Theorem coseq0q4123 15421

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 8107	. . . . 5 |- 0 e. RR
2	1	ltnri 8200	. . . 4 |- -. 0 < 0
3		elioore 10069	. . . . . . 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 15379	. . . . . 6 |- ( pi / 2 ) e. RR
6		reaplt 8696	. . . . . 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 15381	. . . . . . . . . . . . . 14 |- -u ( pi / 2 ) e. RR*
10		3re 9145	. . . . . . . . . . . . . . . 16 |- 3 e. RR
11	10, 5	remulcli 8121	. . . . . . . . . . . . . . 15 |- ( 3 x. ( pi / 2 ) ) e. RR
12	11	rexri 8165	. . . . . . . . . . . . . 14 |- ( 3 x. ( pi / 2 ) ) e. RR*
13		elioo2 10078	. . . . . . . . . . . . . 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 1016	. . . . . . . . . . . 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 8165	. . . . . . . . . . . 12 |- ( pi / 2 ) e. RR*
20		elioo2 10078	. . . . . . . . . . . 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 1183	. . . . . . . . . 10 |- ( ( A e. ( -u ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) /\ A < ( pi / 2 ) ) -> A e. ( -u ( pi / 2 ) (,) ( pi / 2 ) ) )
23		cosq14gt0 15419	. . . . . . . . . 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 4085	. . . . . . 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 1017	. . . . . . . . . . . 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 10078	. . . . . . . . . . . 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 1184	. . . . . . . . . 10 |- ( ( A e. ( -u ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) /\ ( pi / 2 ) < A ) -> A e. ( ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) )
37		cosq23lt0 15420	. . . . . . . . . 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 4083	. . . . . . 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 719	. . . . 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 666	. . 3 |- ( ( A e. ( -u ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) /\ ( cos ` A ) = 0 ) -> -. A # ( pi / 2 ) )
45	3	recnd 8136	. . . 4 |- ( A e. ( -u ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) -> A e. CC )
46		picn 15374	. . . . 5 |- pi e. CC
47		halfcl 9298	. . . . 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 8730	. . . 4 |- ( ( A e. CC /\ ( pi / 2 ) e. CC ) -> ( A = ( pi / 2 ) <-> -. A # ( pi / 2 ) ) )
50	45, 48, 49	syl2an2r 595	. . 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 5599	. . . 4 |- ( A = ( pi / 2 ) -> ( cos ` A ) = ( cos ` ( pi / 2 ) ) )
53		coshalfpi 15384	. . . 4 |- ( cos ` ( pi / 2 ) ) = 0
54	52, 53	eqtrdi 2256	. . 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 596	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 710   /\ w3a 981   = wceq 1373   e. wcel 2178   class class class wbr 4059   ` cfv 5290  (class class class)co 5967   CCcc 7958   RRcr 7959   0cc0 7960   x. cmul 7965   RR*cxr 8141   < clt 8142   -ucneg 8279   # cap 8689   / cdiv 8780   2c2 9122   3c3 9123   (,)cioo 10045   cosccos 12071   picpi 12073

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-mulrcl 8059  ax-addcom 8060  ax-mulcom 8061  ax-addass 8062  ax-mulass 8063  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-1rid 8067  ax-0id 8068  ax-rnegex 8069  ax-precex 8070  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-apti 8075  ax-pre-ltadd 8076  ax-pre-mulgt0 8077  ax-pre-mulext 8078  ax-arch 8079  ax-caucvg 8080  ax-pre-suploc 8081  ax-addf 8082  ax-mulf 8083

This theorem depends on definitions:  df-bi 117  df-stab 833  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-if 3580  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-disj 4036  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-id 4358  df-po 4361  df-iso 4362  df-iord 4431  df-on 4433  df-ilim 4434  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-isom 5299  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-of 6181  df-1st 6249  df-2nd 6250  df-recs 6414  df-irdg 6479  df-frec 6500  df-1o 6525  df-oadd 6529  df-er 6643  df-map 6760  df-pm 6761  df-en 6851  df-dom 6852  df-fin 6853  df-sup 7112  df-inf 7113  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-sub 8280  df-neg 8281  df-reap 8683  df-ap 8690  df-div 8781  df-inn 9072  df-2 9130  df-3 9131  df-4 9132  df-5 9133  df-6 9134  df-7 9135  df-8 9136  df-9 9137  df-n0 9331  df-z 9408  df-uz 9684  df-q 9776  df-rp 9811  df-xneg 9929  df-xadd 9930  df-ioo 10049  df-ioc 10050  df-ico 10051  df-icc 10052  df-fz 10166  df-fzo 10300  df-seqfrec 10630  df-exp 10721  df-fac 10908  df-bc 10930  df-ihash 10958  df-shft 11241  df-cj 11268  df-re 11269  df-im 11270  df-rsqrt 11424  df-abs 11425  df-clim 11705  df-sumdc 11780  df-ef 12074  df-sin 12076  df-cos 12077  df-pi 12079  df-rest 13188  df-topgen 13207  df-psmet 14420  df-xmet 14421  df-met 14422  df-bl 14423  df-mopn 14424  df-top 14585  df-topon 14598  df-bases 14630  df-ntr 14683  df-cn 14775  df-cnp 14776  df-tx 14840  df-cncf 15158  df-limced 15243  df-dvap 15244

This theorem is referenced by:  coseq00topi  15422  coseq0negpitopi  15423