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Mirrors > Home > MPE Home > Th. List > cosf | Structured version Visualization version GIF version |
Description: Domain and codomain of the cosine function. (Contributed by Paul Chapman, 22-Oct-2007.) (Revised by Mario Carneiro, 30-Apr-2014.) |
Ref | Expression |
---|---|
cosf | ⊢ cos:ℂ⟶ℂ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-cos 16010 | . 2 ⊢ cos = (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))) / 2)) | |
2 | ax-icn 11165 | . . . . . 6 ⊢ i ∈ ℂ | |
3 | mulcl 11190 | . . . . . 6 ⊢ ((i ∈ ℂ ∧ 𝑥 ∈ ℂ) → (i · 𝑥) ∈ ℂ) | |
4 | 2, 3 | mpan 688 | . . . . 5 ⊢ (𝑥 ∈ ℂ → (i · 𝑥) ∈ ℂ) |
5 | efcl 16022 | . . . . 5 ⊢ ((i · 𝑥) ∈ ℂ → (exp‘(i · 𝑥)) ∈ ℂ) | |
6 | 4, 5 | syl 17 | . . . 4 ⊢ (𝑥 ∈ ℂ → (exp‘(i · 𝑥)) ∈ ℂ) |
7 | negicn 11457 | . . . . . 6 ⊢ -i ∈ ℂ | |
8 | mulcl 11190 | . . . . . 6 ⊢ ((-i ∈ ℂ ∧ 𝑥 ∈ ℂ) → (-i · 𝑥) ∈ ℂ) | |
9 | 7, 8 | mpan 688 | . . . . 5 ⊢ (𝑥 ∈ ℂ → (-i · 𝑥) ∈ ℂ) |
10 | efcl 16022 | . . . . 5 ⊢ ((-i · 𝑥) ∈ ℂ → (exp‘(-i · 𝑥)) ∈ ℂ) | |
11 | 9, 10 | syl 17 | . . . 4 ⊢ (𝑥 ∈ ℂ → (exp‘(-i · 𝑥)) ∈ ℂ) |
12 | 6, 11 | addcld 11229 | . . 3 ⊢ (𝑥 ∈ ℂ → ((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))) ∈ ℂ) |
13 | 12 | halfcld 12453 | . 2 ⊢ (𝑥 ∈ ℂ → (((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))) / 2) ∈ ℂ) |
14 | 1, 13 | fmpti 7108 | 1 ⊢ cos:ℂ⟶ℂ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2106 ⟶wf 6536 ‘cfv 6540 (class class class)co 7405 ℂcc 11104 ici 11108 + caddc 11109 · cmul 11111 -cneg 11441 / cdiv 11867 2c2 12263 expce 16001 cosccos 16004 |
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 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-inf2 9632 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 |
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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8699 df-pm 8819 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-sup 9433 df-inf 9434 df-oi 9501 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-n0 12469 df-z 12555 df-uz 12819 df-rp 12971 df-ico 13326 df-fz 13481 df-fzo 13624 df-fl 13753 df-seq 13963 df-exp 14024 df-fac 14230 df-hash 14287 df-shft 15010 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-limsup 15411 df-clim 15428 df-rlim 15429 df-sum 15629 df-ef 16007 df-cos 16010 |
This theorem is referenced by: coscl 16066 tanval 16067 recosf1o 26035 resinf1o 26036 ex-co 29680 taupilem3 36188 dvtan 36526 sinmulcos 44567 dvsinexp 44613 dvcosre 44614 dvsinax 44615 dvcosax 44628 itgsinexplem1 44656 dirkercncflem2 44806 fourierdlem56 44864 fourierdlem62 44870 |
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