Theorem coseq0q4123 15339

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 8074	. . . . 5 |- 0 e. RR
2	1	ltnri 8167	. . . 4 |- -. 0 < 0
3		elioore 10036	. . . . . . 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 15297	. . . . . 6 |- ( pi / 2 ) e. RR
6		reaplt 8663	. . . . . 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 15299	. . . . . . . . . . . . . 14 |- -u ( pi / 2 ) e. RR*
10		3re 9112	. . . . . . . . . . . . . . . 16 |- 3 e. RR
11	10, 5	remulcli 8088	. . . . . . . . . . . . . . 15 |- ( 3 x. ( pi / 2 ) ) e. RR
12	11	rexri 8132	. . . . . . . . . . . . . 14 |- ( 3 x. ( pi / 2 ) ) e. RR*
13		elioo2 10045	. . . . . . . . . . . . . 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 8132	. . . . . . . . . . . 12 |- ( pi / 2 ) e. RR*
20		elioo2 10045	. . . . . . . . . . . 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 15337	. . . . . . . . . 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 4071	. . . . . . 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 10045	. . . . . . . . . . . 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 15338	. . . . . . . . . 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 4069	. . . . . . 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 8103	. . . 4 |- ( A e. ( -u ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) -> A e. CC )
46		picn 15292	. . . . 5 |- pi e. CC
47		halfcl 9265	. . . . 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 8697	. . . 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 5578	. . . 4 |- ( A = ( pi / 2 ) -> ( cos ` A ) = ( cos ` ( pi / 2 ) ) )
53		coshalfpi 15302	. . . 4 |- ( cos ` ( pi / 2 ) ) = 0
54	52, 53	eqtrdi 2254	. . 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 2176   class class class wbr 4045   ` cfv 5272  (class class class)co 5946   CCcc 7925   RRcr 7926   0cc0 7927   x. cmul 7932   RR*cxr 8108   < clt 8109   -ucneg 8246   # cap 8656   / cdiv 8747   2c2 9089   3c3 9090   (,)cioo 10012   cosccos 11989   picpi 11991

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4160  ax-sep 4163  ax-nul 4171  ax-pow 4219  ax-pr 4254  ax-un 4481  ax-setind 4586  ax-iinf 4637  ax-cnex 8018  ax-resscn 8019  ax-1cn 8020  ax-1re 8021  ax-icn 8022  ax-addcl 8023  ax-addrcl 8024  ax-mulcl 8025  ax-mulrcl 8026  ax-addcom 8027  ax-mulcom 8028  ax-addass 8029  ax-mulass 8030  ax-distr 8031  ax-i2m1 8032  ax-0lt1 8033  ax-1rid 8034  ax-0id 8035  ax-rnegex 8036  ax-precex 8037  ax-cnre 8038  ax-pre-ltirr 8039  ax-pre-ltwlin 8040  ax-pre-lttrn 8041  ax-pre-apti 8042  ax-pre-ltadd 8043  ax-pre-mulgt0 8044  ax-pre-mulext 8045  ax-arch 8046  ax-caucvg 8047  ax-pre-suploc 8048  ax-addf 8049  ax-mulf 8050

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 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-if 3572  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-disj 4022  df-br 4046  df-opab 4107  df-mpt 4108  df-tr 4144  df-id 4341  df-po 4344  df-iso 4345  df-iord 4414  df-on 4416  df-ilim 4417  df-suc 4419  df-iom 4640  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-res 4688  df-ima 4689  df-iota 5233  df-fun 5274  df-fn 5275  df-f 5276  df-f1 5277  df-fo 5278  df-f1o 5279  df-fv 5280  df-isom 5281  df-riota 5901  df-ov 5949  df-oprab 5950  df-mpo 5951  df-of 6160  df-1st 6228  df-2nd 6229  df-recs 6393  df-irdg 6458  df-frec 6479  df-1o 6504  df-oadd 6508  df-er 6622  df-map 6739  df-pm 6740  df-en 6830  df-dom 6831  df-fin 6832  df-sup 7088  df-inf 7089  df-pnf 8111  df-mnf 8112  df-xr 8113  df-ltxr 8114  df-le 8115  df-sub 8247  df-neg 8248  df-reap 8650  df-ap 8657  df-div 8748  df-inn 9039  df-2 9097  df-3 9098  df-4 9099  df-5 9100  df-6 9101  df-7 9102  df-8 9103  df-9 9104  df-n0 9298  df-z 9375  df-uz 9651  df-q 9743  df-rp 9778  df-xneg 9896  df-xadd 9897  df-ioo 10016  df-ioc 10017  df-ico 10018  df-icc 10019  df-fz 10133  df-fzo 10267  df-seqfrec 10595  df-exp 10686  df-fac 10873  df-bc 10895  df-ihash 10923  df-shft 11159  df-cj 11186  df-re 11187  df-im 11188  df-rsqrt 11342  df-abs 11343  df-clim 11623  df-sumdc 11698  df-ef 11992  df-sin 11994  df-cos 11995  df-pi 11997  df-rest 13106  df-topgen 13125  df-psmet 14338  df-xmet 14339  df-met 14340  df-bl 14341  df-mopn 14342  df-top 14503  df-topon 14516  df-bases 14548  df-ntr 14601  df-cn 14693  df-cnp 14694  df-tx 14758  df-cncf 15076  df-limced 15161  df-dvap 15162

This theorem is referenced by:  coseq00topi  15340  coseq0negpitopi  15341