Theorem sineq0re 16088

Description: A complex number whose sine is zero is real. (Contributed by NM, 17-Aug-2008.) (Revised by Mario Carneiro, 10-May-2014.) (Revised by Jim Kingdon, 23-Sep-2026.)

Assertion

Ref	Expression
sineq0re	|- ( ( A e. CC /\ ( sin ` A ) = 0 ) -> A e. RR )

Proof of Theorem sineq0re

Step	Hyp	Ref	Expression
1		sinval 12495	. . . . . 6 |- ( A e. CC -> ( sin ` A ) = ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) / ( 2 x. _i ) ) )
2	1	eqeq1d 2247	. . . . 5 |- ( A e. CC -> ( ( sin ` A ) = 0 <-> ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) / ( 2 x. _i ) ) = 0 ) )
3		ax-icn 8281	. . . . . . . . 9 |- _i e. CC
4		mulcl 8313	. . . . . . . . 9 |- ( ( _i e. CC /\ A e. CC ) -> ( _i x. A ) e. CC )
5	3, 4	mpan 428	. . . . . . . 8 |- ( A e. CC -> ( _i x. A ) e. CC )
6		efcl 12457	. . . . . . . 8 |- ( ( _i x. A ) e. CC -> ( exp ` ( _i x. A ) ) e. CC )
7	5, 6	syl 14	. . . . . . 7 |- ( A e. CC -> ( exp ` ( _i x. A ) ) e. CC )
8		negicn 8535	. . . . . . . . 9 |- -u _i e. CC
9		mulcl 8313	. . . . . . . . 9 |- ( ( -u _i e. CC /\ A e. CC ) -> ( -u _i x. A ) e. CC )
10	8, 9	mpan 428	. . . . . . . 8 |- ( A e. CC -> ( -u _i x. A ) e. CC )
11		efcl 12457	. . . . . . . 8 |- ( ( -u _i x. A ) e. CC -> ( exp ` ( -u _i x. A ) ) e. CC )
12	10, 11	syl 14	. . . . . . 7 |- ( A e. CC -> ( exp ` ( -u _i x. A ) ) e. CC )
13	7, 12	subcld 8645	. . . . . 6 |- ( A e. CC -> ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) e. CC )
14		2mulicn 9539	. . . . . . 7 |- ( 2 x. _i ) e. CC
15		2muliap0 9541	. . . . . . 7 |- ( 2 x. _i ) # 0
16		diveqap0 9021	. . . . . . 7 |- ( ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) e. CC /\ ( 2 x. _i ) e. CC /\ ( 2 x. _i ) # 0 ) -> ( ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) / ( 2 x. _i ) ) = 0 <-> ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) = 0 ) )
17	14, 15, 16	mp3an23 1370	. . . . . 6 |- ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) e. CC -> ( ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) / ( 2 x. _i ) ) = 0 <-> ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) = 0 ) )
18	13, 17	syl 14	. . . . 5 |- ( A e. CC -> ( ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) / ( 2 x. _i ) ) = 0 <-> ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) = 0 ) )
19	7, 12	subeq0ad 8655	. . . . 5 |- ( A e. CC -> ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) = 0 <-> ( exp ` ( _i x. A ) ) = ( exp ` ( -u _i x. A ) ) ) )
20	2, 18, 19	3bitrd 214	. . . 4 |- ( A e. CC -> ( ( sin ` A ) = 0 <-> ( exp ` ( _i x. A ) ) = ( exp ` ( -u _i x. A ) ) ) )
21		oveq2 6096	. . . . 5 |- ( ( exp ` ( _i x. A ) ) = ( exp ` ( -u _i x. A ) ) -> ( ( exp ` ( _i x. A ) ) x. ( exp ` ( _i x. A ) ) ) = ( ( exp ` ( _i x. A ) ) x. ( exp ` ( -u _i x. A ) ) ) )
22		2cn 9385	. . . . . . . . . . 11 |- 2 e. CC
23		mul12 8463	. . . . . . . . . . 11 |- ( ( _i e. CC /\ 2 e. CC /\ A e. CC ) -> ( _i x. ( 2 x. A ) ) = ( 2 x. ( _i x. A ) ) )
24	3, 22, 23	mp3an12 1368	. . . . . . . . . 10 |- ( A e. CC -> ( _i x. ( 2 x. A ) ) = ( 2 x. ( _i x. A ) ) )
25	5	2timesd 9560	. . . . . . . . . 10 |- ( A e. CC -> ( 2 x. ( _i x. A ) ) = ( ( _i x. A ) + ( _i x. A ) ) )
26	24, 25	eqtrd 2271	. . . . . . . . 9 |- ( A e. CC -> ( _i x. ( 2 x. A ) ) = ( ( _i x. A ) + ( _i x. A ) ) )
27	26	fveq2d 5702	. . . . . . . 8 |- ( A e. CC -> ( exp ` ( _i x. ( 2 x. A ) ) ) = ( exp ` ( ( _i x. A ) + ( _i x. A ) ) ) )
28		efadd 12468	. . . . . . . . 9 |- ( ( ( _i x. A ) e. CC /\ ( _i x. A ) e. CC ) -> ( exp ` ( ( _i x. A ) + ( _i x. A ) ) ) = ( ( exp ` ( _i x. A ) ) x. ( exp ` ( _i x. A ) ) ) )
29	5, 5, 28	syl2anc 415	. . . . . . . 8 |- ( A e. CC -> ( exp ` ( ( _i x. A ) + ( _i x. A ) ) ) = ( ( exp ` ( _i x. A ) ) x. ( exp ` ( _i x. A ) ) ) )
30	27, 29	eqtr2d 2272	. . . . . . 7 |- ( A e. CC -> ( ( exp ` ( _i x. A ) ) x. ( exp ` ( _i x. A ) ) ) = ( exp ` ( _i x. ( 2 x. A ) ) ) )
31		efadd 12468	. . . . . . . . 9 |- ( ( ( _i x. A ) e. CC /\ ( -u _i x. A ) e. CC ) -> ( exp ` ( ( _i x. A ) + ( -u _i x. A ) ) ) = ( ( exp ` ( _i x. A ) ) x. ( exp ` ( -u _i x. A ) ) ) )
32	5, 10, 31	syl2anc 415	. . . . . . . 8 |- ( A e. CC -> ( exp ` ( ( _i x. A ) + ( -u _i x. A ) ) ) = ( ( exp ` ( _i x. A ) ) x. ( exp ` ( -u _i x. A ) ) ) )
33	3	negidi 8603	. . . . . . . . . . . 12 |- ( _i + -u _i ) = 0
34	33	oveq1i 6098	. . . . . . . . . . 11 |- ( ( _i + -u _i ) x. A ) = ( 0 x. A )
35		adddir 8324	. . . . . . . . . . . 12 |- ( ( _i e. CC /\ -u _i e. CC /\ A e. CC ) -> ( ( _i + -u _i ) x. A ) = ( ( _i x. A ) + ( -u _i x. A ) ) )
36	3, 8, 35	mp3an12 1368	. . . . . . . . . . 11 |- ( A e. CC -> ( ( _i + -u _i ) x. A ) = ( ( _i x. A ) + ( -u _i x. A ) ) )
37		mul02 8722	. . . . . . . . . . 11 |- ( A e. CC -> ( 0 x. A ) = 0 )
38	34, 36, 37	3eqtr3a 2295	. . . . . . . . . 10 |- ( A e. CC -> ( ( _i x. A ) + ( -u _i x. A ) ) = 0 )
39	38	fveq2d 5702	. . . . . . . . 9 |- ( A e. CC -> ( exp ` ( ( _i x. A ) + ( -u _i x. A ) ) ) = ( exp ` 0 ) )
40		ef0 12465	. . . . . . . . 9 |- ( exp ` 0 ) = 1
41	39, 40	eqtrdi 2287	. . . . . . . 8 |- ( A e. CC -> ( exp ` ( ( _i x. A ) + ( -u _i x. A ) ) ) = 1 )
42	32, 41	eqtr3d 2273	. . . . . . 7 |- ( A e. CC -> ( ( exp ` ( _i x. A ) ) x. ( exp ` ( -u _i x. A ) ) ) = 1 )
43	30, 42	eqeq12d 2253	. . . . . 6 |- ( A e. CC -> ( ( ( exp ` ( _i x. A ) ) x. ( exp ` ( _i x. A ) ) ) = ( ( exp ` ( _i x. A ) ) x. ( exp ` ( -u _i x. A ) ) ) <-> ( exp ` ( _i x. ( 2 x. A ) ) ) = 1 ) )
44		fveq2 5698	. . . . . 6 |- ( ( exp ` ( _i x. ( 2 x. A ) ) ) = 1 -> ( abs ` ( exp ` ( _i x. ( 2 x. A ) ) ) ) = ( abs ` 1 ) )
45	43, 44	biimtrdi 163	. . . . 5 |- ( A e. CC -> ( ( ( exp ` ( _i x. A ) ) x. ( exp ` ( _i x. A ) ) ) = ( ( exp ` ( _i x. A ) ) x. ( exp ` ( -u _i x. A ) ) ) -> ( abs ` ( exp ` ( _i x. ( 2 x. A ) ) ) ) = ( abs ` 1 ) ) )
46	21, 45	syl5 32	. . . 4 |- ( A e. CC -> ( ( exp ` ( _i x. A ) ) = ( exp ` ( -u _i x. A ) ) -> ( abs ` ( exp ` ( _i x. ( 2 x. A ) ) ) ) = ( abs ` 1 ) ) )
47	20, 46	sylbid 150	. . 3 |- ( A e. CC -> ( ( sin ` A ) = 0 -> ( abs ` ( exp ` ( _i x. ( 2 x. A ) ) ) ) = ( abs ` 1 ) ) )
48		abs1 11861	. . . . 5 |- ( abs ` 1 ) = 1
49	48	eqeq2i 2249	. . . 4 |- ( ( abs ` ( exp ` ( _i x. ( 2 x. A ) ) ) ) = ( abs ` 1 ) <-> ( abs ` ( exp ` ( _i x. ( 2 x. A ) ) ) ) = 1 )
50		2re 9384	. . . . . 6 |- 2 e. RR
51		2ap0 9407	. . . . . 6 |- 2 # 0
52		mulreap 11652	. . . . . 6 |- ( ( A e. CC /\ 2 e. RR /\ 2 # 0 ) -> ( A e. RR <-> ( 2 x. A ) e. RR ) )
53	50, 51, 52	mp3an23 1370	. . . . 5 |- ( A e. CC -> ( A e. RR <-> ( 2 x. A ) e. RR ) )
54		mulcl 8313	. . . . . . 7 |- ( ( 2 e. CC /\ A e. CC ) -> ( 2 x. A ) e. CC )
55	22, 54	mpan 428	. . . . . 6 |- ( A e. CC -> ( 2 x. A ) e. CC )
56		absefib 12564	. . . . . 6 |- ( ( 2 x. A ) e. CC -> ( ( 2 x. A ) e. RR <-> ( abs ` ( exp ` ( _i x. ( 2 x. A ) ) ) ) = 1 ) )
57	55, 56	syl 14	. . . . 5 |- ( A e. CC -> ( ( 2 x. A ) e. RR <-> ( abs ` ( exp ` ( _i x. ( 2 x. A ) ) ) ) = 1 ) )
58	53, 57	bitr2d 189	. . . 4 |- ( A e. CC -> ( ( abs ` ( exp ` ( _i x. ( 2 x. A ) ) ) ) = 1 <-> A e. RR ) )
59	49, 58	bitrid 192	. . 3 |- ( A e. CC -> ( ( abs ` ( exp ` ( _i x. ( 2 x. A ) ) ) ) = ( abs ` 1 ) <-> A e. RR ) )
60	47, 59	sylibd 149	. 2 |- ( A e. CC -> ( ( sin ` A ) = 0 -> A e. RR ) )
61	60	imp 124	1 |- ( ( A e. CC /\ ( sin ` A ) = 0 ) -> A e. RR )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   class class class wbr 4131   ` cfv 5380  (class class class)co 6088   CCcc 8184   RRcr 8185   0cc0 8186   1c1 8187   _ici 8188   + caddc 8189   x. cmul 8191   - cmin 8505   -ucneg 8506   # cap 8918   / cdiv 9011   2c2 9365   abscabs 11786   expce 12435   sincsin 12437

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

This proof depends on definitions:  df-bi 117  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-1st 6378  df-2nd 6379  df-recs 6580  df-irdg 6645  df-frec 6666  df-1o 6691  df-oadd 6695  df-er 6811  df-en 7028  df-dom 7029  df-fin 7030  df-sup 7331  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-ico 10314  df-fz 10430  df-fzo 10568  df-seqfrec 10907  df-exp 10998  df-fac 11187  df-bc 11209  df-ihash 11238  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-sin 12443  df-cos 12444

This theorem is used by: (None)