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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 10888 | . . 3 ⊢ (𝐴 ∈ ℂ → -𝐴 ∈ ℂ) | |
2 | sinval 15477 | . . 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 15477 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (sin‘𝐴) = (((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) / (2 · i))) | |
5 | 4 | negeqd 10882 | . . . 4 ⊢ (𝐴 ∈ ℂ → -(sin‘𝐴) = -(((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) / (2 · i))) |
6 | ax-icn 10598 | . . . . . . . 8 ⊢ i ∈ ℂ | |
7 | mulcl 10623 | . . . . . . . 8 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i · 𝐴) ∈ ℂ) | |
8 | 6, 7 | mpan 688 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (i · 𝐴) ∈ ℂ) |
9 | efcl 15438 | . . . . . . 7 ⊢ ((i · 𝐴) ∈ ℂ → (exp‘(i · 𝐴)) ∈ ℂ) | |
10 | 8, 9 | syl 17 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (exp‘(i · 𝐴)) ∈ ℂ) |
11 | negicn 10889 | . . . . . . . 8 ⊢ -i ∈ ℂ | |
12 | mulcl 10623 | . . . . . . . 8 ⊢ ((-i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (-i · 𝐴) ∈ ℂ) | |
13 | 11, 12 | mpan 688 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (-i · 𝐴) ∈ ℂ) |
14 | efcl 15438 | . . . . . . 7 ⊢ ((-i · 𝐴) ∈ ℂ → (exp‘(-i · 𝐴)) ∈ ℂ) | |
15 | 13, 14 | syl 17 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (exp‘(-i · 𝐴)) ∈ ℂ) |
16 | 10, 15 | subcld 10999 | . . . . 5 ⊢ (𝐴 ∈ ℂ → ((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) ∈ ℂ) |
17 | 2mulicn 11863 | . . . . . 6 ⊢ (2 · i) ∈ ℂ | |
18 | 2muline0 11864 | . . . . . 6 ⊢ (2 · i) ≠ 0 | |
19 | divneg 11334 | . . . . . 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 1449 | . . . . 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 2858 | . . 3 ⊢ (𝐴 ∈ ℂ → -(sin‘𝐴) = (-((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) / (2 · i))) |
23 | mulneg12 11080 | . . . . . . . . 9 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (-i · 𝐴) = (i · -𝐴)) | |
24 | 6, 23 | mpan 688 | . . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (-i · 𝐴) = (i · -𝐴)) |
25 | 24 | eqcomd 2829 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (i · -𝐴) = (-i · 𝐴)) |
26 | 25 | fveq2d 6676 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (exp‘(i · -𝐴)) = (exp‘(-i · 𝐴))) |
27 | mul2neg 11081 | . . . . . . . 8 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (-i · -𝐴) = (i · 𝐴)) | |
28 | 6, 27 | mpan 688 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (-i · -𝐴) = (i · 𝐴)) |
29 | 28 | fveq2d 6676 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (exp‘(-i · -𝐴)) = (exp‘(i · 𝐴))) |
30 | 26, 29 | oveq12d 7176 | . . . . 5 ⊢ (𝐴 ∈ ℂ → ((exp‘(i · -𝐴)) − (exp‘(-i · -𝐴))) = ((exp‘(-i · 𝐴)) − (exp‘(i · 𝐴)))) |
31 | 10, 15 | negsubdi2d 11015 | . . . . 5 ⊢ (𝐴 ∈ ℂ → -((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) = ((exp‘(-i · 𝐴)) − (exp‘(i · 𝐴)))) |
32 | 30, 31 | eqtr4d 2861 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((exp‘(i · -𝐴)) − (exp‘(-i · -𝐴))) = -((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴)))) |
33 | 32 | oveq1d 7173 | . . 3 ⊢ (𝐴 ∈ ℂ → (((exp‘(i · -𝐴)) − (exp‘(-i · -𝐴))) / (2 · i)) = (-((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) / (2 · i))) |
34 | 22, 33 | eqtr4d 2861 | . 2 ⊢ (𝐴 ∈ ℂ → -(sin‘𝐴) = (((exp‘(i · -𝐴)) − (exp‘(-i · -𝐴))) / (2 · i))) |
35 | 3, 34 | eqtr4d 2861 | 1 ⊢ (𝐴 ∈ ℂ → (sin‘-𝐴) = -(sin‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ‘cfv 6357 (class class class)co 7158 ℂcc 10537 0cc0 10539 ici 10541 · cmul 10544 − cmin 10872 -cneg 10873 / cdiv 11299 2c2 11695 expce 15417 sincsin 15419 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 ax-addf 10618 ax-mulf 10619 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-pm 8411 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-sup 8908 df-inf 8909 df-oi 8976 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-ico 12747 df-fz 12896 df-fzo 13037 df-fl 13165 df-seq 13373 df-exp 13433 df-fac 13637 df-hash 13694 df-shft 14428 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-limsup 14830 df-clim 14847 df-rlim 14848 df-sum 15045 df-ef 15423 df-sin 15425 |
This theorem is referenced by: tanneg 15503 sin0 15504 efmival 15508 sinsub 15523 cossub 15524 sincossq 15531 sin2pim 25073 reasinsin 25476 atantan 25503 sinccvglem 32917 dirkertrigeqlem2 42391 fourierdlem43 42442 fourierdlem44 42443 sqwvfoura 42520 |
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