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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 11401 | . . 3 ⊢ (𝐴 ∈ ℂ → -𝐴 ∈ ℂ) | |
2 | sinval 16004 | . . 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 16004 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (sin‘𝐴) = (((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) / (2 · i))) | |
5 | 4 | negeqd 11395 | . . . 4 ⊢ (𝐴 ∈ ℂ → -(sin‘𝐴) = -(((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) / (2 · i))) |
6 | ax-icn 11110 | . . . . . . . 8 ⊢ i ∈ ℂ | |
7 | mulcl 11135 | . . . . . . . 8 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i · 𝐴) ∈ ℂ) | |
8 | 6, 7 | mpan 688 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (i · 𝐴) ∈ ℂ) |
9 | efcl 15965 | . . . . . . 7 ⊢ ((i · 𝐴) ∈ ℂ → (exp‘(i · 𝐴)) ∈ ℂ) | |
10 | 8, 9 | syl 17 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (exp‘(i · 𝐴)) ∈ ℂ) |
11 | negicn 11402 | . . . . . . . 8 ⊢ -i ∈ ℂ | |
12 | mulcl 11135 | . . . . . . . 8 ⊢ ((-i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (-i · 𝐴) ∈ ℂ) | |
13 | 11, 12 | mpan 688 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (-i · 𝐴) ∈ ℂ) |
14 | efcl 15965 | . . . . . . 7 ⊢ ((-i · 𝐴) ∈ ℂ → (exp‘(-i · 𝐴)) ∈ ℂ) | |
15 | 13, 14 | syl 17 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (exp‘(-i · 𝐴)) ∈ ℂ) |
16 | 10, 15 | subcld 11512 | . . . . 5 ⊢ (𝐴 ∈ ℂ → ((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) ∈ ℂ) |
17 | 2mulicn 12376 | . . . . . 6 ⊢ (2 · i) ∈ ℂ | |
18 | 2muline0 12377 | . . . . . 6 ⊢ (2 · i) ≠ 0 | |
19 | divneg 11847 | . . . . . 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 1453 | . . . . 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 2776 | . . 3 ⊢ (𝐴 ∈ ℂ → -(sin‘𝐴) = (-((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) / (2 · i))) |
23 | mulneg12 11593 | . . . . . . . . 9 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (-i · 𝐴) = (i · -𝐴)) | |
24 | 6, 23 | mpan 688 | . . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (-i · 𝐴) = (i · -𝐴)) |
25 | 24 | eqcomd 2742 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (i · -𝐴) = (-i · 𝐴)) |
26 | 25 | fveq2d 6846 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (exp‘(i · -𝐴)) = (exp‘(-i · 𝐴))) |
27 | mul2neg 11594 | . . . . . . . 8 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (-i · -𝐴) = (i · 𝐴)) | |
28 | 6, 27 | mpan 688 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (-i · -𝐴) = (i · 𝐴)) |
29 | 28 | fveq2d 6846 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (exp‘(-i · -𝐴)) = (exp‘(i · 𝐴))) |
30 | 26, 29 | oveq12d 7375 | . . . . 5 ⊢ (𝐴 ∈ ℂ → ((exp‘(i · -𝐴)) − (exp‘(-i · -𝐴))) = ((exp‘(-i · 𝐴)) − (exp‘(i · 𝐴)))) |
31 | 10, 15 | negsubdi2d 11528 | . . . . 5 ⊢ (𝐴 ∈ ℂ → -((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) = ((exp‘(-i · 𝐴)) − (exp‘(i · 𝐴)))) |
32 | 30, 31 | eqtr4d 2779 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((exp‘(i · -𝐴)) − (exp‘(-i · -𝐴))) = -((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴)))) |
33 | 32 | oveq1d 7372 | . . 3 ⊢ (𝐴 ∈ ℂ → (((exp‘(i · -𝐴)) − (exp‘(-i · -𝐴))) / (2 · i)) = (-((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) / (2 · i))) |
34 | 22, 33 | eqtr4d 2779 | . 2 ⊢ (𝐴 ∈ ℂ → -(sin‘𝐴) = (((exp‘(i · -𝐴)) − (exp‘(-i · -𝐴))) / (2 · i))) |
35 | 3, 34 | eqtr4d 2779 | 1 ⊢ (𝐴 ∈ ℂ → (sin‘-𝐴) = -(sin‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 ‘cfv 6496 (class class class)co 7357 ℂcc 11049 0cc0 11051 ici 11053 · cmul 11056 − cmin 11385 -cneg 11386 / cdiv 11812 2c2 12208 expce 15944 sincsin 15946 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-inf2 9577 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-er 8648 df-pm 8768 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-sup 9378 df-inf 9379 df-oi 9446 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-n0 12414 df-z 12500 df-uz 12764 df-rp 12916 df-ico 13270 df-fz 13425 df-fzo 13568 df-fl 13697 df-seq 13907 df-exp 13968 df-fac 14174 df-hash 14231 df-shft 14952 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-limsup 15353 df-clim 15370 df-rlim 15371 df-sum 15571 df-ef 15950 df-sin 15952 |
This theorem is referenced by: tanneg 16030 sin0 16031 efmival 16035 sinsub 16050 cossub 16051 sincossq 16058 sin2pim 25842 reasinsin 26246 atantan 26273 sinccvglem 34260 dirkertrigeqlem2 44330 fourierdlem43 44381 fourierdlem44 44382 sqwvfoura 44459 |
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