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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 11462 | . . . . . 6 ⊢ -i ∈ ℂ | |
2 | mulcl 11193 | . . . . . 6 ⊢ ((-i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (-i · 𝐴) ∈ ℂ) | |
3 | 1, 2 | mpan 687 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (-i · 𝐴) ∈ ℂ) |
4 | efcl 16030 | . . . . 5 ⊢ ((-i · 𝐴) ∈ ℂ → (exp‘(-i · 𝐴)) ∈ ℂ) | |
5 | 3, 4 | syl 17 | . . . 4 ⊢ (𝐴 ∈ ℂ → (exp‘(-i · 𝐴)) ∈ ℂ) |
6 | ax-icn 11168 | . . . . . 6 ⊢ i ∈ ℂ | |
7 | mulcl 11193 | . . . . . 6 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i · 𝐴) ∈ ℂ) | |
8 | 6, 7 | mpan 687 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (i · 𝐴) ∈ ℂ) |
9 | efcl 16030 | . . . . 5 ⊢ ((i · 𝐴) ∈ ℂ → (exp‘(i · 𝐴)) ∈ ℂ) | |
10 | 8, 9 | syl 17 | . . . 4 ⊢ (𝐴 ∈ ℂ → (exp‘(i · 𝐴)) ∈ ℂ) |
11 | mulneg12 11653 | . . . . . . . 8 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (-i · 𝐴) = (i · -𝐴)) | |
12 | 6, 11 | mpan 687 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (-i · 𝐴) = (i · -𝐴)) |
13 | 12 | eqcomd 2732 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (i · -𝐴) = (-i · 𝐴)) |
14 | 13 | fveq2d 6888 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (exp‘(i · -𝐴)) = (exp‘(-i · 𝐴))) |
15 | mul2neg 11654 | . . . . . . 7 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (-i · -𝐴) = (i · 𝐴)) | |
16 | 6, 15 | mpan 687 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (-i · -𝐴) = (i · 𝐴)) |
17 | 16 | fveq2d 6888 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (exp‘(-i · -𝐴)) = (exp‘(i · 𝐴))) |
18 | 14, 17 | oveq12d 7422 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((exp‘(i · -𝐴)) + (exp‘(-i · -𝐴))) = ((exp‘(-i · 𝐴)) + (exp‘(i · 𝐴)))) |
19 | 5, 10, 18 | comraddd 11429 | . . 3 ⊢ (𝐴 ∈ ℂ → ((exp‘(i · -𝐴)) + (exp‘(-i · -𝐴))) = ((exp‘(i · 𝐴)) + (exp‘(-i · 𝐴)))) |
20 | 19 | oveq1d 7419 | . 2 ⊢ (𝐴 ∈ ℂ → (((exp‘(i · -𝐴)) + (exp‘(-i · -𝐴))) / 2) = (((exp‘(i · 𝐴)) + (exp‘(-i · 𝐴))) / 2)) |
21 | negcl 11461 | . . 3 ⊢ (𝐴 ∈ ℂ → -𝐴 ∈ ℂ) | |
22 | cosval 16071 | . . 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 16071 | . 2 ⊢ (𝐴 ∈ ℂ → (cos‘𝐴) = (((exp‘(i · 𝐴)) + (exp‘(-i · 𝐴))) / 2)) | |
25 | 20, 23, 24 | 3eqtr4d 2776 | 1 ⊢ (𝐴 ∈ ℂ → (cos‘-𝐴) = (cos‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ‘cfv 6536 (class class class)co 7404 ℂcc 11107 ici 11111 + caddc 11112 · cmul 11114 -cneg 11446 / cdiv 11872 2c2 12268 expce 16009 cosccos 16012 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-inf2 9635 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-er 8702 df-pm 8822 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-sup 9436 df-inf 9437 df-oi 9504 df-card 9933 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-div 11873 df-nn 12214 df-2 12276 df-3 12277 df-n0 12474 df-z 12560 df-uz 12824 df-rp 12978 df-ico 13333 df-fz 13488 df-fzo 13631 df-fl 13760 df-seq 13970 df-exp 14031 df-fac 14237 df-hash 14294 df-shft 15018 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-limsup 15419 df-clim 15436 df-rlim 15437 df-sum 15637 df-ef 16015 df-cos 16018 |
This theorem is referenced by: tanneg 16096 efmival 16101 sinsub 16116 cossub 16117 sincossq 16124 cosneghalfpi 26356 cos2pim 26372 ptolemy 26382 coseq0negpitopi 26389 tanord 26423 argregt0 26495 argrege0 26496 atantan 26806 |
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