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Theorem coscn 15765

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 12366	. 2 ⊢ cos = (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))) / 2))
2		eqid 2234	. . . . . . . 8 ⊢ (MetOpen‘(abs ∘ − )) = (MetOpen‘(abs ∘ − ))
3	2	addcncntop 15557	. . . . . . . . 9 ⊢ + ∈ (((MetOpen‘(abs ∘ − )) ×_t (MetOpen‘(abs ∘ − ))) Cn (MetOpen‘(abs ∘ − )))
4	3	a1i 9	. . . . . . . 8 ⊢ (⊤ → + ∈ (((MetOpen‘(abs ∘ − )) ×_t (MetOpen‘(abs ∘ − ))) Cn (MetOpen‘(abs ∘ − ))))
5		efcn 15763	. . . . . . . . . 10 ⊢ exp ∈ (ℂ–cn→ℂ)
6	5	a1i 9	. . . . . . . . 9 ⊢ (⊤ → exp ∈ (ℂ–cn→ℂ))
7		ax-icn 8238	. . . . . . . . . 10 ⊢ i ∈ ℂ
8		eqid 2234	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℂ ↦ (i · 𝑥)) = (𝑥 ∈ ℂ ↦ (i · 𝑥))
9	8	mulc1cncf 15584	. . . . . . . . . 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 15593	. . . . . . . 8 ⊢ (⊤ → (𝑥 ∈ ℂ ↦ (exp‘(i · 𝑥))) ∈ (ℂ–cn→ℂ))
12		negicn 8491	. . . . . . . . . 10 ⊢ -i ∈ ℂ
13		eqid 2234	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℂ ↦ (-i · 𝑥)) = (𝑥 ∈ ℂ ↦ (-i · 𝑥))
14	13	mulc1cncf 15584	. . . . . . . . . 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 15593	. . . . . . . 8 ⊢ (⊤ → (𝑥 ∈ ℂ ↦ (exp‘(-i · 𝑥))) ∈ (ℂ–cn→ℂ))
17	2, 4, 11, 16	cncfmpt2fcntop 15594	. . . . . . 7 ⊢ (⊤ → (𝑥 ∈ ℂ ↦ ((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥)))) ∈ (ℂ–cn→ℂ))
18		cncff 15572	. . . . . . 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 2234	. . . . . . 7 ⊢ (𝑥 ∈ ℂ ↦ ((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥)))) = (𝑥 ∈ ℂ ↦ ((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))))
21	20	fmpt 5832	. . . . . 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 2235	. . . . 5 ⊢ (⊤ → (𝑥 ∈ ℂ ↦ ((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥)))) = (𝑥 ∈ ℂ ↦ ((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥)))))
24		eqidd 2235	. . . . 5 ⊢ (⊤ → (𝑦 ∈ ℂ ↦ (𝑦 / 2)) = (𝑦 ∈ ℂ ↦ (𝑦 / 2)))
25		oveq1 6065	. . . . 5 ⊢ (𝑦 = ((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))) → (𝑦 / 2) = (((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))) / 2))
26	22, 23, 24, 25	fmptcof 5849	. . . 4 ⊢ (⊤ → ((𝑦 ∈ ℂ ↦ (𝑦 / 2)) ∘ (𝑥 ∈ ℂ ↦ ((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))))) = (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))) / 2)))
27		2cn 9328	. . . . . . 7 ⊢ 2 ∈ ℂ
28		2ap0 9350	. . . . . . 7 ⊢ 2 # 0
29		eqid 2234	. . . . . . . 8 ⊢ (𝑦 ∈ ℂ ↦ (𝑦 / 2)) = (𝑦 ∈ ℂ ↦ (𝑦 / 2))
30	29	divccncfap 15585	. . . . . . 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 15586	. . . 4 ⊢ (⊤ → ((𝑦 ∈ ℂ ↦ (𝑦 / 2)) ∘ (𝑥 ∈ ℂ ↦ ((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))))) ∈ (ℂ–cn→ℂ))
34	26, 33	eqeltrrd 2312	. . 3 ⊢ (⊤ → (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))) / 2)) ∈ (ℂ–cn→ℂ))
35	34	mptru 1407	. 2 ⊢ (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))) / 2)) ∈ (ℂ–cn→ℂ)
36	1, 35	eqeltri 2307	1 ⊢ cos ∈ (ℂ–cn→ℂ)

Colors of variables: wff set class

Syntax hints: ⊤wtru 1399 ∈ wcel 2205 ∀wral 2522 class class class wbr 4114 ↦ cmpt 4176 ∘ ccom 4758 ⟶wf 5353 ‘cfv 5357 (class class class)co 6058 ℂcc 8141 0cc0 8143 ici 8145 + caddc 8146 · cmul 8148 − cmin 8461 -cneg 8462 # cap 8873 / cdiv 8966 2c2 9308 abscabs 11711 expce 12357 cosccos 12360 MetOpencmopn 14819 Cn ccn 15180 ×_t ctx 15247 –cn→ccncf 15565

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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2207 ax-14 2208 ax-ext 2216 ax-coll 4230 ax-sep 4233 ax-nul 4241 ax-pow 4292 ax-pr 4327 ax-un 4559 ax-setind 4664 ax-iinf 4715 ax-cnex 8234 ax-resscn 8235 ax-1cn 8236 ax-1re 8237 ax-icn 8238 ax-addcl 8239 ax-addrcl 8240 ax-mulcl 8241 ax-mulrcl 8242 ax-addcom 8243 ax-mulcom 8244 ax-addass 8245 ax-mulass 8246 ax-distr 8247 ax-i2m1 8248 ax-0lt1 8249 ax-1rid 8250 ax-0id 8251 ax-rnegex 8252 ax-precex 8253 ax-cnre 8254 ax-pre-ltirr 8255 ax-pre-ltwlin 8256 ax-pre-lttrn 8257 ax-pre-apti 8258 ax-pre-ltadd 8259 ax-pre-mulgt0 8260 ax-pre-mulext 8261 ax-arch 8262 ax-caucvg 8263 ax-addf 8265 ax-mulf 8266

This theorem depends on definitions: df-bi 117 df-stab 839 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ne 2415 df-nel 2510 df-ral 2527 df-rex 2528 df-reu 2529 df-rmo 2530 df-rab 2531 df-v 2817 df-sbc 3046 df-csb 3142 df-dif 3216 df-un 3218 df-in 3220 df-ss 3227 df-nul 3513 df-if 3625 df-pw 3676 df-sn 3700 df-pr 3701 df-op 3703 df-uni 3920 df-int 3955 df-iun 3998 df-disj 4091 df-br 4115 df-opab 4177 df-mpt 4178 df-tr 4214 df-id 4419 df-po 4422 df-iso 4423 df-iord 4492 df-on 4494 df-ilim 4495 df-suc 4497 df-iom 4718 df-xp 4760 df-rel 4761 df-cnv 4762 df-co 4763 df-dm 4764 df-rn 4765 df-res 4766 df-ima 4767 df-iota 5317 df-fun 5359 df-fn 5360 df-f 5361 df-f1 5362 df-fo 5363 df-f1o 5364 df-fv 5365 df-isom 5366 df-riota 6011 df-ov 6061 df-oprab 6062 df-mpo 6063 df-of 6275 df-1st 6347 df-2nd 6348 df-recs 6549 df-irdg 6614 df-frec 6635 df-1o 6660 df-oadd 6664 df-er 6780 df-map 6897 df-pm 6898 df-en 6989 df-dom 6990 df-fin 6991 df-sup 7288 df-inf 7289 df-pnf 8326 df-mnf 8327 df-xr 8328 df-ltxr 8329 df-le 8330 df-sub 8463 df-neg 8464 df-reap 8867 df-ap 8874 df-div 8967 df-inn 9258 df-2 9316 df-3 9317 df-4 9318 df-n0 9517 df-z 9598 df-uz 9875 df-q 9973 df-rp 10008 df-xneg 10127 df-xadd 10128 df-ico 10249 df-fz 10365 df-fzo 10502 df-seqfrec 10837 df-exp 10928 df-fac 11116 df-bc 11138 df-ihash 11167 df-shft 11528 df-cj 11555 df-re 11556 df-im 11557 df-rsqrt 11712 df-abs 11713 df-clim 11993 df-sumdc 12068 df-ef 12363 df-cos 12366 df-rest 13542 df-topgen 13561 df-psmet 14821 df-xmet 14822 df-met 14823 df-bl 14824 df-mopn 14825 df-top 14993 df-topon 15006 df-bases 15038 df-ntr 15091 df-cn 15183 df-cnp 15184 df-tx 15248 df-cncf 15566 df-limced 15651 df-dvap 15652

This theorem is referenced by: cosz12 15775 ioocosf1o 15849

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