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Theorem coscn 16008

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 12444	. 2 ⊢ cos = (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) + (exp‘( -i · 𝑥))) / 2))
2		eqid 2238	. . . . . . . 8 ⊢ (MetOpen‘(abs ∘ − )) = (MetOpen‘(abs ∘ − ))
3	2	addcncntop 15800	. . . . . . . . 9 ⊢ + ∈ (((MetOpen‘(abs ∘ − )) ×_t (MetOpen‘(abs ∘ − ))) Cn (MetOpen‘(abs ∘ − )))
4	3	a1i 9	. . . . . . . 8 ⊢ (⊤ → + ∈ (((MetOpen‘(abs ∘ − )) ×_t (MetOpen‘(abs ∘ − ))) Cn (MetOpen‘(abs ∘ − ))))
5		efcn 16006	. . . . . . . . . 10 ⊢ exp ∈ (ℂ–cn→ℂ)
6	5	a1i 9	. . . . . . . . 9 ⊢ (⊤ → exp ∈ (ℂ–cn→ℂ))
7		ax-icn 8281	. . . . . . . . . 10 ⊢ i ∈ ℂ
8		eqid 2238	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℂ ↦ (i · 𝑥)) = (𝑥 ∈ ℂ ↦ (i · 𝑥))
9	8	mulc1cncf 15827	. . . . . . . . . 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 15836	. . . . . . . 8 ⊢ (⊤ → (𝑥 ∈ ℂ ↦ (exp‘(i · 𝑥))) ∈ (ℂ–cn→ℂ))
12		negicn 8535	. . . . . . . . . 10 ⊢ -i ∈ ℂ
13		eqid 2238	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℂ ↦ ( -i · 𝑥)) = (𝑥 ∈ ℂ ↦ ( -i · 𝑥))
14	13	mulc1cncf 15827	. . . . . . . . . 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 15836	. . . . . . . 8 ⊢ (⊤ → (𝑥 ∈ ℂ ↦ (exp‘( -i · 𝑥))) ∈ (ℂ–cn→ℂ))
17	2, 4, 11, 16	cncfmpt2fcntop 15837	. . . . . . 7 ⊢ (⊤ → (𝑥 ∈ ℂ ↦ ((exp‘(i · 𝑥)) + (exp‘( -i · 𝑥)))) ∈ (ℂ–cn→ℂ))
18		cncff 15815	. . . . . . 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 2238	. . . . . . 7 ⊢ (𝑥 ∈ ℂ ↦ ((exp‘(i · 𝑥)) + (exp‘( -i · 𝑥)))) = (𝑥 ∈ ℂ ↦ ((exp‘(i · 𝑥)) + (exp‘( -i · 𝑥))))
21	20	fmpt 5861	. . . . . 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 2239	. . . . 5 ⊢ (⊤ → (𝑥 ∈ ℂ ↦ ((exp‘(i · 𝑥)) + (exp‘( -i · 𝑥)))) = (𝑥 ∈ ℂ ↦ ((exp‘(i · 𝑥)) + (exp‘( -i · 𝑥)))))
24		eqidd 2239	. . . . 5 ⊢ (⊤ → (𝑦 ∈ ℂ ↦ (𝑦 / 2)) = (𝑦 ∈ ℂ ↦ (𝑦 / 2)))
25		oveq1 6095	. . . . 5 ⊢ (𝑦 = ((exp‘(i · 𝑥)) + (exp‘( -i · 𝑥))) → (𝑦 / 2) = (((exp‘(i · 𝑥)) + (exp‘( -i · 𝑥))) / 2))
26	22, 23, 24, 25	fmptcof 5878	. . . 4 ⊢ (⊤ → ((𝑦 ∈ ℂ ↦ (𝑦 / 2)) ∘ (𝑥 ∈ ℂ ↦ ((exp‘(i · 𝑥)) + (exp‘( -i · 𝑥))))) = (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) + (exp‘( -i · 𝑥))) / 2)))
27		2cn 9385	. . . . . . 7 ⊢ 2 ∈ ℂ
28		2ap0 9407	. . . . . . 7 ⊢ 2 # 0
29		eqid 2238	. . . . . . . 8 ⊢ (𝑦 ∈ ℂ ↦ (𝑦 / 2)) = (𝑦 ∈ ℂ ↦ (𝑦 / 2))
30	29	divccncfap 15828	. . . . . . 7 ⊢ ((2 ∈ ℂ ∧ 2 # 0) → (𝑦 ∈ ℂ ↦ (𝑦 / 2)) ∈ (ℂ–cn→ℂ))
31	27, 28, 30	mp2an 430	. . . . . 6 ⊢ (𝑦 ∈ ℂ ↦ (𝑦 / 2)) ∈ (ℂ–cn→ℂ)
32	31	a1i 9	. . . . 5 ⊢ (⊤ → (𝑦 ∈ ℂ ↦ (𝑦 / 2)) ∈ (ℂ–cn→ℂ))
33	17, 32	cncfco 15829	. . . 4 ⊢ (⊤ → ((𝑦 ∈ ℂ ↦ (𝑦 / 2)) ∘ (𝑥 ∈ ℂ ↦ ((exp‘(i · 𝑥)) + (exp‘( -i · 𝑥))))) ∈ (ℂ–cn→ℂ))
34	26, 33	eqeltrrd 2316	. . 3 ⊢ (⊤ → (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) + (exp‘( -i · 𝑥))) / 2)) ∈ (ℂ–cn→ℂ))
35	34	mptru 1411	. 2 ⊢ (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) + (exp‘( -i · 𝑥))) / 2)) ∈ (ℂ–cn→ℂ)
36	1, 35	eqeltri 2311	1 ⊢ cos ∈ (ℂ–cn→ℂ)

Colors of variables: wff set class

This proof depends on syntax axioms: ⊤wtru 1403 ∈ wcel 2209 ∀wral 2528 class class class wbr 4131 ↦ cmpt 4193 ∘ ccom 4780 ⟶wf 5376 ‘cfv 5380 (class class class)co 6088 ℂcc 8184 0cc0 8186 ici 8188 + caddc 8189 · cmul 8191 − cmin 8505 -cneg 8506 # cap 8918 / cdiv 9011 2c2 9365 abscabs 11786 expce 12435 cosccos 12438 MetOpencmopn 14988 Cn ccn 15423 ×_t ctx 15490 –cn→ccncf 15808

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4248 ax-sep 4251 ax-nul 4261 ax-pow 4313 ax-pr 4348 ax-un 4580 ax-setind 4686 ax-iinf 4737 ax-cnex 8277 ax-resscn 8278 ax-1cn 8279 ax-1re 8280 ax-icn 8281 ax-addcl 8282 ax-addrcl 8283 ax-mulcl 8284 ax-mulrcl 8285 ax-addcom 8286 ax-mulcom 8287 ax-addass 8288 ax-mulass 8289 ax-distr 8290 ax-i2m1 8291 ax-0lt1 8292 ax-1rid 8293 ax-0id 8294 ax-rnegex 8295 ax-precex 8296 ax-cnre 8297 ax-pre-ltirr 8298 ax-pre-ltwlin 8299 ax-pre-lttrn 8300 ax-pre-apti 8301 ax-pre-ltadd 8302 ax-pre-mulgt0 8303 ax-pre-mulext 8304 ax-arch 8305 ax-caucvg 8306 ax-addf 8308 ax-mulf 8309

This proof depends on definitions: df-bi 117 df-stab 843 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rmo 2536 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3639 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-iun 4014 df-disj 4107 df-br 4132 df-opab 4194 df-mpt 4195 df-tr 4232 df-id 4440 df-po 4443 df-iso 4444 df-iord 4513 df-on 4515 df-ilim 4516 df-suc 4518 df-iom 4740 df-xp 4782 df-rel 4783 df-cnv 4784 df-co 4785 df-dm 4786 df-rn 4787 df-res 4788 df-ima 4789 df-iota 5340 df-fun 5382 df-fn 5383 df-f 5384 df-f1 5385 df-fo 5386 df-f1o 5387 df-fv 5388 df-isom 5389 df-riota 6041 df-ov 6091 df-oprab 6092 df-mpo 6093 df-of 6305 df-1st 6378 df-2nd 6379 df-recs 6580 df-irdg 6645 df-frec 6666 df-1o 6691 df-oadd 6695 df-er 6811 df-map 6928 df-pm 6929 df-en 7028 df-dom 7029 df-fin 7030 df-sup 7331 df-inf 7332 df-pnf 8369 df-mnf 8370 df-xr 8371 df-ltxr 8372 df-le 8373 df-sub 8507 df-neg 8508 df-reap 8912 df-ap 8919 df-div 9012 df-inn 9315 df-2 9373 df-3 9374 df-4 9375 df-n0 9576 df-z 9657 df-uz 9939 df-q 10037 df-rp 10073 df-xneg 10192 df-xadd 10193 df-ico 10314 df-fz 10430 df-fzo 10568 df-seqfrec 10907 df-exp 10998 df-fac 11187 df-bc 11209 df-ihash 11238 df-shft 11603 df-cj 11630 df-re 11631 df-im 11632 df-rsqrt 11787 df-abs 11788 df-clim 12071 df-sumdc 12146 df-ef 12441 df-cos 12444 df-rest 13655 df-topgen 13674 df-psmet 14990 df-xmet 14991 df-met 14992 df-bl 14993 df-mopn 14994 df-top 15236 df-topon 15249 df-bases 15281 df-ntr 15334 df-cn 15426 df-cnp 15427 df-tx 15491 df-cncf 15809 df-limced 15894 df-dvap 15895

This theorem is used by: cosz12 16019 ioocosf1o 16094

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