-sep 5088 ax-nul 5095 ax-pow 5150 ax-pr 5214 ax-un 7310 ax-inf2 8939 ax-cnex 10428 ax-resscn 10429 ax-1cn 10430 ax-icn 10431 ax-addcl 10432 ax-addrcl 10433 ax-mulcl 10434 ax-mulrcl 10435 ax-mulcom 10436 ax-addass 10437 ax-mulass 10438 ax-distr 10439 ax-i2m1 10440 ax-1ne0 10441 ax-1rid 10442 ax-rnegex 10443 ax-rrecex 10444 ax-cnre 10445 ax-pre-lttri 10446 ax-pre-lttrn 10447 ax-pre-ltadd 10448 ax-pre-mulgt0 10449 ax-pre-sup 10450 ax-addf 10451 ax-mulf 10452 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1079 df-3an 1080 df-tru 1523 df-fal 1533 df-ex 1760 df-nf 1764 df-sb 2041 df-mo 2574 df-eu 2610 df-clab 2774 df-cleq 2786 df-clel 2861 df-nfc 2933 df-ne 2983 df-nel 3089 df-ral 3108 df-rex 3109 df-reu 3110 df-rmo 3111 df-rab 3112 df-v 3434 df-sbc 3702 df-csb 3807 df-dif 3857 df-un 3859 df-in 3861 df-ss 3869 df-pss 3871 df-nul 4207 df-if 4376 df-pw 4449 df-sn 4467 df-pr 4469 df-tp 4471 df-op 4473 df-uni 4740 df-int 4777 df-iun 4821 df-br 4957 df-opab 5019 df-mpt 5036 df-tr 5058 df-id 5340 df-eprel 5345 df-po 5354 df-so 5355 df-fr 5394 df-se 5395 df-we 5396 df-xp 5441 df-rel 5442 df-cnv 5443 df-co 5444 df-dm 5445 df-rn 5446 df-res 5447 df-ima 5448 df-pred 6015 df-ord 6061 df-on 6062 df-lim 6063 df-suc 6064 df-iota 6181 df-fun 6219 df-fn 6220 df-f 6221 df-f1 6222 df-fo 6223 df-f1o 6224 df-fv 6225 df-isom 6226 df-riota 6968 df-ov 7010 df-oprab 7011 df-mpo 7012 df-om 7428 df-1st 7536 df-2nd 7537 df-wrecs 7789 df-recs 7851 df-rdg 7889 df-1o 7944 df-oadd 7948 df-er 8130 df-pm 8250 df-en 8348 df-dom 8349 df-sdom 8350 df-fin 8351 df-sup 8742 df-inf 8743 df-oi 8810 df-card 9203 df-pnf 10512 df-mnf 10513 df-xr 10514 df-ltxr 10515 df-le 10516 df-sub 10708 df-neg 10709 df-div 11135 df-nn 11476 df-2 11537 df-3 11538 df-n0 11735 df-z 11819 df-uz 12083 df-rp 12229 df-ico 12583 df-fz 12732 df-fzo 12873 df-fl 13000 df-seq 13208 df-exp 13268 df-fac 13472 df-hash 13529 df-shft 14248 df-cj 14280 df-re 14281 df-im 14282 df-sqrt 14416 df-abs 14417 df-limsup 14650 df-clim 14667 df-rlim 14668 df-sum 14865 df-ef 15242 df-sin 15244 |
This theorem is referenced by: tanneg 15322 sin0 15323 efmival 15327 sinsub 15342 cossub 15343 sincossq 15350 sin2pim 24742 reasinsin 25143 atantan 25170 sinccvglem 32468 dirkertrigeqlem2 41880 fourierdlem43 41931 fourierdlem44 41932 sqwvfoura 42009 |
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