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Theorem sincn 15113

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 11834	. 2 ⊢ sin = (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) − (exp‘(-i · 𝑥))) / (2 · i)))
2		eqid 2196	. . . . . . . 8 ⊢ (MetOpen‘(abs ∘ − )) = (MetOpen‘(abs ∘ − ))
3	2	subcncntop 14907	. . . . . . . . 9 ⊢ − ∈ (((MetOpen‘(abs ∘ − )) ×_t (MetOpen‘(abs ∘ − ))) Cn (MetOpen‘(abs ∘ − )))
4	3	a1i 9	. . . . . . . 8 ⊢ (⊤ → − ∈ (((MetOpen‘(abs ∘ − )) ×_t (MetOpen‘(abs ∘ − ))) Cn (MetOpen‘(abs ∘ − ))))
5		efcn 15112	. . . . . . . . . 10 ⊢ exp ∈ (ℂ–cn→ℂ)
6	5	a1i 9	. . . . . . . . 9 ⊢ (⊤ → exp ∈ (ℂ–cn→ℂ))
7		ax-icn 7993	. . . . . . . . . 10 ⊢ i ∈ ℂ
8		eqid 2196	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℂ ↦ (i · 𝑥)) = (𝑥 ∈ ℂ ↦ (i · 𝑥))
9	8	mulc1cncf 14933	. . . . . . . . . 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 14942	. . . . . . . 8 ⊢ (⊤ → (𝑥 ∈ ℂ ↦ (exp‘(i · 𝑥))) ∈ (ℂ–cn→ℂ))
12		negicn 8246	. . . . . . . . . 10 ⊢ -i ∈ ℂ
13		eqid 2196	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℂ ↦ (-i · 𝑥)) = (𝑥 ∈ ℂ ↦ (-i · 𝑥))
14	13	mulc1cncf 14933	. . . . . . . . . 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 14942	. . . . . . . 8 ⊢ (⊤ → (𝑥 ∈ ℂ ↦ (exp‘(-i · 𝑥))) ∈ (ℂ–cn→ℂ))
17	2, 4, 11, 16	cncfmpt2fcntop 14943	. . . . . . 7 ⊢ (⊤ → (𝑥 ∈ ℂ ↦ ((exp‘(i · 𝑥)) − (exp‘(-i · 𝑥)))) ∈ (ℂ–cn→ℂ))
18		cncff 14921	. . . . . . 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 2196	. . . . . . 7 ⊢ (𝑥 ∈ ℂ ↦ ((exp‘(i · 𝑥)) − (exp‘(-i · 𝑥)))) = (𝑥 ∈ ℂ ↦ ((exp‘(i · 𝑥)) − (exp‘(-i · 𝑥))))
21	20	fmpt 5715	. . . . . 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 2197	. . . . 5 ⊢ (⊤ → (𝑥 ∈ ℂ ↦ ((exp‘(i · 𝑥)) − (exp‘(-i · 𝑥)))) = (𝑥 ∈ ℂ ↦ ((exp‘(i · 𝑥)) − (exp‘(-i · 𝑥)))))
24		eqidd 2197	. . . . 5 ⊢ (⊤ → (𝑦 ∈ ℂ ↦ (𝑦 / (2 · i))) = (𝑦 ∈ ℂ ↦ (𝑦 / (2 · i))))
25		oveq1 5932	. . . . 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 5732	. . . 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 9232	. . . . . . 7 ⊢ (2 · i) ∈ ℂ
28		2muliap0 9234	. . . . . . 7 ⊢ (2 · i) # 0
29		eqid 2196	. . . . . . . 8 ⊢ (𝑦 ∈ ℂ ↦ (𝑦 / (2 · i))) = (𝑦 ∈ ℂ ↦ (𝑦 / (2 · i)))
30	29	divccncfap 14934	. . . . . . 7 ⊢ (((2 · i) ∈ ℂ ∧ (2 · i) # 0) → (𝑦 ∈ ℂ ↦ (𝑦 / (2 · i))) ∈ (ℂ–cn→ℂ))
31	27, 28, 30	mp2an 426	. . . . . 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 14935	. . . 4 ⊢ (⊤ → ((𝑦 ∈ ℂ ↦ (𝑦 / (2 · i))) ∘ (𝑥 ∈ ℂ ↦ ((exp‘(i · 𝑥)) − (exp‘(-i · 𝑥))))) ∈ (ℂ–cn→ℂ))
34	26, 33	eqeltrrd 2274	. . 3 ⊢ (⊤ → (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) − (exp‘(-i · 𝑥))) / (2 · i))) ∈ (ℂ–cn→ℂ))
35	34	mptru 1373	. 2 ⊢ (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) − (exp‘(-i · 𝑥))) / (2 · i))) ∈ (ℂ–cn→ℂ)
36	1, 35	eqeltri 2269	1 ⊢ sin ∈ (ℂ–cn→ℂ)

Colors of variables: wff set class

Syntax hints: ⊤wtru 1365 ∈ wcel 2167 ∀wral 2475 class class class wbr 4034 ↦ cmpt 4095 ∘ ccom 4668 ⟶wf 5255 ‘cfv 5259 (class class class)co 5925 ℂcc 7896 0cc0 7898 ici 7900 · cmul 7903 − cmin 8216 -cneg 8217 # cap 8627 / cdiv 8718 2c2 9060 abscabs 11181 expce 11826 sincsin 11828 MetOpencmopn 14175 Cn ccn 14529 ×_t ctx 14596 –cn→ccncf 14914

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 615 ax-in2 616 ax-io 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-coll 4149 ax-sep 4152 ax-nul 4160 ax-pow 4208 ax-pr 4243 ax-un 4469 ax-setind 4574 ax-iinf 4625 ax-cnex 7989 ax-resscn 7990 ax-1cn 7991 ax-1re 7992 ax-icn 7993 ax-addcl 7994 ax-addrcl 7995 ax-mulcl 7996 ax-mulrcl 7997 ax-addcom 7998 ax-mulcom 7999 ax-addass 8000 ax-mulass 8001 ax-distr 8002 ax-i2m1 8003 ax-0lt1 8004 ax-1rid 8005 ax-0id 8006 ax-rnegex 8007 ax-precex 8008 ax-cnre 8009 ax-pre-ltirr 8010 ax-pre-ltwlin 8011 ax-pre-lttrn 8012 ax-pre-apti 8013 ax-pre-ltadd 8014 ax-pre-mulgt0 8015 ax-pre-mulext 8016 ax-arch 8017 ax-caucvg 8018 ax-addf 8020 ax-mulf 8021

This theorem depends on definitions: df-bi 117 df-stab 832 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-nel 2463 df-ral 2480 df-rex 2481 df-reu 2482 df-rmo 2483 df-rab 2484 df-v 2765 df-sbc 2990 df-csb 3085 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-nul 3452 df-if 3563 df-pw 3608 df-sn 3629 df-pr 3630 df-op 3632 df-uni 3841 df-int 3876 df-iun 3919 df-disj 4012 df-br 4035 df-opab 4096 df-mpt 4097 df-tr 4133 df-id 4329 df-po 4332 df-iso 4333 df-iord 4402 df-on 4404 df-ilim 4405 df-suc 4407 df-iom 4628 df-xp 4670 df-rel 4671 df-cnv 4672 df-co 4673 df-dm 4674 df-rn 4675 df-res 4676 df-ima 4677 df-iota 5220 df-fun 5261 df-fn 5262 df-f 5263 df-f1 5264 df-fo 5265 df-f1o 5266 df-fv 5267 df-isom 5268 df-riota 5880 df-ov 5928 df-oprab 5929 df-mpo 5930 df-of 6139 df-1st 6207 df-2nd 6208 df-recs 6372 df-irdg 6437 df-frec 6458 df-1o 6483 df-oadd 6487 df-er 6601 df-map 6718 df-pm 6719 df-en 6809 df-dom 6810 df-fin 6811 df-sup 7059 df-inf 7060 df-pnf 8082 df-mnf 8083 df-xr 8084 df-ltxr 8085 df-le 8086 df-sub 8218 df-neg 8219 df-reap 8621 df-ap 8628 df-div 8719 df-inn 9010 df-2 9068 df-3 9069 df-4 9070 df-n0 9269 df-z 9346 df-uz 9621 df-q 9713 df-rp 9748 df-xneg 9866 df-xadd 9867 df-ico 9988 df-fz 10103 df-fzo 10237 df-seqfrec 10559 df-exp 10650 df-fac 10837 df-bc 10859 df-ihash 10887 df-shft 10999 df-cj 11026 df-re 11027 df-im 11028 df-rsqrt 11182 df-abs 11183 df-clim 11463 df-sumdc 11538 df-ef 11832 df-sin 11834 df-rest 12945 df-topgen 12964 df-psmet 14177 df-xmet 14178 df-met 14179 df-bl 14180 df-mopn 14181 df-top 14342 df-topon 14355 df-bases 14387 df-ntr 14440 df-cn 14532 df-cnp 14533 df-tx 14597 df-cncf 14915 df-limced 15000 df-dvap 15001

This theorem is referenced by: (None)

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