Theorem coseq0q4123 13395

Description: Location of the zeroes of cosine in ( -u ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ). (Contributed by Jim Kingdon, 14-Mar-2024.)

Assertion

Ref	Expression
coseq0q4123	|- ( A e. ( -u ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) -> ( ( cos ` A ) = 0 <-> A = ( pi / 2 ) ) )

Proof of Theorem coseq0q4123

Step	Hyp	Ref	Expression
1		0re 7899	. . . . 5 |- 0 e. RR
2	1	ltnri 7991	. . . 4 |- -. 0 < 0
3		elioore 9848	. . . . . . 7 |- ( A e. ( -u ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) -> A e. RR )
4	3	adantr 274	. . . . . 6 |- ( ( A e. ( -u ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) /\ ( cos ` A ) = 0 ) -> A e. RR )
5		halfpire 13353	. . . . . 6 |- ( pi / 2 ) e. RR
6		reaplt 8486	. . . . . 6 |- ( ( A e. RR /\ ( pi / 2 ) e. RR ) -> ( A # ( pi / 2 ) <-> ( A < ( pi / 2 ) \/ ( pi / 2 ) < A ) ) )
7	4, 5, 6	sylancl 410	. . . . 5 |- ( ( A e. ( -u ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) /\ ( cos ` A ) = 0 ) -> ( A # ( pi / 2 ) <-> ( A < ( pi / 2 ) \/ ( pi / 2 ) < A ) ) )
8	3	adantr 274	. . . . . . . . . . 11 |- ( ( A e. ( -u ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) /\ A < ( pi / 2 ) ) -> A e. RR )
9		neghalfpirx 13355	. . . . . . . . . . . . . 14 |- -u ( pi / 2 ) e. RR*
10		3re 8931	. . . . . . . . . . . . . . . 16 |- 3 e. RR
11	10, 5	remulcli 7913	. . . . . . . . . . . . . . 15 |- ( 3 x. ( pi / 2 ) ) e. RR
12	11	rexri 7956	. . . . . . . . . . . . . 14 |- ( 3 x. ( pi / 2 ) ) e. RR*
13		elioo2 9857	. . . . . . . . . . . . . 14 |- ( ( -u ( pi / 2 ) e. RR* /\ ( 3 x. ( pi / 2 ) ) e. RR* ) -> ( A e. ( -u ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) <-> ( A e. RR /\ -u ( pi / 2 ) < A /\ A < ( 3 x. ( pi / 2 ) ) ) ) )
14	9, 12, 13	mp2an 423	. . . . . . . . . . . . 13 |- ( A e. ( -u ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) <-> ( A e. RR /\ -u ( pi / 2 ) < A /\ A < ( 3 x. ( pi / 2 ) ) ) )
15	14	simp2bi 1003	. . . . . . . . . . . 12 |- ( A e. ( -u ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) -> -u ( pi / 2 ) < A )
16	15	adantr 274	. . . . . . . . . . 11 |- ( ( A e. ( -u ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) /\ A < ( pi / 2 ) ) -> -u ( pi / 2 ) < A )
17		simpr 109	. . . . . . . . . . 11 |- ( ( A e. ( -u ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) /\ A < ( pi / 2 ) ) -> A < ( pi / 2 ) )
18	9	a1i 9	. . . . . . . . . . . 12 |- ( ( A e. ( -u ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) /\ A < ( pi / 2 ) ) -> -u ( pi / 2 ) e. RR* )
19	5	rexri 7956	. . . . . . . . . . . 12 |- ( pi / 2 ) e. RR*
20		elioo2 9857	. . . . . . . . . . . 12 |- ( ( -u ( pi / 2 ) e. RR* /\ ( pi / 2 ) e. RR* ) -> ( A e. ( -u ( pi / 2 ) (,) ( pi / 2 ) ) <-> ( A e. RR /\ -u ( pi / 2 ) < A /\ A < ( pi / 2 ) ) ) )
21	18, 19, 20	sylancl 410	. . . . . . . . . . 11 |- ( ( A e. ( -u ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) /\ A < ( pi / 2 ) ) -> ( A e. ( -u ( pi / 2 ) (,) ( pi / 2 ) ) <-> ( A e. RR /\ -u ( pi / 2 ) < A /\ A < ( pi / 2 ) ) ) )
22	8, 16, 17, 21	mpbir3and 1170	. . . . . . . . . 10 |- ( ( A e. ( -u ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) /\ A < ( pi / 2 ) ) -> A e. ( -u ( pi / 2 ) (,) ( pi / 2 ) ) )
23		cosq14gt0 13393	. . . . . . . . . 10 |- ( A e. ( -u ( pi / 2 ) (,) ( pi / 2 ) ) -> 0 < ( cos ` A ) )
24	22, 23	syl 14	. . . . . . . . 9 |- ( ( A e. ( -u ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) /\ A < ( pi / 2 ) ) -> 0 < ( cos ` A ) )
25	24	adantlr 469	. . . . . . . 8 |- ( ( ( A e. ( -u ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) /\ ( cos ` A ) = 0 ) /\ A < ( pi / 2 ) ) -> 0 < ( cos ` A ) )
26		simplr 520	. . . . . . . 8 |- ( ( ( A e. ( -u ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) /\ ( cos ` A ) = 0 ) /\ A < ( pi / 2 ) ) -> ( cos ` A ) = 0 )
27	25, 26	breqtrd 4008	. . . . . . 7 |- ( ( ( A e. ( -u ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) /\ ( cos ` A ) = 0 ) /\ A < ( pi / 2 ) ) -> 0 < 0 )
28	27	ex 114	. . . . . 6 |- ( ( A e. ( -u ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) /\ ( cos ` A ) = 0 ) -> ( A < ( pi / 2 ) -> 0 < 0 ) )
29		simplr 520	. . . . . . . 8 |- ( ( ( A e. ( -u ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) /\ ( cos ` A ) = 0 ) /\ ( pi / 2 ) < A ) -> ( cos ` A ) = 0 )
30	3	adantr 274	. . . . . . . . . . 11 |- ( ( A e. ( -u ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) /\ ( pi / 2 ) < A ) -> A e. RR )
31		simpr 109	. . . . . . . . . . 11 |- ( ( A e. ( -u ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) /\ ( pi / 2 ) < A ) -> ( pi / 2 ) < A )
32	14	simp3bi 1004	. . . . . . . . . . . 12 |- ( A e. ( -u ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) -> A < ( 3 x. ( pi / 2 ) ) )
33	32	adantr 274	. . . . . . . . . . 11 |- ( ( A e. ( -u ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) /\ ( pi / 2 ) < A ) -> A < ( 3 x. ( pi / 2 ) ) )
34		elioo2 9857	. . . . . . . . . . . 12 |- ( ( ( pi / 2 ) e. RR* /\ ( 3 x. ( pi / 2 ) ) e. RR* ) -> ( A e. ( ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) <-> ( A e. RR /\ ( pi / 2 ) < A /\ A < ( 3 x. ( pi / 2 ) ) ) ) )
35	19, 12, 34	mp2an 423	. . . . . . . . . . 11 |- ( A e. ( ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) <-> ( A e. RR /\ ( pi / 2 ) < A /\ A < ( 3 x. ( pi / 2 ) ) ) )
36	30, 31, 33, 35	syl3anbrc 1171	. . . . . . . . . 10 |- ( ( A e. ( -u ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) /\ ( pi / 2 ) < A ) -> A e. ( ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) )
37		cosq23lt0 13394	. . . . . . . . . 10 |- ( A e. ( ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) -> ( cos ` A ) < 0 )
38	36, 37	syl 14	. . . . . . . . 9 |- ( ( A e. ( -u ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) /\ ( pi / 2 ) < A ) -> ( cos ` A ) < 0 )
39	38	adantlr 469	. . . . . . . 8 |- ( ( ( A e. ( -u ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) /\ ( cos ` A ) = 0 ) /\ ( pi / 2 ) < A ) -> ( cos ` A ) < 0 )
40	29, 39	eqbrtrrd 4006	. . . . . . 7 |- ( ( ( A e. ( -u ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) /\ ( cos ` A ) = 0 ) /\ ( pi / 2 ) < A ) -> 0 < 0 )
41	40	ex 114	. . . . . 6 |- ( ( A e. ( -u ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) /\ ( cos ` A ) = 0 ) -> ( ( pi / 2 ) < A -> 0 < 0 ) )
42	28, 41	jaod 707	. . . . 5 |- ( ( A e. ( -u ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) /\ ( cos ` A ) = 0 ) -> ( ( A < ( pi / 2 ) \/ ( pi / 2 ) < A ) -> 0 < 0 ) )
43	7, 42	sylbid 149	. . . 4 |- ( ( A e. ( -u ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) /\ ( cos ` A ) = 0 ) -> ( A # ( pi / 2 ) -> 0 < 0 ) )
44	2, 43	mtoi 654	. . 3 |- ( ( A e. ( -u ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) /\ ( cos ` A ) = 0 ) -> -. A # ( pi / 2 ) )
45	3	recnd 7927	. . . 4 |- ( A e. ( -u ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) -> A e. CC )
46		picn 13348	. . . . 5 |- pi e. CC
47		halfcl 9083	. . . . 5 |- ( pi e. CC -> ( pi / 2 ) e. CC )
48	46, 47	mp1i 10	. . . 4 |- ( ( A e. ( -u ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) /\ ( cos ` A ) = 0 ) -> ( pi / 2 ) e. CC )
49		apti 8520	. . . 4 |- ( ( A e. CC /\ ( pi / 2 ) e. CC ) -> ( A = ( pi / 2 ) <-> -. A # ( pi / 2 ) ) )
50	45, 48, 49	syl2an2r 585	. . 3 |- ( ( A e. ( -u ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) /\ ( cos ` A ) = 0 ) -> ( A = ( pi / 2 ) <-> -. A # ( pi / 2 ) ) )
51	44, 50	mpbird 166	. 2 |- ( ( A e. ( -u ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) /\ ( cos ` A ) = 0 ) -> A = ( pi / 2 ) )
52		fveq2 5486	. . . 4 |- ( A = ( pi / 2 ) -> ( cos ` A ) = ( cos ` ( pi / 2 ) ) )
53		coshalfpi 13358	. . . 4 |- ( cos ` ( pi / 2 ) ) = 0
54	52, 53	eqtrdi 2215	. . 3 |- ( A = ( pi / 2 ) -> ( cos ` A ) = 0 )
55	54	adantl 275	. 2 |- ( ( A e. ( -u ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) /\ A = ( pi / 2 ) ) -> ( cos ` A ) = 0 )
56	51, 55	impbida 586	1 |- ( A e. ( -u ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) -> ( ( cos ` A ) = 0 <-> A = ( pi / 2 ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 698   /\ w3a 968   = wceq 1343   e. wcel 2136   class class class wbr 3982   ` cfv 5188  (class class class)co 5842   CCcc 7751   RRcr 7752   0cc0 7753   x. cmul 7758   RR*cxr 7932   < clt 7933   -ucneg 8070   # cap 8479   / cdiv 8568   2c2 8908   3c3 8909   (,)cioo 9824   cosccos 11586   picpi 11588

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871  ax-arch 7872  ax-caucvg 7873  ax-pre-suploc 7874  ax-addf 7875  ax-mulf 7876

This theorem depends on definitions:  df-bi 116  df-stab 821  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-disj 3960  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-ilim 4347  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-isom 5197  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-of 6050  df-1st 6108  df-2nd 6109  df-recs 6273  df-irdg 6338  df-frec 6359  df-1o 6384  df-oadd 6388  df-er 6501  df-map 6616  df-pm 6617  df-en 6707  df-dom 6708  df-fin 6709  df-sup 6949  df-inf 6950  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-inn 8858  df-2 8916  df-3 8917  df-4 8918  df-5 8919  df-6 8920  df-7 8921  df-8 8922  df-9 8923  df-n0 9115  df-z 9192  df-uz 9467  df-q 9558  df-rp 9590  df-xneg 9708  df-xadd 9709  df-ioo 9828  df-ioc 9829  df-ico 9830  df-icc 9831  df-fz 9945  df-fzo 10078  df-seqfrec 10381  df-exp 10455  df-fac 10639  df-bc 10661  df-ihash 10689  df-shft 10757  df-cj 10784  df-re 10785  df-im 10786  df-rsqrt 10940  df-abs 10941  df-clim 11220  df-sumdc 11295  df-ef 11589  df-sin 11591  df-cos 11592  df-pi 11594  df-rest 12558  df-topgen 12577  df-psmet 12627  df-xmet 12628  df-met 12629  df-bl 12630  df-mopn 12631  df-top 12636  df-topon 12649  df-bases 12681  df-ntr 12736  df-cn 12828  df-cnp 12829  df-tx 12893  df-cncf 13198  df-limced 13265  df-dvap 13266

This theorem is referenced by:  coseq00topi  13396  coseq0negpitopi  13397