Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 10160 | . . 3 ⊢ (𝐴 ∈ ℂ → -𝐴 ∈ ℂ) | |
2 | sinval 14691 | . . 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 14691 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (sin‘𝐴) = (((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) / (2 · i))) | |
5 | 4 | negeqd 10154 | . . . 4 ⊢ (𝐴 ∈ ℂ → -(sin‘𝐴) = -(((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) / (2 · i))) |
6 | ax-icn 9874 | . . . . . . . 8 ⊢ i ∈ ℂ | |
7 | mulcl 9899 | . . . . . . . 8 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i · 𝐴) ∈ ℂ) | |
8 | 6, 7 | mpan 702 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (i · 𝐴) ∈ ℂ) |
9 | efcl 14652 | . . . . . . 7 ⊢ ((i · 𝐴) ∈ ℂ → (exp‘(i · 𝐴)) ∈ ℂ) | |
10 | 8, 9 | syl 17 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (exp‘(i · 𝐴)) ∈ ℂ) |
11 | negicn 10161 | . . . . . . . 8 ⊢ -i ∈ ℂ | |
12 | mulcl 9899 | . . . . . . . 8 ⊢ ((-i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (-i · 𝐴) ∈ ℂ) | |
13 | 11, 12 | mpan 702 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (-i · 𝐴) ∈ ℂ) |
14 | efcl 14652 | . . . . . . 7 ⊢ ((-i · 𝐴) ∈ ℂ → (exp‘(-i · 𝐴)) ∈ ℂ) | |
15 | 13, 14 | syl 17 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (exp‘(-i · 𝐴)) ∈ ℂ) |
16 | 10, 15 | subcld 10271 | . . . . 5 ⊢ (𝐴 ∈ ℂ → ((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) ∈ ℂ) |
17 | 2mulicn 11132 | . . . . . 6 ⊢ (2 · i) ∈ ℂ | |
18 | 2muline0 11133 | . . . . . 6 ⊢ (2 · i) ≠ 0 | |
19 | divneg 10598 | . . . . . 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 1408 | . . . . 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 2644 | . . 3 ⊢ (𝐴 ∈ ℂ → -(sin‘𝐴) = (-((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) / (2 · i))) |
23 | mulneg12 10347 | . . . . . . . . 9 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (-i · 𝐴) = (i · -𝐴)) | |
24 | 6, 23 | mpan 702 | . . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (-i · 𝐴) = (i · -𝐴)) |
25 | 24 | eqcomd 2616 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (i · -𝐴) = (-i · 𝐴)) |
26 | 25 | fveq2d 6107 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (exp‘(i · -𝐴)) = (exp‘(-i · 𝐴))) |
27 | mul2neg 10348 | . . . . . . . 8 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (-i · -𝐴) = (i · 𝐴)) | |
28 | 6, 27 | mpan 702 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (-i · -𝐴) = (i · 𝐴)) |
29 | 28 | fveq2d 6107 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (exp‘(-i · -𝐴)) = (exp‘(i · 𝐴))) |
30 | 26, 29 | oveq12d 6567 | . . . . 5 ⊢ (𝐴 ∈ ℂ → ((exp‘(i · -𝐴)) − (exp‘(-i · -𝐴))) = ((exp‘(-i · 𝐴)) − (exp‘(i · 𝐴)))) |
31 | 10, 15 | negsubdi2d 10287 | . . . . 5 ⊢ (𝐴 ∈ ℂ → -((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) = ((exp‘(-i · 𝐴)) − (exp‘(i · 𝐴)))) |
32 | 30, 31 | eqtr4d 2647 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((exp‘(i · -𝐴)) − (exp‘(-i · -𝐴))) = -((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴)))) |
33 | 32 | oveq1d 6564 | . . 3 ⊢ (𝐴 ∈ ℂ → (((exp‘(i · -𝐴)) − (exp‘(-i · -𝐴))) / (2 · i)) = (-((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) / (2 · i))) |
34 | 22, 33 | eqtr4d 2647 | . 2 ⊢ (𝐴 ∈ ℂ → -(sin‘𝐴) = (((exp‘(i · -𝐴)) − (exp‘(-i · -𝐴))) / (2 · i))) |
35 | 3, 34 | eqtr4d 2647 | 1 ⊢ (𝐴 ∈ ℂ → (sin‘-𝐴) = -(sin‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 0cc0 9815 ici 9817 · cmul 9820 − cmin 10145 -cneg 10146 / cdiv 10563 2c2 10947 expce 14631 sincsin 14633 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-ico 12052 df-fz 12198 df-fzo 12335 df-fl 12455 df-seq 12664 df-exp 12723 df-fac 12923 df-hash 12980 df-shft 13655 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-limsup 14050 df-clim 14067 df-rlim 14068 df-sum 14265 df-ef 14637 df-sin 14639 |
This theorem is referenced by: tanneg 14717 sin0 14718 efmival 14722 sinsub 14737 cossub 14738 sincossq 14745 sin2pim 24041 reasinsin 24423 atantan 24450 sinccvglem 30820 dirkertrigeqlem2 38992 fourierdlem43 39043 fourierdlem44 39044 sqwvfoura 39121 |
Copyright terms: Public domain | W3C validator |