Theorem coseq0q4123 15524

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 8157	. . . . 5 |- 0 e. RR
2	1	ltnri 8250	. . . 4 |- -. 0 < 0
3		elioore 10120	. . . . . . 7 |- ( A e. ( -u ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) -> A e. RR )
4	3	adantr 276	. . . . . 6 |- ( ( A e. ( -u ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) /\ ( cos ` A ) = 0 ) -> A e. RR )
5		halfpire 15482	. . . . . 6 |- ( pi / 2 ) e. RR
6		reaplt 8746	. . . . . 6 |- ( ( A e. RR /\ ( pi / 2 ) e. RR ) -> ( A # ( pi / 2 ) <-> ( A < ( pi / 2 ) \/ ( pi / 2 ) < A ) ) )
7	4, 5, 6	sylancl 413	. . . . 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 276	. . . . . . . . . . 11 |- ( ( A e. ( -u ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) /\ A < ( pi / 2 ) ) -> A e. RR )
9		neghalfpirx 15484	. . . . . . . . . . . . . 14 |- -u ( pi / 2 ) e. RR*
10		3re 9195	. . . . . . . . . . . . . . . 16 |- 3 e. RR
11	10, 5	remulcli 8171	. . . . . . . . . . . . . . 15 |- ( 3 x. ( pi / 2 ) ) e. RR
12	11	rexri 8215	. . . . . . . . . . . . . 14 |- ( 3 x. ( pi / 2 ) ) e. RR*
13		elioo2 10129	. . . . . . . . . . . . . 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 426	. . . . . . . . . . . . 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 1037	. . . . . . . . . . . 12 |- ( A e. ( -u ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) -> -u ( pi / 2 ) < A )
16	15	adantr 276	. . . . . . . . . . 11 |- ( ( A e. ( -u ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) /\ A < ( pi / 2 ) ) -> -u ( pi / 2 ) < A )
17		simpr 110	. . . . . . . . . . 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 8215	. . . . . . . . . . . 12 |- ( pi / 2 ) e. RR*
20		elioo2 10129	. . . . . . . . . . . 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 413	. . . . . . . . . . 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 1204	. . . . . . . . . 10 |- ( ( A e. ( -u ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) /\ A < ( pi / 2 ) ) -> A e. ( -u ( pi / 2 ) (,) ( pi / 2 ) ) )
23		cosq14gt0 15522	. . . . . . . . . 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 477	. . . . . . . 8 |- ( ( ( A e. ( -u ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) /\ ( cos ` A ) = 0 ) /\ A < ( pi / 2 ) ) -> 0 < ( cos ` A ) )
26		simplr 528	. . . . . . . 8 |- ( ( ( A e. ( -u ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) /\ ( cos ` A ) = 0 ) /\ A < ( pi / 2 ) ) -> ( cos ` A ) = 0 )
27	25, 26	breqtrd 4109	. . . . . . 7 |- ( ( ( A e. ( -u ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) /\ ( cos ` A ) = 0 ) /\ A < ( pi / 2 ) ) -> 0 < 0 )
28	27	ex 115	. . . . . 6 |- ( ( A e. ( -u ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) /\ ( cos ` A ) = 0 ) -> ( A < ( pi / 2 ) -> 0 < 0 ) )
29		simplr 528	. . . . . . . 8 |- ( ( ( A e. ( -u ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) /\ ( cos ` A ) = 0 ) /\ ( pi / 2 ) < A ) -> ( cos ` A ) = 0 )
30	3	adantr 276	. . . . . . . . . . 11 |- ( ( A e. ( -u ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) /\ ( pi / 2 ) < A ) -> A e. RR )
31		simpr 110	. . . . . . . . . . 11 |- ( ( A e. ( -u ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) /\ ( pi / 2 ) < A ) -> ( pi / 2 ) < A )
32	14	simp3bi 1038	. . . . . . . . . . . 12 |- ( A e. ( -u ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) -> A < ( 3 x. ( pi / 2 ) ) )
33	32	adantr 276	. . . . . . . . . . 11 |- ( ( A e. ( -u ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) /\ ( pi / 2 ) < A ) -> A < ( 3 x. ( pi / 2 ) ) )
34		elioo2 10129	. . . . . . . . . . . 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 426	. . . . . . . . . . 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 1205	. . . . . . . . . 10 |- ( ( A e. ( -u ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) /\ ( pi / 2 ) < A ) -> A e. ( ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) )
37		cosq23lt0 15523	. . . . . . . . . 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 477	. . . . . . . 8 |- ( ( ( A e. ( -u ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) /\ ( cos ` A ) = 0 ) /\ ( pi / 2 ) < A ) -> ( cos ` A ) < 0 )
40	29, 39	eqbrtrrd 4107	. . . . . . 7 |- ( ( ( A e. ( -u ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) /\ ( cos ` A ) = 0 ) /\ ( pi / 2 ) < A ) -> 0 < 0 )
41	40	ex 115	. . . . . 6 |- ( ( A e. ( -u ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) /\ ( cos ` A ) = 0 ) -> ( ( pi / 2 ) < A -> 0 < 0 ) )
42	28, 41	jaod 722	. . . . 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 150	. . . 4 |- ( ( A e. ( -u ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) /\ ( cos ` A ) = 0 ) -> ( A # ( pi / 2 ) -> 0 < 0 ) )
44	2, 43	mtoi 668	. . 3 |- ( ( A e. ( -u ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) /\ ( cos ` A ) = 0 ) -> -. A # ( pi / 2 ) )
45	3	recnd 8186	. . . 4 |- ( A e. ( -u ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) -> A e. CC )
46		picn 15477	. . . . 5 |- pi e. CC
47		halfcl 9348	. . . . 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 8780	. . . 4 |- ( ( A e. CC /\ ( pi / 2 ) e. CC ) -> ( A = ( pi / 2 ) <-> -. A # ( pi / 2 ) ) )
50	45, 48, 49	syl2an2r 597	. . 3 |- ( ( A e. ( -u ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) /\ ( cos ` A ) = 0 ) -> ( A = ( pi / 2 ) <-> -. A # ( pi / 2 ) ) )
51	44, 50	mpbird 167	. 2 |- ( ( A e. ( -u ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) /\ ( cos ` A ) = 0 ) -> A = ( pi / 2 ) )
52		fveq2 5629	. . . 4 |- ( A = ( pi / 2 ) -> ( cos ` A ) = ( cos ` ( pi / 2 ) ) )
53		coshalfpi 15487	. . . 4 |- ( cos ` ( pi / 2 ) ) = 0
54	52, 53	eqtrdi 2278	. . 3 |- ( A = ( pi / 2 ) -> ( cos ` A ) = 0 )
55	54	adantl 277	. 2 |- ( ( A e. ( -u ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) /\ A = ( pi / 2 ) ) -> ( cos ` A ) = 0 )
56	51, 55	impbida 598	1 |- ( A e. ( -u ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) -> ( ( cos ` A ) = 0 <-> A = ( pi / 2 ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 713   /\ w3a 1002   = wceq 1395   e. wcel 2200   class class class wbr 4083   ` cfv 5318  (class class class)co 6007   CCcc 8008   RRcr 8009   0cc0 8010   x. cmul 8015   RR*cxr 8191   < clt 8192   -ucneg 8329   # cap 8739   / cdiv 8830   2c2 9172   3c3 9173   (,)cioo 10096   cosccos 12172   picpi 12174

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8101  ax-resscn 8102  ax-1cn 8103  ax-1re 8104  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-mulrcl 8109  ax-addcom 8110  ax-mulcom 8111  ax-addass 8112  ax-mulass 8113  ax-distr 8114  ax-i2m1 8115  ax-0lt1 8116  ax-1rid 8117  ax-0id 8118  ax-rnegex 8119  ax-precex 8120  ax-cnre 8121  ax-pre-ltirr 8122  ax-pre-ltwlin 8123  ax-pre-lttrn 8124  ax-pre-apti 8125  ax-pre-ltadd 8126  ax-pre-mulgt0 8127  ax-pre-mulext 8128  ax-arch 8129  ax-caucvg 8130  ax-pre-suploc 8131  ax-addf 8132  ax-mulf 8133

This theorem depends on definitions:  df-bi 117  df-stab 836  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-disj 4060  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-isom 5327  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-of 6224  df-1st 6292  df-2nd 6293  df-recs 6457  df-irdg 6522  df-frec 6543  df-1o 6568  df-oadd 6572  df-er 6688  df-map 6805  df-pm 6806  df-en 6896  df-dom 6897  df-fin 6898  df-sup 7162  df-inf 7163  df-pnf 8194  df-mnf 8195  df-xr 8196  df-ltxr 8197  df-le 8198  df-sub 8330  df-neg 8331  df-reap 8733  df-ap 8740  df-div 8831  df-inn 9122  df-2 9180  df-3 9181  df-4 9182  df-5 9183  df-6 9184  df-7 9185  df-8 9186  df-9 9187  df-n0 9381  df-z 9458  df-uz 9734  df-q 9827  df-rp 9862  df-xneg 9980  df-xadd 9981  df-ioo 10100  df-ioc 10101  df-ico 10102  df-icc 10103  df-fz 10217  df-fzo 10351  df-seqfrec 10682  df-exp 10773  df-fac 10960  df-bc 10982  df-ihash 11010  df-shft 11342  df-cj 11369  df-re 11370  df-im 11371  df-rsqrt 11525  df-abs 11526  df-clim 11806  df-sumdc 11881  df-ef 12175  df-sin 12177  df-cos 12178  df-pi 12180  df-rest 13290  df-topgen 13309  df-psmet 14523  df-xmet 14524  df-met 14525  df-bl 14526  df-mopn 14527  df-top 14688  df-topon 14701  df-bases 14733  df-ntr 14786  df-cn 14878  df-cnp 14879  df-tx 14943  df-cncf 15261  df-limced 15346  df-dvap 15347

This theorem is referenced by:  coseq00topi  15525  coseq0negpitopi  15526