Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > coscn | GIF version |
Description: Cosine is continuous. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Mario Carneiro, 3-Sep-2014.) |
Ref | Expression |
---|---|
coscn | ⊢ cos ∈ (ℂ–cn→ℂ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-cos 11360 | . 2 ⊢ cos = (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))) / 2)) | |
2 | eqid 2139 | . . . . . . . 8 ⊢ (MetOpen‘(abs ∘ − )) = (MetOpen‘(abs ∘ − )) | |
3 | 2 | addcncntop 12724 | . . . . . . . . 9 ⊢ + ∈ (((MetOpen‘(abs ∘ − )) ×t (MetOpen‘(abs ∘ − ))) Cn (MetOpen‘(abs ∘ − ))) |
4 | 3 | a1i 9 | . . . . . . . 8 ⊢ (⊤ → + ∈ (((MetOpen‘(abs ∘ − )) ×t (MetOpen‘(abs ∘ − ))) Cn (MetOpen‘(abs ∘ − )))) |
5 | efcn 12860 | . . . . . . . . . 10 ⊢ exp ∈ (ℂ–cn→ℂ) | |
6 | 5 | a1i 9 | . . . . . . . . 9 ⊢ (⊤ → exp ∈ (ℂ–cn→ℂ)) |
7 | ax-icn 7718 | . . . . . . . . . 10 ⊢ i ∈ ℂ | |
8 | eqid 2139 | . . . . . . . . . . 11 ⊢ (𝑥 ∈ ℂ ↦ (i · 𝑥)) = (𝑥 ∈ ℂ ↦ (i · 𝑥)) | |
9 | 8 | mulc1cncf 12748 | . . . . . . . . . 10 ⊢ (i ∈ ℂ → (𝑥 ∈ ℂ ↦ (i · 𝑥)) ∈ (ℂ–cn→ℂ)) |
10 | 7, 9 | mp1i 10 | . . . . . . . . 9 ⊢ (⊤ → (𝑥 ∈ ℂ ↦ (i · 𝑥)) ∈ (ℂ–cn→ℂ)) |
11 | 6, 10 | cncfmpt1f 12756 | . . . . . . . 8 ⊢ (⊤ → (𝑥 ∈ ℂ ↦ (exp‘(i · 𝑥))) ∈ (ℂ–cn→ℂ)) |
12 | negicn 7966 | . . . . . . . . . 10 ⊢ -i ∈ ℂ | |
13 | eqid 2139 | . . . . . . . . . . 11 ⊢ (𝑥 ∈ ℂ ↦ (-i · 𝑥)) = (𝑥 ∈ ℂ ↦ (-i · 𝑥)) | |
14 | 13 | mulc1cncf 12748 | . . . . . . . . . 10 ⊢ (-i ∈ ℂ → (𝑥 ∈ ℂ ↦ (-i · 𝑥)) ∈ (ℂ–cn→ℂ)) |
15 | 12, 14 | mp1i 10 | . . . . . . . . 9 ⊢ (⊤ → (𝑥 ∈ ℂ ↦ (-i · 𝑥)) ∈ (ℂ–cn→ℂ)) |
16 | 6, 15 | cncfmpt1f 12756 | . . . . . . . 8 ⊢ (⊤ → (𝑥 ∈ ℂ ↦ (exp‘(-i · 𝑥))) ∈ (ℂ–cn→ℂ)) |
17 | 2, 4, 11, 16 | cncfmpt2fcntop 12757 | . . . . . . 7 ⊢ (⊤ → (𝑥 ∈ ℂ ↦ ((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥)))) ∈ (ℂ–cn→ℂ)) |
18 | cncff 12736 | . . . . . . 7 ⊢ ((𝑥 ∈ ℂ ↦ ((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥)))) ∈ (ℂ–cn→ℂ) → (𝑥 ∈ ℂ ↦ ((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥)))):ℂ⟶ℂ) | |
19 | 17, 18 | syl 14 | . . . . . 6 ⊢ (⊤ → (𝑥 ∈ ℂ ↦ ((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥)))):ℂ⟶ℂ) |
20 | eqid 2139 | . . . . . . 7 ⊢ (𝑥 ∈ ℂ ↦ ((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥)))) = (𝑥 ∈ ℂ ↦ ((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥)))) | |
21 | 20 | fmpt 5570 | . . . . . 6 ⊢ (∀𝑥 ∈ ℂ ((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))) ∈ ℂ ↔ (𝑥 ∈ ℂ ↦ ((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥)))):ℂ⟶ℂ) |
22 | 19, 21 | sylibr 133 | . . . . 5 ⊢ (⊤ → ∀𝑥 ∈ ℂ ((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))) ∈ ℂ) |
23 | eqidd 2140 | . . . . 5 ⊢ (⊤ → (𝑥 ∈ ℂ ↦ ((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥)))) = (𝑥 ∈ ℂ ↦ ((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))))) | |
24 | eqidd 2140 | . . . . 5 ⊢ (⊤ → (𝑦 ∈ ℂ ↦ (𝑦 / 2)) = (𝑦 ∈ ℂ ↦ (𝑦 / 2))) | |
25 | oveq1 5781 | . . . . 5 ⊢ (𝑦 = ((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))) → (𝑦 / 2) = (((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))) / 2)) | |
26 | 22, 23, 24, 25 | fmptcof 5587 | . . . 4 ⊢ (⊤ → ((𝑦 ∈ ℂ ↦ (𝑦 / 2)) ∘ (𝑥 ∈ ℂ ↦ ((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))))) = (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))) / 2))) |
27 | 2cn 8794 | . . . . . . 7 ⊢ 2 ∈ ℂ | |
28 | 2ap0 8816 | . . . . . . 7 ⊢ 2 # 0 | |
29 | eqid 2139 | . . . . . . . 8 ⊢ (𝑦 ∈ ℂ ↦ (𝑦 / 2)) = (𝑦 ∈ ℂ ↦ (𝑦 / 2)) | |
30 | 29 | divccncfap 12749 | . . . . . . 7 ⊢ ((2 ∈ ℂ ∧ 2 # 0) → (𝑦 ∈ ℂ ↦ (𝑦 / 2)) ∈ (ℂ–cn→ℂ)) |
31 | 27, 28, 30 | mp2an 422 | . . . . . 6 ⊢ (𝑦 ∈ ℂ ↦ (𝑦 / 2)) ∈ (ℂ–cn→ℂ) |
32 | 31 | a1i 9 | . . . . 5 ⊢ (⊤ → (𝑦 ∈ ℂ ↦ (𝑦 / 2)) ∈ (ℂ–cn→ℂ)) |
33 | 17, 32 | cncfco 12750 | . . . 4 ⊢ (⊤ → ((𝑦 ∈ ℂ ↦ (𝑦 / 2)) ∘ (𝑥 ∈ ℂ ↦ ((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))))) ∈ (ℂ–cn→ℂ)) |
34 | 26, 33 | eqeltrrd 2217 | . . 3 ⊢ (⊤ → (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))) / 2)) ∈ (ℂ–cn→ℂ)) |
35 | 34 | mptru 1340 | . 2 ⊢ (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))) / 2)) ∈ (ℂ–cn→ℂ) |
36 | 1, 35 | eqeltri 2212 | 1 ⊢ cos ∈ (ℂ–cn→ℂ) |
Colors of variables: wff set class |
Syntax hints: ⊤wtru 1332 ∈ wcel 1480 ∀wral 2416 class class class wbr 3929 ↦ cmpt 3989 ∘ ccom 4543 ⟶wf 5119 ‘cfv 5123 (class class class)co 5774 ℂcc 7621 0cc0 7623 ici 7625 + caddc 7626 · cmul 7628 − cmin 7936 -cneg 7937 # cap 8346 / cdiv 8435 2c2 8774 abscabs 10772 expce 11351 cosccos 11354 MetOpencmopn 12157 Cn ccn 12357 ×t ctx 12424 –cn→ccncf 12729 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 ax-cnex 7714 ax-resscn 7715 ax-1cn 7716 ax-1re 7717 ax-icn 7718 ax-addcl 7719 ax-addrcl 7720 ax-mulcl 7721 ax-mulrcl 7722 ax-addcom 7723 ax-mulcom 7724 ax-addass 7725 ax-mulass 7726 ax-distr 7727 ax-i2m1 7728 ax-0lt1 7729 ax-1rid 7730 ax-0id 7731 ax-rnegex 7732 ax-precex 7733 ax-cnre 7734 ax-pre-ltirr 7735 ax-pre-ltwlin 7736 ax-pre-lttrn 7737 ax-pre-apti 7738 ax-pre-ltadd 7739 ax-pre-mulgt0 7740 ax-pre-mulext 7741 ax-arch 7742 ax-caucvg 7743 ax-addf 7745 ax-mulf 7746 |
This theorem depends on definitions: df-bi 116 df-stab 816 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-disj 3907 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-po 4218 df-iso 4219 df-iord 4288 df-on 4290 df-ilim 4291 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-isom 5132 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-of 5982 df-1st 6038 df-2nd 6039 df-recs 6202 df-irdg 6267 df-frec 6288 df-1o 6313 df-oadd 6317 df-er 6429 df-map 6544 df-pm 6545 df-en 6635 df-dom 6636 df-fin 6637 df-sup 6871 df-inf 6872 df-pnf 7805 df-mnf 7806 df-xr 7807 df-ltxr 7808 df-le 7809 df-sub 7938 df-neg 7939 df-reap 8340 df-ap 8347 df-div 8436 df-inn 8724 df-2 8782 df-3 8783 df-4 8784 df-n0 8981 df-z 9058 df-uz 9330 df-q 9415 df-rp 9445 df-xneg 9562 df-xadd 9563 df-ico 9680 df-fz 9794 df-fzo 9923 df-seqfrec 10222 df-exp 10296 df-fac 10475 df-bc 10497 df-ihash 10525 df-shft 10590 df-cj 10617 df-re 10618 df-im 10619 df-rsqrt 10773 df-abs 10774 df-clim 11051 df-sumdc 11126 df-ef 11357 df-cos 11360 df-rest 12125 df-topgen 12144 df-psmet 12159 df-xmet 12160 df-met 12161 df-bl 12162 df-mopn 12163 df-top 12168 df-topon 12181 df-bases 12213 df-ntr 12268 df-cn 12360 df-cnp 12361 df-tx 12425 df-cncf 12730 df-limced 12797 df-dvap 12798 |
This theorem is referenced by: cosz12 12864 |
Copyright terms: Public domain | W3C validator |