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Theorem coscn 15452

Description: Cosine is continuous. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Mario Carneiro, 3-Sep-2014.)

**Assertion**
Ref	Expression
coscn	⊢ cos ∈ (ℂ–cn→ℂ)

Proof of Theorem coscn Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-cos 12170	. 2 ⊢ cos = (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))) / 2))
2		eqid 2229	. . . . . . . 8 ⊢ (MetOpen‘(abs ∘ − )) = (MetOpen‘(abs ∘ − ))
3	2	addcncntop 15244	. . . . . . . . 9 ⊢ + ∈ (((MetOpen‘(abs ∘ − )) ×_t (MetOpen‘(abs ∘ − ))) Cn (MetOpen‘(abs ∘ − )))
4	3	a1i 9	. . . . . . . 8 ⊢ (⊤ → + ∈ (((MetOpen‘(abs ∘ − )) ×_t (MetOpen‘(abs ∘ − ))) Cn (MetOpen‘(abs ∘ − ))))
5		efcn 15450	. . . . . . . . . 10 ⊢ exp ∈ (ℂ–cn→ℂ)
6	5	a1i 9	. . . . . . . . 9 ⊢ (⊤ → exp ∈ (ℂ–cn→ℂ))
7		ax-icn 8102	. . . . . . . . . 10 ⊢ i ∈ ℂ
8		eqid 2229	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℂ ↦ (i · 𝑥)) = (𝑥 ∈ ℂ ↦ (i · 𝑥))
9	8	mulc1cncf 15271	. . . . . . . . . 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 15280	. . . . . . . 8 ⊢ (⊤ → (𝑥 ∈ ℂ ↦ (exp‘(i · 𝑥))) ∈ (ℂ–cn→ℂ))
12		negicn 8355	. . . . . . . . . 10 ⊢ -i ∈ ℂ
13		eqid 2229	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℂ ↦ (-i · 𝑥)) = (𝑥 ∈ ℂ ↦ (-i · 𝑥))
14	13	mulc1cncf 15271	. . . . . . . . . 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 15280	. . . . . . . 8 ⊢ (⊤ → (𝑥 ∈ ℂ ↦ (exp‘(-i · 𝑥))) ∈ (ℂ–cn→ℂ))
17	2, 4, 11, 16	cncfmpt2fcntop 15281	. . . . . . 7 ⊢ (⊤ → (𝑥 ∈ ℂ ↦ ((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥)))) ∈ (ℂ–cn→ℂ))
18		cncff 15259	. . . . . . 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 2229	. . . . . . 7 ⊢ (𝑥 ∈ ℂ ↦ ((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥)))) = (𝑥 ∈ ℂ ↦ ((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))))
21	20	fmpt 5787	. . . . . 6 ⊢ (∀𝑥 ∈ ℂ ((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))) ∈ ℂ ↔ (𝑥 ∈ ℂ ↦ ((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥)))):ℂ⟶ℂ)
22	19, 21	sylibr 134	. . . . 5 ⊢ (⊤ → ∀𝑥 ∈ ℂ ((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))) ∈ ℂ)
23		eqidd 2230	. . . . 5 ⊢ (⊤ → (𝑥 ∈ ℂ ↦ ((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥)))) = (𝑥 ∈ ℂ ↦ ((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥)))))
24		eqidd 2230	. . . . 5 ⊢ (⊤ → (𝑦 ∈ ℂ ↦ (𝑦 / 2)) = (𝑦 ∈ ℂ ↦ (𝑦 / 2)))
25		oveq1 6014	. . . . 5 ⊢ (𝑦 = ((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))) → (𝑦 / 2) = (((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))) / 2))
26	22, 23, 24, 25	fmptcof 5804	. . . 4 ⊢ (⊤ → ((𝑦 ∈ ℂ ↦ (𝑦 / 2)) ∘ (𝑥 ∈ ℂ ↦ ((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))))) = (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))) / 2)))
27		2cn 9189	. . . . . . 7 ⊢ 2 ∈ ℂ
28		2ap0 9211	. . . . . . 7 ⊢ 2 # 0
29		eqid 2229	. . . . . . . 8 ⊢ (𝑦 ∈ ℂ ↦ (𝑦 / 2)) = (𝑦 ∈ ℂ ↦ (𝑦 / 2))
30	29	divccncfap 15272	. . . . . . 7 ⊢ ((2 ∈ ℂ ∧ 2 # 0) → (𝑦 ∈ ℂ ↦ (𝑦 / 2)) ∈ (ℂ–cn→ℂ))
31	27, 28, 30	mp2an 426	. . . . . 6 ⊢ (𝑦 ∈ ℂ ↦ (𝑦 / 2)) ∈ (ℂ–cn→ℂ)
32	31	a1i 9	. . . . 5 ⊢ (⊤ → (𝑦 ∈ ℂ ↦ (𝑦 / 2)) ∈ (ℂ–cn→ℂ))
33	17, 32	cncfco 15273	. . . 4 ⊢ (⊤ → ((𝑦 ∈ ℂ ↦ (𝑦 / 2)) ∘ (𝑥 ∈ ℂ ↦ ((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))))) ∈ (ℂ–cn→ℂ))
34	26, 33	eqeltrrd 2307	. . 3 ⊢ (⊤ → (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))) / 2)) ∈ (ℂ–cn→ℂ))
35	34	mptru 1404	. 2 ⊢ (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))) / 2)) ∈ (ℂ–cn→ℂ)
36	1, 35	eqeltri 2302	1 ⊢ cos ∈ (ℂ–cn→ℂ)

Colors of variables: wff set class

Syntax hints: ⊤wtru 1396 ∈ wcel 2200 ∀wral 2508 class class class wbr 4083 ↦ cmpt 4145 ∘ ccom 4723 ⟶wf 5314 ‘cfv 5318 (class class class)co 6007 ℂcc 8005 0cc0 8007 ici 8009 + caddc 8010 · cmul 8012 − cmin 8325 -cneg 8326 # cap 8736 / cdiv 8827 2c2 9169 abscabs 11516 expce 12161 cosccos 12164 MetOpencmopn 14513 Cn ccn 14867 ×_t ctx 14934 –cn→ccncf 15252

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4199 ax-sep 4202 ax-nul 4210 ax-pow 4258 ax-pr 4293 ax-un 4524 ax-setind 4629 ax-iinf 4680 ax-cnex 8098 ax-resscn 8099 ax-1cn 8100 ax-1re 8101 ax-icn 8102 ax-addcl 8103 ax-addrcl 8104 ax-mulcl 8105 ax-mulrcl 8106 ax-addcom 8107 ax-mulcom 8108 ax-addass 8109 ax-mulass 8110 ax-distr 8111 ax-i2m1 8112 ax-0lt1 8113 ax-1rid 8114 ax-0id 8115 ax-rnegex 8116 ax-precex 8117 ax-cnre 8118 ax-pre-ltirr 8119 ax-pre-ltwlin 8120 ax-pre-lttrn 8121 ax-pre-apti 8122 ax-pre-ltadd 8123 ax-pre-mulgt0 8124 ax-pre-mulext 8125 ax-arch 8126 ax-caucvg 8127 ax-addf 8129 ax-mulf 8130

This theorem depends on definitions: df-bi 117 df-stab 836 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rmo 2516 df-rab 2517 df-v 2801 df-sbc 3029 df-csb 3125 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-nul 3492 df-if 3603 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3889 df-int 3924 df-iun 3967 df-disj 4060 df-br 4084 df-opab 4146 df-mpt 4147 df-tr 4183 df-id 4384 df-po 4387 df-iso 4388 df-iord 4457 df-on 4459 df-ilim 4460 df-suc 4462 df-iom 4683 df-xp 4725 df-rel 4726 df-cnv 4727 df-co 4728 df-dm 4729 df-rn 4730 df-res 4731 df-ima 4732 df-iota 5278 df-fun 5320 df-fn 5321 df-f 5322 df-f1 5323 df-fo 5324 df-f1o 5325 df-fv 5326 df-isom 5327 df-riota 5960 df-ov 6010 df-oprab 6011 df-mpo 6012 df-of 6224 df-1st 6292 df-2nd 6293 df-recs 6457 df-irdg 6522 df-frec 6543 df-1o 6568 df-oadd 6572 df-er 6688 df-map 6805 df-pm 6806 df-en 6896 df-dom 6897 df-fin 6898 df-sup 7159 df-inf 7160 df-pnf 8191 df-mnf 8192 df-xr 8193 df-ltxr 8194 df-le 8195 df-sub 8327 df-neg 8328 df-reap 8730 df-ap 8737 df-div 8828 df-inn 9119 df-2 9177 df-3 9178 df-4 9179 df-n0 9378 df-z 9455 df-uz 9731 df-q 9823 df-rp 9858 df-xneg 9976 df-xadd 9977 df-ico 10098 df-fz 10213 df-fzo 10347 df-seqfrec 10678 df-exp 10769 df-fac 10956 df-bc 10978 df-ihash 11006 df-shft 11334 df-cj 11361 df-re 11362 df-im 11363 df-rsqrt 11517 df-abs 11518 df-clim 11798 df-sumdc 11873 df-ef 12167 df-cos 12170 df-rest 13282 df-topgen 13301 df-psmet 14515 df-xmet 14516 df-met 14517 df-bl 14518 df-mopn 14519 df-top 14680 df-topon 14693 df-bases 14725 df-ntr 14778 df-cn 14870 df-cnp 14871 df-tx 14935 df-cncf 15253 df-limced 15338 df-dvap 15339

This theorem is referenced by: cosz12 15462 ioocosf1o 15536

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