Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 15414 | . 2 ⊢ cos = (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))) / 2)) | |
2 | ax-icn 10585 | . . . . . 6 ⊢ i ∈ ℂ | |
3 | mulcl 10610 | . . . . . 6 ⊢ ((i ∈ ℂ ∧ 𝑥 ∈ ℂ) → (i · 𝑥) ∈ ℂ) | |
4 | 2, 3 | mpan 686 | . . . . 5 ⊢ (𝑥 ∈ ℂ → (i · 𝑥) ∈ ℂ) |
5 | efcl 15426 | . . . . 5 ⊢ ((i · 𝑥) ∈ ℂ → (exp‘(i · 𝑥)) ∈ ℂ) | |
6 | 4, 5 | syl 17 | . . . 4 ⊢ (𝑥 ∈ ℂ → (exp‘(i · 𝑥)) ∈ ℂ) |
7 | negicn 10876 | . . . . . 6 ⊢ -i ∈ ℂ | |
8 | mulcl 10610 | . . . . . 6 ⊢ ((-i ∈ ℂ ∧ 𝑥 ∈ ℂ) → (-i · 𝑥) ∈ ℂ) | |
9 | 7, 8 | mpan 686 | . . . . 5 ⊢ (𝑥 ∈ ℂ → (-i · 𝑥) ∈ ℂ) |
10 | efcl 15426 | . . . . 5 ⊢ ((-i · 𝑥) ∈ ℂ → (exp‘(-i · 𝑥)) ∈ ℂ) | |
11 | 9, 10 | syl 17 | . . . 4 ⊢ (𝑥 ∈ ℂ → (exp‘(-i · 𝑥)) ∈ ℂ) |
12 | 6, 11 | addcld 10649 | . . 3 ⊢ (𝑥 ∈ ℂ → ((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))) ∈ ℂ) |
13 | 12 | halfcld 11871 | . 2 ⊢ (𝑥 ∈ ℂ → (((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))) / 2) ∈ ℂ) |
14 | 1, 13 | fmpti 6869 | 1 ⊢ cos:ℂ⟶ℂ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2105 ⟶wf 6345 ‘cfv 6349 (class class class)co 7145 ℂcc 10524 ici 10528 + caddc 10529 · cmul 10531 -cneg 10860 / cdiv 11286 2c2 11681 expce 15405 cosccos 15408 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 ax-inf2 9093 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 ax-mulf 10606 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4833 df-int 4870 df-iun 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-se 5509 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7569 df-1st 7680 df-2nd 7681 df-wrecs 7938 df-recs 7999 df-rdg 8037 df-1o 8093 df-oadd 8097 df-er 8279 df-pm 8399 df-en 8499 df-dom 8500 df-sdom 8501 df-fin 8502 df-sup 8895 df-inf 8896 df-oi 8963 df-card 9357 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11628 df-2 11689 df-3 11690 df-n0 11887 df-z 11971 df-uz 12233 df-rp 12380 df-ico 12734 df-fz 12883 df-fzo 13024 df-fl 13152 df-seq 13360 df-exp 13420 df-fac 13624 df-hash 13681 df-shft 14416 df-cj 14448 df-re 14449 df-im 14450 df-sqrt 14584 df-abs 14585 df-limsup 14818 df-clim 14835 df-rlim 14836 df-sum 15033 df-ef 15411 df-cos 15414 |
This theorem is referenced by: coscl 15470 tanval 15471 recosf1o 25046 resinf1o 25047 ex-co 28145 taupilem3 34483 dvtan 34824 sinmulcos 42026 dvsinexp 42075 dvcosre 42076 dvsinax 42077 dvcosax 42091 itgsinexplem1 42119 dirkercncflem2 42270 fourierdlem56 42328 fourierdlem62 42334 |
Copyright terms: Public domain | W3C validator |