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Theorem coscn 15500

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 12217	. 2 ⊢ cos = (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))) / 2))
2		eqid 2231	. . . . . . . 8 ⊢ (MetOpen‘(abs ∘ − )) = (MetOpen‘(abs ∘ − ))
3	2	addcncntop 15292	. . . . . . . . 9 ⊢ + ∈ (((MetOpen‘(abs ∘ − )) ×_t (MetOpen‘(abs ∘ − ))) Cn (MetOpen‘(abs ∘ − )))
4	3	a1i 9	. . . . . . . 8 ⊢ (⊤ → + ∈ (((MetOpen‘(abs ∘ − )) ×_t (MetOpen‘(abs ∘ − ))) Cn (MetOpen‘(abs ∘ − ))))
5		efcn 15498	. . . . . . . . . 10 ⊢ exp ∈ (ℂ–cn→ℂ)
6	5	a1i 9	. . . . . . . . 9 ⊢ (⊤ → exp ∈ (ℂ–cn→ℂ))
7		ax-icn 8127	. . . . . . . . . 10 ⊢ i ∈ ℂ
8		eqid 2231	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℂ ↦ (i · 𝑥)) = (𝑥 ∈ ℂ ↦ (i · 𝑥))
9	8	mulc1cncf 15319	. . . . . . . . . 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 15328	. . . . . . . 8 ⊢ (⊤ → (𝑥 ∈ ℂ ↦ (exp‘(i · 𝑥))) ∈ (ℂ–cn→ℂ))
12		negicn 8380	. . . . . . . . . 10 ⊢ -i ∈ ℂ
13		eqid 2231	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℂ ↦ (-i · 𝑥)) = (𝑥 ∈ ℂ ↦ (-i · 𝑥))
14	13	mulc1cncf 15319	. . . . . . . . . 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 15328	. . . . . . . 8 ⊢ (⊤ → (𝑥 ∈ ℂ ↦ (exp‘(-i · 𝑥))) ∈ (ℂ–cn→ℂ))
17	2, 4, 11, 16	cncfmpt2fcntop 15329	. . . . . . 7 ⊢ (⊤ → (𝑥 ∈ ℂ ↦ ((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥)))) ∈ (ℂ–cn→ℂ))
18		cncff 15307	. . . . . . 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 2231	. . . . . . 7 ⊢ (𝑥 ∈ ℂ ↦ ((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥)))) = (𝑥 ∈ ℂ ↦ ((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))))
21	20	fmpt 5797	. . . . . 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 2232	. . . . 5 ⊢ (⊤ → (𝑥 ∈ ℂ ↦ ((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥)))) = (𝑥 ∈ ℂ ↦ ((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥)))))
24		eqidd 2232	. . . . 5 ⊢ (⊤ → (𝑦 ∈ ℂ ↦ (𝑦 / 2)) = (𝑦 ∈ ℂ ↦ (𝑦 / 2)))
25		oveq1 6025	. . . . 5 ⊢ (𝑦 = ((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))) → (𝑦 / 2) = (((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))) / 2))
26	22, 23, 24, 25	fmptcof 5814	. . . 4 ⊢ (⊤ → ((𝑦 ∈ ℂ ↦ (𝑦 / 2)) ∘ (𝑥 ∈ ℂ ↦ ((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))))) = (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))) / 2)))
27		2cn 9214	. . . . . . 7 ⊢ 2 ∈ ℂ
28		2ap0 9236	. . . . . . 7 ⊢ 2 # 0
29		eqid 2231	. . . . . . . 8 ⊢ (𝑦 ∈ ℂ ↦ (𝑦 / 2)) = (𝑦 ∈ ℂ ↦ (𝑦 / 2))
30	29	divccncfap 15320	. . . . . . 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 15321	. . . 4 ⊢ (⊤ → ((𝑦 ∈ ℂ ↦ (𝑦 / 2)) ∘ (𝑥 ∈ ℂ ↦ ((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))))) ∈ (ℂ–cn→ℂ))
34	26, 33	eqeltrrd 2309	. . 3 ⊢ (⊤ → (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))) / 2)) ∈ (ℂ–cn→ℂ))
35	34	mptru 1406	. 2 ⊢ (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))) / 2)) ∈ (ℂ–cn→ℂ)
36	1, 35	eqeltri 2304	1 ⊢ cos ∈ (ℂ–cn→ℂ)

Colors of variables: wff set class

Syntax hints: ⊤wtru 1398 ∈ wcel 2202 ∀wral 2510 class class class wbr 4088 ↦ cmpt 4150 ∘ ccom 4729 ⟶wf 5322 ‘cfv 5326 (class class class)co 6018 ℂcc 8030 0cc0 8032 ici 8034 + caddc 8035 · cmul 8037 − cmin 8350 -cneg 8351 # cap 8761 / cdiv 8852 2c2 9194 abscabs 11562 expce 12208 cosccos 12211 MetOpencmopn 14561 Cn ccn 14915 ×_t ctx 14982 –cn→ccncf 15300

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 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-13 2204 ax-14 2205 ax-ext 2213 ax-coll 4204 ax-sep 4207 ax-nul 4215 ax-pow 4264 ax-pr 4299 ax-un 4530 ax-setind 4635 ax-iinf 4686 ax-cnex 8123 ax-resscn 8124 ax-1cn 8125 ax-1re 8126 ax-icn 8127 ax-addcl 8128 ax-addrcl 8129 ax-mulcl 8130 ax-mulrcl 8131 ax-addcom 8132 ax-mulcom 8133 ax-addass 8134 ax-mulass 8135 ax-distr 8136 ax-i2m1 8137 ax-0lt1 8138 ax-1rid 8139 ax-0id 8140 ax-rnegex 8141 ax-precex 8142 ax-cnre 8143 ax-pre-ltirr 8144 ax-pre-ltwlin 8145 ax-pre-lttrn 8146 ax-pre-apti 8147 ax-pre-ltadd 8148 ax-pre-mulgt0 8149 ax-pre-mulext 8150 ax-arch 8151 ax-caucvg 8152 ax-addf 8154 ax-mulf 8155

This theorem depends on definitions: df-bi 117 df-stab 838 df-dc 842 df-3or 1005 df-3an 1006 df-tru 1400 df-fal 1403 df-nf 1509 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-ne 2403 df-nel 2498 df-ral 2515 df-rex 2516 df-reu 2517 df-rmo 2518 df-rab 2519 df-v 2804 df-sbc 3032 df-csb 3128 df-dif 3202 df-un 3204 df-in 3206 df-ss 3213 df-nul 3495 df-if 3606 df-pw 3654 df-sn 3675 df-pr 3676 df-op 3678 df-uni 3894 df-int 3929 df-iun 3972 df-disj 4065 df-br 4089 df-opab 4151 df-mpt 4152 df-tr 4188 df-id 4390 df-po 4393 df-iso 4394 df-iord 4463 df-on 4465 df-ilim 4466 df-suc 4468 df-iom 4689 df-xp 4731 df-rel 4732 df-cnv 4733 df-co 4734 df-dm 4735 df-rn 4736 df-res 4737 df-ima 4738 df-iota 5286 df-fun 5328 df-fn 5329 df-f 5330 df-f1 5331 df-fo 5332 df-f1o 5333 df-fv 5334 df-isom 5335 df-riota 5971 df-ov 6021 df-oprab 6022 df-mpo 6023 df-of 6235 df-1st 6303 df-2nd 6304 df-recs 6471 df-irdg 6536 df-frec 6557 df-1o 6582 df-oadd 6586 df-er 6702 df-map 6819 df-pm 6820 df-en 6910 df-dom 6911 df-fin 6912 df-sup 7183 df-inf 7184 df-pnf 8216 df-mnf 8217 df-xr 8218 df-ltxr 8219 df-le 8220 df-sub 8352 df-neg 8353 df-reap 8755 df-ap 8762 df-div 8853 df-inn 9144 df-2 9202 df-3 9203 df-4 9204 df-n0 9403 df-z 9480 df-uz 9756 df-q 9854 df-rp 9889 df-xneg 10007 df-xadd 10008 df-ico 10129 df-fz 10244 df-fzo 10378 df-seqfrec 10711 df-exp 10802 df-fac 10989 df-bc 11011 df-ihash 11039 df-shft 11380 df-cj 11407 df-re 11408 df-im 11409 df-rsqrt 11563 df-abs 11564 df-clim 11844 df-sumdc 11919 df-ef 12214 df-cos 12217 df-rest 13329 df-topgen 13348 df-psmet 14563 df-xmet 14564 df-met 14565 df-bl 14566 df-mopn 14567 df-top 14728 df-topon 14741 df-bases 14773 df-ntr 14826 df-cn 14918 df-cnp 14919 df-tx 14983 df-cncf 15301 df-limced 15386 df-dvap 15387

This theorem is referenced by: cosz12 15510 ioocosf1o 15584

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