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Mirrors > Home > MPE Home > Th. List > cosneg | Structured version Visualization version GIF version |
Description: The cosines of a number and its negative are the same. (Contributed by NM, 30-Apr-2005.) |
Ref | Expression |
---|---|
cosneg | ⊢ (𝐴 ∈ ℂ → (cos‘-𝐴) = (cos‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | negicn 10736 | . . . . . 6 ⊢ -i ∈ ℂ | |
2 | mulcl 10470 | . . . . . 6 ⊢ ((-i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (-i · 𝐴) ∈ ℂ) | |
3 | 1, 2 | mpan 686 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (-i · 𝐴) ∈ ℂ) |
4 | efcl 15269 | . . . . 5 ⊢ ((-i · 𝐴) ∈ ℂ → (exp‘(-i · 𝐴)) ∈ ℂ) | |
5 | 3, 4 | syl 17 | . . . 4 ⊢ (𝐴 ∈ ℂ → (exp‘(-i · 𝐴)) ∈ ℂ) |
6 | ax-icn 10445 | . . . . . 6 ⊢ i ∈ ℂ | |
7 | mulcl 10470 | . . . . . 6 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i · 𝐴) ∈ ℂ) | |
8 | 6, 7 | mpan 686 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (i · 𝐴) ∈ ℂ) |
9 | efcl 15269 | . . . . 5 ⊢ ((i · 𝐴) ∈ ℂ → (exp‘(i · 𝐴)) ∈ ℂ) | |
10 | 8, 9 | syl 17 | . . . 4 ⊢ (𝐴 ∈ ℂ → (exp‘(i · 𝐴)) ∈ ℂ) |
11 | mulneg12 10928 | . . . . . . . 8 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (-i · 𝐴) = (i · -𝐴)) | |
12 | 6, 11 | mpan 686 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (-i · 𝐴) = (i · -𝐴)) |
13 | 12 | eqcomd 2800 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (i · -𝐴) = (-i · 𝐴)) |
14 | 13 | fveq2d 6545 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (exp‘(i · -𝐴)) = (exp‘(-i · 𝐴))) |
15 | mul2neg 10929 | . . . . . . 7 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (-i · -𝐴) = (i · 𝐴)) | |
16 | 6, 15 | mpan 686 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (-i · -𝐴) = (i · 𝐴)) |
17 | 16 | fveq2d 6545 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (exp‘(-i · -𝐴)) = (exp‘(i · 𝐴))) |
18 | 14, 17 | oveq12d 7037 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((exp‘(i · -𝐴)) + (exp‘(-i · -𝐴))) = ((exp‘(-i · 𝐴)) + (exp‘(i · 𝐴)))) |
19 | 5, 10, 18 | comraddd 10703 | . . 3 ⊢ (𝐴 ∈ ℂ → ((exp‘(i · -𝐴)) + (exp‘(-i · -𝐴))) = ((exp‘(i · 𝐴)) + (exp‘(-i · 𝐴)))) |
20 | 19 | oveq1d 7034 | . 2 ⊢ (𝐴 ∈ ℂ → (((exp‘(i · -𝐴)) + (exp‘(-i · -𝐴))) / 2) = (((exp‘(i · 𝐴)) + (exp‘(-i · 𝐴))) / 2)) |
21 | negcl 10735 | . . 3 ⊢ (𝐴 ∈ ℂ → -𝐴 ∈ ℂ) | |
22 | cosval 15309 | . . 3 ⊢ (-𝐴 ∈ ℂ → (cos‘-𝐴) = (((exp‘(i · -𝐴)) + (exp‘(-i · -𝐴))) / 2)) | |
23 | 21, 22 | syl 17 | . 2 ⊢ (𝐴 ∈ ℂ → (cos‘-𝐴) = (((exp‘(i · -𝐴)) + (exp‘(-i · -𝐴))) / 2)) |
24 | cosval 15309 | . 2 ⊢ (𝐴 ∈ ℂ → (cos‘𝐴) = (((exp‘(i · 𝐴)) + (exp‘(-i · 𝐴))) / 2)) | |
25 | 20, 23, 24 | 3eqtr4d 2840 | 1 ⊢ (𝐴 ∈ ℂ → (cos‘-𝐴) = (cos‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1522 ∈ wcel 2080 ‘cfv 6228 (class class class)co 7019 ℂcc 10384 ici 10388 + caddc 10389 · cmul 10391 -cneg 10720 / cdiv 11147 2c2 11542 expce 15248 cosccos 15251 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1778 ax-4 1792 ax-5 1889 ax-6 1948 ax-7 1993 ax-8 2082 ax-9 2090 ax-10 2111 ax-11 2125 ax-12 2140 ax-13 2343 ax-ext 2768 ax-rep 5084 ax-sep 5097 ax-nul 5104 ax-pow 5160 ax-pr 5224 ax-un 7322 ax-inf2 8953 ax-cnex 10442 ax-resscn 10443 ax-1cn 10444 ax-icn 10445 ax-addcl 10446 ax-addrcl 10447 ax-mulcl 10448 ax-mulrcl 10449 ax-mulcom 10450 ax-addass 10451 ax-mulass 10452 ax-distr 10453 ax-i2m1 10454 ax-1ne0 10455 ax-1rid 10456 ax-rnegex 10457 ax-rrecex 10458 ax-cnre 10459 ax-pre-lttri 10460 ax-pre-lttrn 10461 ax-pre-ltadd 10462 ax-pre-mulgt0 10463 ax-pre-sup 10464 ax-addf 10465 ax-mulf 10466 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-fal 1535 df-ex 1763 df-nf 1767 df-sb 2042 df-mo 2575 df-eu 2611 df-clab 2775 df-cleq 2787 df-clel 2862 df-nfc 2934 df-ne 2984 df-nel 3090 df-ral 3109 df-rex 3110 df-reu 3111 df-rmo 3112 df-rab 3113 df-v 3438 df-sbc 3708 df-csb 3814 df-dif 3864 df-un 3866 df-in 3868 df-ss 3876 df-pss 3878 df-nul 4214 df-if 4384 df-pw 4457 df-sn 4475 df-pr 4477 df-tp 4479 df-op 4481 df-uni 4748 df-int 4785 df-iun 4829 df-br 4965 df-opab 5027 df-mpt 5044 df-tr 5067 df-id 5351 df-eprel 5356 df-po 5365 df-so 5366 df-fr 5405 df-se 5406 df-we 5407 df-xp 5452 df-rel 5453 df-cnv 5454 df-co 5455 df-dm 5456 df-rn 5457 df-res 5458 df-ima 5459 df-pred 6026 df-ord 6072 df-on 6073 df-lim 6074 df-suc 6075 df-iota 6192 df-fun 6230 df-fn 6231 df-f 6232 df-f1 6233 df-fo 6234 df-f1o 6235 df-fv 6236 df-isom 6237 df-riota 6980 df-ov 7022 df-oprab 7023 df-mpo 7024 df-om 7440 df-1st 7548 df-2nd 7549 df-wrecs 7801 df-recs 7863 df-rdg 7901 df-1o 7956 df-oadd 7960 df-er 8142 df-pm 8262 df-en 8361 df-dom 8362 df-sdom 8363 df-fin 8364 df-sup 8755 df-inf 8756 df-oi 8823 df-card 9217 df-pnf 10526 df-mnf 10527 df-xr 10528 df-ltxr 10529 df-le 10530 df-sub 10721 df-neg 10722 df-div 11148 df-nn 11489 df-2 11550 df-3 11551 df-n0 11748 df-z 11832 df-uz 12094 df-rp 12240 df-ico 12594 df-fz 12743 df-fzo 12884 df-fl 13012 df-seq 13220 df-exp 13280 df-fac 13484 df-hash 13541 df-shft 14260 df-cj 14292 df-re 14293 df-im 14294 df-sqrt 14428 df-abs 14429 df-limsup 14662 df-clim 14679 df-rlim 14680 df-sum 14877 df-ef 15254 df-cos 15257 |
This theorem is referenced by: tanneg 15334 efmival 15339 sinsub 15354 cossub 15355 sincossq 15362 cosneghalfpi 24739 cos2pim 24755 ptolemy 24765 coseq0negpitopi 24772 tanord 24803 argregt0 24874 argrege0 24875 atantan 25182 |
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