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Theorem sincn 15870

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

**Assertion**
Ref	Expression
sincn	⊢ sin ∈ (ℂ–cn→ℂ)

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

Step	Hyp	Ref	Expression
1		df-sin 12417	. 2 ⊢ sin = (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) − (exp‘(-i · 𝑥))) / (2 · i)))
2		eqid 2238	. . . . . . . 8 ⊢ (MetOpen‘(abs ∘ − )) = (MetOpen‘(abs ∘ − ))
3	2	subcncntop 15664	. . . . . . . . 9 ⊢ − ∈ (((MetOpen‘(abs ∘ − )) ×_t (MetOpen‘(abs ∘ − ))) Cn (MetOpen‘(abs ∘ − )))
4	3	a1i 9	. . . . . . . 8 ⊢ (⊤ → − ∈ (((MetOpen‘(abs ∘ − )) ×_t (MetOpen‘(abs ∘ − ))) Cn (MetOpen‘(abs ∘ − ))))
5		efcn 15869	. . . . . . . . . 10 ⊢ exp ∈ (ℂ–cn→ℂ)
6	5	a1i 9	. . . . . . . . 9 ⊢ (⊤ → exp ∈ (ℂ–cn→ℂ))
7		ax-icn 8274	. . . . . . . . . 10 ⊢ i ∈ ℂ
8		eqid 2238	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℂ ↦ (i · 𝑥)) = (𝑥 ∈ ℂ ↦ (i · 𝑥))
9	8	mulc1cncf 15690	. . . . . . . . . 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 15699	. . . . . . . 8 ⊢ (⊤ → (𝑥 ∈ ℂ ↦ (exp‘(i · 𝑥))) ∈ (ℂ–cn→ℂ))
12		negicn 8527	. . . . . . . . . 10 ⊢ -i ∈ ℂ
13		eqid 2238	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℂ ↦ (-i · 𝑥)) = (𝑥 ∈ ℂ ↦ (-i · 𝑥))
14	13	mulc1cncf 15690	. . . . . . . . . 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 15699	. . . . . . . 8 ⊢ (⊤ → (𝑥 ∈ ℂ ↦ (exp‘(-i · 𝑥))) ∈ (ℂ–cn→ℂ))
17	2, 4, 11, 16	cncfmpt2fcntop 15700	. . . . . . 7 ⊢ (⊤ → (𝑥 ∈ ℂ ↦ ((exp‘(i · 𝑥)) − (exp‘(-i · 𝑥)))) ∈ (ℂ–cn→ℂ))
18		cncff 15678	. . . . . . 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 5858	. . . . . 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 · i))) = (𝑦 ∈ ℂ ↦ (𝑦 / (2 · i))))
25		oveq1 6092	. . . . 5 ⊢ (𝑦 = ((exp‘(i · 𝑥)) − (exp‘(-i · 𝑥))) → (𝑦 / (2 · i)) = (((exp‘(i · 𝑥)) − (exp‘(-i · 𝑥))) / (2 · i)))
26	22, 23, 24, 25	fmptcof 5875	. . . 4 ⊢ (⊤ → ((𝑦 ∈ ℂ ↦ (𝑦 / (2 · i))) ∘ (𝑥 ∈ ℂ ↦ ((exp‘(i · 𝑥)) − (exp‘(-i · 𝑥))))) = (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) − (exp‘(-i · 𝑥))) / (2 · i))))
27		2mulicn 9527	. . . . . . 7 ⊢ (2 · i) ∈ ℂ
28		2muliap0 9529	. . . . . . 7 ⊢ (2 · i) # 0
29		eqid 2238	. . . . . . . 8 ⊢ (𝑦 ∈ ℂ ↦ (𝑦 / (2 · i))) = (𝑦 ∈ ℂ ↦ (𝑦 / (2 · i)))
30	29	divccncfap 15691	. . . . . . 7 ⊢ (((2 · i) ∈ ℂ ∧ (2 · i) # 0) → (𝑦 ∈ ℂ ↦ (𝑦 / (2 · i))) ∈ (ℂ–cn→ℂ))
31	27, 28, 30	mp2an 430	. . . . . 6 ⊢ (𝑦 ∈ ℂ ↦ (𝑦 / (2 · i))) ∈ (ℂ–cn→ℂ)
32	31	a1i 9	. . . . 5 ⊢ (⊤ → (𝑦 ∈ ℂ ↦ (𝑦 / (2 · i))) ∈ (ℂ–cn→ℂ))
33	17, 32	cncfco 15692	. . . 4 ⊢ (⊤ → ((𝑦 ∈ ℂ ↦ (𝑦 / (2 · i))) ∘ (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 · i))) ∈ (ℂ–cn→ℂ))
35	34	mptru 1411	. 2 ⊢ (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) − (exp‘(-i · 𝑥))) / (2 · i))) ∈ (ℂ–cn→ℂ)
36	1, 35	eqeltri 2311	1 ⊢ sin ∈ (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 4130 ↦ cmpt 4192 ∘ ccom 4778 ⟶wf 5373 ‘cfv 5377 (class class class)co 6085 ℂcc 8177 0cc0 8179 ici 8181 · cmul 8184 − cmin 8497 -cneg 8498 # cap 8909 / cdiv 9002 2c2 9355 abscabs 11763 expce 12409 sincsin 12411 MetOpencmopn 14878 Cn ccn 15286 ×_t ctx 15353 –cn→ccncf 15671

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 4246 ax-sep 4249 ax-nul 4259 ax-pow 4311 ax-pr 4346 ax-un 4578 ax-setind 4684 ax-iinf 4735 ax-cnex 8270 ax-resscn 8271 ax-1cn 8272 ax-1re 8273 ax-icn 8274 ax-addcl 8275 ax-addrcl 8276 ax-mulcl 8277 ax-mulrcl 8278 ax-addcom 8279 ax-mulcom 8280 ax-addass 8281 ax-mulass 8282 ax-distr 8283 ax-i2m1 8284 ax-0lt1 8285 ax-1rid 8286 ax-0id 8287 ax-rnegex 8288 ax-precex 8289 ax-cnre 8290 ax-pre-ltirr 8291 ax-pre-ltwlin 8292 ax-pre-lttrn 8293 ax-pre-apti 8294 ax-pre-ltadd 8295 ax-pre-mulgt0 8296 ax-pre-mulext 8297 ax-arch 8298 ax-caucvg 8299 ax-addf 8301 ax-mulf 8302

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 4131 df-opab 4193 df-mpt 4194 df-tr 4230 df-id 4438 df-po 4441 df-iso 4442 df-iord 4511 df-on 4513 df-ilim 4514 df-suc 4516 df-iom 4738 df-xp 4780 df-rel 4781 df-cnv 4782 df-co 4783 df-dm 4784 df-rn 4785 df-res 4786 df-ima 4787 df-iota 5337 df-fun 5379 df-fn 5380 df-f 5381 df-f1 5382 df-fo 5383 df-f1o 5384 df-fv 5385 df-isom 5386 df-riota 6038 df-ov 6088 df-oprab 6089 df-mpo 6090 df-of 6302 df-1st 6374 df-2nd 6375 df-recs 6576 df-irdg 6641 df-frec 6662 df-1o 6687 df-oadd 6691 df-er 6807 df-map 6924 df-pm 6925 df-en 7023 df-dom 7024 df-fin 7025 df-sup 7324 df-inf 7325 df-pnf 8362 df-mnf 8363 df-xr 8364 df-ltxr 8365 df-le 8366 df-sub 8499 df-neg 8500 df-reap 8903 df-ap 8910 df-div 9003 df-inn 9305 df-2 9363 df-3 9364 df-4 9365 df-n0 9564 df-z 9645 df-uz 9922 df-q 10020 df-rp 10055 df-xneg 10174 df-xadd 10175 df-ico 10296 df-fz 10412 df-fzo 10550 df-seqfrec 10885 df-exp 10976 df-fac 11164 df-bc 11186 df-ihash 11215 df-shft 11580 df-cj 11607 df-re 11608 df-im 11609 df-rsqrt 11764 df-abs 11765 df-clim 12045 df-sumdc 12120 df-ef 12415 df-sin 12417 df-rest 13595 df-topgen 13614 df-psmet 14880 df-xmet 14881 df-met 14882 df-bl 14883 df-mopn 14884 df-top 15099 df-topon 15112 df-bases 15144 df-ntr 15197 df-cn 15289 df-cnp 15290 df-tx 15354 df-cncf 15672 df-limced 15757 df-dvap 15758

This theorem is used by: (None)

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