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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 15427 | . 2 ⊢ cos = (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))) / 2)) | |
2 | ax-icn 10599 | . . . . . 6 ⊢ i ∈ ℂ | |
3 | mulcl 10624 | . . . . . 6 ⊢ ((i ∈ ℂ ∧ 𝑥 ∈ ℂ) → (i · 𝑥) ∈ ℂ) | |
4 | 2, 3 | mpan 688 | . . . . 5 ⊢ (𝑥 ∈ ℂ → (i · 𝑥) ∈ ℂ) |
5 | efcl 15439 | . . . . 5 ⊢ ((i · 𝑥) ∈ ℂ → (exp‘(i · 𝑥)) ∈ ℂ) | |
6 | 4, 5 | syl 17 | . . . 4 ⊢ (𝑥 ∈ ℂ → (exp‘(i · 𝑥)) ∈ ℂ) |
7 | negicn 10890 | . . . . . 6 ⊢ -i ∈ ℂ | |
8 | mulcl 10624 | . . . . . 6 ⊢ ((-i ∈ ℂ ∧ 𝑥 ∈ ℂ) → (-i · 𝑥) ∈ ℂ) | |
9 | 7, 8 | mpan 688 | . . . . 5 ⊢ (𝑥 ∈ ℂ → (-i · 𝑥) ∈ ℂ) |
10 | efcl 15439 | . . . . 5 ⊢ ((-i · 𝑥) ∈ ℂ → (exp‘(-i · 𝑥)) ∈ ℂ) | |
11 | 9, 10 | syl 17 | . . . 4 ⊢ (𝑥 ∈ ℂ → (exp‘(-i · 𝑥)) ∈ ℂ) |
12 | 6, 11 | addcld 10663 | . . 3 ⊢ (𝑥 ∈ ℂ → ((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))) ∈ ℂ) |
13 | 12 | halfcld 11885 | . 2 ⊢ (𝑥 ∈ ℂ → (((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))) / 2) ∈ ℂ) |
14 | 1, 13 | fmpti 6879 | 1 ⊢ cos:ℂ⟶ℂ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2113 ⟶wf 6354 ‘cfv 6358 (class class class)co 7159 ℂcc 10538 ici 10542 + caddc 10543 · cmul 10545 -cneg 10874 / cdiv 11300 2c2 11695 expce 15418 cosccos 15421 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-inf2 9107 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 ax-pre-sup 10618 ax-addf 10619 ax-mulf 10620 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-fal 1549 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-se 5518 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-oadd 8109 df-er 8292 df-pm 8412 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-sup 8909 df-inf 8910 df-oi 8977 df-card 9371 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-div 11301 df-nn 11642 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 14429 df-cj 14461 df-re 14462 df-im 14463 df-sqrt 14597 df-abs 14598 df-limsup 14831 df-clim 14848 df-rlim 14849 df-sum 15046 df-ef 15424 df-cos 15427 |
This theorem is referenced by: coscl 15483 tanval 15484 recosf1o 25122 resinf1o 25123 ex-co 28220 taupilem3 34604 dvtan 34946 sinmulcos 42152 dvsinexp 42201 dvcosre 42202 dvsinax 42203 dvcosax 42217 itgsinexplem1 42245 dirkercncflem2 42396 fourierdlem56 42454 fourierdlem62 42460 |
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