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Theorem sincn 26352

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 15976	. 2 ⊢ sin = (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) − (exp‘(-i · 𝑥))) / (2 · i)))
2		eqid 2729	. . . . . . . 8 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
3	2	subcn 24753	. . . . . . . . 9 ⊢ − ∈ (((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld)) Cn (TopOpen‘ℂ_fld))
4	3	a1i 11	. . . . . . . 8 ⊢ (⊤ → − ∈ (((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld)) Cn (TopOpen‘ℂ_fld)))
5		efcn 26351	. . . . . . . . . 10 ⊢ exp ∈ (ℂ–cn→ℂ)
6	5	a1i 11	. . . . . . . . 9 ⊢ (⊤ → exp ∈ (ℂ–cn→ℂ))
7		ax-icn 11068	. . . . . . . . . 10 ⊢ i ∈ ℂ
8		eqid 2729	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℂ ↦ (i · 𝑥)) = (𝑥 ∈ ℂ ↦ (i · 𝑥))
9	8	mulc1cncf 24796	. . . . . . . . . 10 ⊢ (i ∈ ℂ → (𝑥 ∈ ℂ ↦ (i · 𝑥)) ∈ (ℂ–cn→ℂ))
10	7, 9	mp1i 13	. . . . . . . . 9 ⊢ (⊤ → (𝑥 ∈ ℂ ↦ (i · 𝑥)) ∈ (ℂ–cn→ℂ))
11	6, 10	cncfmpt1f 24805	. . . . . . . 8 ⊢ (⊤ → (𝑥 ∈ ℂ ↦ (exp‘(i · 𝑥))) ∈ (ℂ–cn→ℂ))
12		negicn 11364	. . . . . . . . . 10 ⊢ -i ∈ ℂ
13		eqid 2729	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℂ ↦ (-i · 𝑥)) = (𝑥 ∈ ℂ ↦ (-i · 𝑥))
14	13	mulc1cncf 24796	. . . . . . . . . 10 ⊢ (-i ∈ ℂ → (𝑥 ∈ ℂ ↦ (-i · 𝑥)) ∈ (ℂ–cn→ℂ))
15	12, 14	mp1i 13	. . . . . . . . 9 ⊢ (⊤ → (𝑥 ∈ ℂ ↦ (-i · 𝑥)) ∈ (ℂ–cn→ℂ))
16	6, 15	cncfmpt1f 24805	. . . . . . . 8 ⊢ (⊤ → (𝑥 ∈ ℂ ↦ (exp‘(-i · 𝑥))) ∈ (ℂ–cn→ℂ))
17	2, 4, 11, 16	cncfmpt2f 24806	. . . . . . 7 ⊢ (⊤ → (𝑥 ∈ ℂ ↦ ((exp‘(i · 𝑥)) − (exp‘(-i · 𝑥)))) ∈ (ℂ–cn→ℂ))
18		cncff 24784	. . . . . . 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 17	. . . . . 6 ⊢ (⊤ → (𝑥 ∈ ℂ ↦ ((exp‘(i · 𝑥)) − (exp‘(-i · 𝑥)))):ℂ⟶ℂ)
20		eqid 2729	. . . . . . 7 ⊢ (𝑥 ∈ ℂ ↦ ((exp‘(i · 𝑥)) − (exp‘(-i · 𝑥)))) = (𝑥 ∈ ℂ ↦ ((exp‘(i · 𝑥)) − (exp‘(-i · 𝑥))))
21	20	fmpt 7044	. . . . . 6 ⊢ (∀𝑥 ∈ ℂ ((exp‘(i · 𝑥)) − (exp‘(-i · 𝑥))) ∈ ℂ ↔ (𝑥 ∈ ℂ ↦ ((exp‘(i · 𝑥)) − (exp‘(-i · 𝑥)))):ℂ⟶ℂ)
22	19, 21	sylibr 234	. . . . 5 ⊢ (⊤ → ∀𝑥 ∈ ℂ ((exp‘(i · 𝑥)) − (exp‘(-i · 𝑥))) ∈ ℂ)
23		eqidd 2730	. . . . 5 ⊢ (⊤ → (𝑥 ∈ ℂ ↦ ((exp‘(i · 𝑥)) − (exp‘(-i · 𝑥)))) = (𝑥 ∈ ℂ ↦ ((exp‘(i · 𝑥)) − (exp‘(-i · 𝑥)))))
24		eqidd 2730	. . . . 5 ⊢ (⊤ → (𝑦 ∈ ℂ ↦ (𝑦 / (2 · i))) = (𝑦 ∈ ℂ ↦ (𝑦 / (2 · i))))
25		oveq1 7356	. . . . 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 7064	. . . 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 12348	. . . . . . 7 ⊢ (2 · i) ∈ ℂ
28		2muline0 12349	. . . . . . 7 ⊢ (2 · i) ≠ 0
29		eqid 2729	. . . . . . . 8 ⊢ (𝑦 ∈ ℂ ↦ (𝑦 / (2 · i))) = (𝑦 ∈ ℂ ↦ (𝑦 / (2 · i)))
30	29	divccncf 24797	. . . . . . 7 ⊢ (((2 · i) ∈ ℂ ∧ (2 · i) ≠ 0) → (𝑦 ∈ ℂ ↦ (𝑦 / (2 · i))) ∈ (ℂ–cn→ℂ))
31	27, 28, 30	mp2an 692	. . . . . 6 ⊢ (𝑦 ∈ ℂ ↦ (𝑦 / (2 · i))) ∈ (ℂ–cn→ℂ)
32	31	a1i 11	. . . . 5 ⊢ (⊤ → (𝑦 ∈ ℂ ↦ (𝑦 / (2 · i))) ∈ (ℂ–cn→ℂ))
33	17, 32	cncfco 24798	. . . 4 ⊢ (⊤ → ((𝑦 ∈ ℂ ↦ (𝑦 / (2 · i))) ∘ (𝑥 ∈ ℂ ↦ ((exp‘(i · 𝑥)) − (exp‘(-i · 𝑥))))) ∈ (ℂ–cn→ℂ))
34	26, 33	eqeltrrd 2829	. . 3 ⊢ (⊤ → (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) − (exp‘(-i · 𝑥))) / (2 · i))) ∈ (ℂ–cn→ℂ))
35	34	mptru 1547	. 2 ⊢ (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) − (exp‘(-i · 𝑥))) / (2 · i))) ∈ (ℂ–cn→ℂ)
36	1, 35	eqeltri 2824	1 ⊢ sin ∈ (ℂ–cn→ℂ)

Colors of variables: wff setvar class

Syntax hints: ⊤wtru 1541 ∈ wcel 2109 ≠ wne 2925 ∀wral 3044 ↦ cmpt 5173 ∘ ccom 5623 ⟶wf 6478 ‘cfv 6482 (class class class)co 7349 ℂcc 11007 0cc0 11009 ici 11011 · cmul 11014 − cmin 11347 -cneg 11348 / cdiv 11777 2c2 12183 expce 15968 sincsin 15970 TopOpenctopn 17325 ℂ_fldccnfld 21261 Cn ccn 23109 ×_t ctx 23445 –cn→ccncf 24767

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5218 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-inf2 9537 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-pre-sup 11087 ax-addf 11088

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3343 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-uni 4859 df-int 4897 df-iun 4943 df-iin 4944 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-se 5573 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6249 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-isom 6491 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-of 7613 df-om 7800 df-1st 7924 df-2nd 7925 df-supp 8094 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-1o 8388 df-2o 8389 df-er 8625 df-map 8755 df-pm 8756 df-ixp 8825 df-en 8873 df-dom 8874 df-sdom 8875 df-fin 8876 df-fsupp 9252 df-fi 9301 df-sup 9332 df-inf 9333 df-oi 9402 df-card 9835 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-sub 11349 df-neg 11350 df-div 11778 df-nn 12129 df-2 12191 df-3 12192 df-4 12193 df-5 12194 df-6 12195 df-7 12196 df-8 12197 df-9 12198 df-n0 12385 df-z 12472 df-dec 12592 df-uz 12736 df-q 12850 df-rp 12894 df-xneg 13014 df-xadd 13015 df-xmul 13016 df-ico 13254 df-icc 13255 df-fz 13411 df-fzo 13558 df-fl 13696 df-seq 13909 df-exp 13969 df-fac 14181 df-bc 14210 df-hash 14238 df-shft 14974 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-limsup 15378 df-clim 15395 df-rlim 15396 df-sum 15594 df-ef 15974 df-sin 15976 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-starv 17176 df-sca 17177 df-vsca 17178 df-ip 17179 df-tset 17180 df-ple 17181 df-ds 17183 df-unif 17184 df-hom 17185 df-cco 17186 df-rest 17326 df-topn 17327 df-0g 17345 df-gsum 17346 df-topgen 17347 df-pt 17348 df-prds 17351 df-xrs 17406 df-qtop 17411 df-imas 17412 df-xps 17414 df-mre 17488 df-mrc 17489 df-acs 17491 df-mgm 18514 df-sgrp 18593 df-mnd 18609 df-submnd 18658 df-mulg 18947 df-cntz 19196 df-cmn 19661 df-psmet 21253 df-xmet 21254 df-met 21255 df-bl 21256 df-mopn 21257 df-fbas 21258 df-fg 21259 df-cnfld 21262 df-top 22779 df-topon 22796 df-topsp 22818 df-bases 22831 df-cld 22904 df-ntr 22905 df-cls 22906 df-nei 22983 df-lp 23021 df-perf 23022 df-cn 23112 df-cnp 23113 df-haus 23200 df-tx 23447 df-hmeo 23640 df-fil 23731 df-fm 23823 df-flim 23824 df-flf 23825 df-xms 24206 df-ms 24207 df-tms 24208 df-cncf 24769 df-limc 25765 df-dv 25766

This theorem is referenced by: pilem3 26361 resuppsinopn 42336 resincncf 45856 itgsin0pilem1 45931 ibliccsinexp 45932 itgsinexplem1 45935 itgsinexp 45936 itgcoscmulx 45950 itgsincmulx 45955 dirkeritg 46083 dirkercncflem2 46085 dirkercncflem4 46087 fourierdlem21 46109 fourierdlem22 46110 fourierdlem39 46127 fourierdlem62 46149 fourierdlem68 46155 fourierdlem73 46160 fourierdlem76 46163 fourierdlem78 46165 fourierdlem83 46170 sqwvfourb 46210

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