Theorem cos02pilt1 15574

Description: Cosine is less than one between zero and 2 x. pi. (Contributed by Jim Kingdon, 19-Mar-2024.)

Assertion

Ref	Expression
cos02pilt1	|- ( A e. ( 0 (,) ( 2 x. pi ) ) -> ( cos ` A ) < 1 )

Proof of Theorem cos02pilt1

Step	Hyp	Ref	Expression
1		elioore 10146	. . . . . . 7 |- ( A e. ( 0 (,) ( 2 x. pi ) ) -> A e. RR )
2	1	adantr 276	. . . . . 6 |- ( ( A e. ( 0 (,) ( 2 x. pi ) ) /\ ( pi / 2 ) < A ) -> A e. RR )
3	2	adantr 276	. . . . 5 |- ( ( ( A e. ( 0 (,) ( 2 x. pi ) ) /\ ( pi / 2 ) < A ) /\ pi < A ) -> A e. RR )
4		pire 15509	. . . . . . 7 |- pi e. RR
5	4	a1i 9	. . . . . 6 |- ( ( ( A e. ( 0 (,) ( 2 x. pi ) ) /\ ( pi / 2 ) < A ) /\ pi < A ) -> pi e. RR )
6		simpr 110	. . . . . 6 |- ( ( ( A e. ( 0 (,) ( 2 x. pi ) ) /\ ( pi / 2 ) < A ) /\ pi < A ) -> pi < A )
7	5, 3, 6	ltled 8297	. . . . 5 |- ( ( ( A e. ( 0 (,) ( 2 x. pi ) ) /\ ( pi / 2 ) < A ) /\ pi < A ) -> pi <_ A )
8		0xr 8225	. . . . . . . 8 |- 0 e. RR*
9		2re 9212	. . . . . . . . . 10 |- 2 e. RR
10	9, 4	remulcli 8192	. . . . . . . . 9 |- ( 2 x. pi ) e. RR
11	10	rexri 8236	. . . . . . . 8 |- ( 2 x. pi ) e. RR*
12		elioo2 10155	. . . . . . . 8 |- ( ( 0 e. RR* /\ ( 2 x. pi ) e. RR* ) -> ( A e. ( 0 (,) ( 2 x. pi ) ) <-> ( A e. RR /\ 0 < A /\ A < ( 2 x. pi ) ) ) )
13	8, 11, 12	mp2an 426	. . . . . . 7 |- ( A e. ( 0 (,) ( 2 x. pi ) ) <-> ( A e. RR /\ 0 < A /\ A < ( 2 x. pi ) ) )
14	13	simp3bi 1040	. . . . . 6 |- ( A e. ( 0 (,) ( 2 x. pi ) ) -> A < ( 2 x. pi ) )
15	14	ad2antrr 488	. . . . 5 |- ( ( ( A e. ( 0 (,) ( 2 x. pi ) ) /\ ( pi / 2 ) < A ) /\ pi < A ) -> A < ( 2 x. pi ) )
16		elico2 10171	. . . . . 6 |- ( ( pi e. RR /\ ( 2 x. pi ) e. RR* ) -> ( A e. ( pi [,) ( 2 x. pi ) ) <-> ( A e. RR /\ pi <_ A /\ A < ( 2 x. pi ) ) ) )
17	4, 11, 16	mp2an 426	. . . . 5 |- ( A e. ( pi [,) ( 2 x. pi ) ) <-> ( A e. RR /\ pi <_ A /\ A < ( 2 x. pi ) ) )
18	3, 7, 15, 17	syl3anbrc 1207	. . . 4 |- ( ( ( A e. ( 0 (,) ( 2 x. pi ) ) /\ ( pi / 2 ) < A ) /\ pi < A ) -> A e. ( pi [,) ( 2 x. pi ) ) )
19		cosq34lt1 15573	. . . 4 |- ( A e. ( pi [,) ( 2 x. pi ) ) -> ( cos ` A ) < 1 )
20	18, 19	syl 14	. . 3 |- ( ( ( A e. ( 0 (,) ( 2 x. pi ) ) /\ ( pi / 2 ) < A ) /\ pi < A ) -> ( cos ` A ) < 1 )
21	2	adantr 276	. . . . 5 |- ( ( ( A e. ( 0 (,) ( 2 x. pi ) ) /\ ( pi / 2 ) < A ) /\ A < ( 3 x. ( pi / 2 ) ) ) -> A e. RR )
22		simplr 529	. . . . 5 |- ( ( ( A e. ( 0 (,) ( 2 x. pi ) ) /\ ( pi / 2 ) < A ) /\ A < ( 3 x. ( pi / 2 ) ) ) -> ( pi / 2 ) < A )
23		simpr 110	. . . . 5 |- ( ( ( A e. ( 0 (,) ( 2 x. pi ) ) /\ ( pi / 2 ) < A ) /\ A < ( 3 x. ( pi / 2 ) ) ) -> A < ( 3 x. ( pi / 2 ) ) )
24		halfpire 15515	. . . . . . 7 |- ( pi / 2 ) e. RR
25	24	rexri 8236	. . . . . 6 |- ( pi / 2 ) e. RR*
26		3re 9216	. . . . . . . 8 |- 3 e. RR
27	26, 24	remulcli 8192	. . . . . . 7 |- ( 3 x. ( pi / 2 ) ) e. RR
28	27	rexri 8236	. . . . . 6 |- ( 3 x. ( pi / 2 ) ) e. RR*
29		elioo2 10155	. . . . . 6 |- ( ( ( 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 ) ) ) ) )
30	25, 28, 29	mp2an 426	. . . . 5 |- ( A e. ( ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) <-> ( A e. RR /\ ( pi / 2 ) < A /\ A < ( 3 x. ( pi / 2 ) ) ) )
31	21, 22, 23, 30	syl3anbrc 1207	. . . 4 |- ( ( ( A e. ( 0 (,) ( 2 x. pi ) ) /\ ( pi / 2 ) < A ) /\ A < ( 3 x. ( pi / 2 ) ) ) -> A e. ( ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) )
32		elioore 10146	. . . . . 6 |- ( A e. ( ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) -> A e. RR )
33	32	recoscld 12284	. . . . 5 |- ( A e. ( ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) -> ( cos ` A ) e. RR )
34		0red 8179	. . . . 5 |- ( A e. ( ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) -> 0 e. RR )
35		1red 8193	. . . . 5 |- ( A e. ( ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) -> 1 e. RR )
36		cosq23lt0 15556	. . . . 5 |- ( A e. ( ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) -> ( cos ` A ) < 0 )
37		0lt1 8305	. . . . . 6 |- 0 < 1
38	37	a1i 9	. . . . 5 |- ( A e. ( ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) -> 0 < 1 )
39	33, 34, 35, 36, 38	lttrd 8304	. . . 4 |- ( A e. ( ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) -> ( cos ` A ) < 1 )
40	31, 39	syl 14	. . 3 |- ( ( ( A e. ( 0 (,) ( 2 x. pi ) ) /\ ( pi / 2 ) < A ) /\ A < ( 3 x. ( pi / 2 ) ) ) -> ( cos ` A ) < 1 )
41		2lt3 9313	. . . . . 6 |- 2 < 3
42		2pos 9233	. . . . . . . 8 |- 0 < 2
43	9, 42	pm3.2i 272	. . . . . . 7 |- ( 2 e. RR /\ 0 < 2 )
44		3pos 9236	. . . . . . . 8 |- 0 < 3
45	26, 44	pm3.2i 272	. . . . . . 7 |- ( 3 e. RR /\ 0 < 3 )
46		pipos 15511	. . . . . . . 8 |- 0 < pi
47	4, 46	pm3.2i 272	. . . . . . 7 |- ( pi e. RR /\ 0 < pi )
48		ltdiv2 9066	. . . . . . 7 |- ( ( ( 2 e. RR /\ 0 < 2 ) /\ ( 3 e. RR /\ 0 < 3 ) /\ ( pi e. RR /\ 0 < pi ) ) -> ( 2 < 3 <-> ( pi / 3 ) < ( pi / 2 ) ) )
49	43, 45, 47, 48	mp3an 1373	. . . . . 6 |- ( 2 < 3 <-> ( pi / 3 ) < ( pi / 2 ) )
50	41, 49	mpbi 145	. . . . 5 |- ( pi / 3 ) < ( pi / 2 )
51		ltdivmul 9055	. . . . . 6 |- ( ( pi e. RR /\ ( pi / 2 ) e. RR /\ ( 3 e. RR /\ 0 < 3 ) ) -> ( ( pi / 3 ) < ( pi / 2 ) <-> pi < ( 3 x. ( pi / 2 ) ) ) )
52	4, 24, 45, 51	mp3an 1373	. . . . 5 |- ( ( pi / 3 ) < ( pi / 2 ) <-> pi < ( 3 x. ( pi / 2 ) ) )
53	50, 52	mpbi 145	. . . 4 |- pi < ( 3 x. ( pi / 2 ) )
54		axltwlin 8246	. . . . 5 |- ( ( pi e. RR /\ ( 3 x. ( pi / 2 ) ) e. RR /\ A e. RR ) -> ( pi < ( 3 x. ( pi / 2 ) ) -> ( pi < A \/ A < ( 3 x. ( pi / 2 ) ) ) ) )
55	4, 27, 2, 54	mp3an12i 1377	. . . 4 |- ( ( A e. ( 0 (,) ( 2 x. pi ) ) /\ ( pi / 2 ) < A ) -> ( pi < ( 3 x. ( pi / 2 ) ) -> ( pi < A \/ A < ( 3 x. ( pi / 2 ) ) ) ) )
56	53, 55	mpi 15	. . 3 |- ( ( A e. ( 0 (,) ( 2 x. pi ) ) /\ ( pi / 2 ) < A ) -> ( pi < A \/ A < ( 3 x. ( pi / 2 ) ) ) )
57	20, 40, 56	mpjaodan 805	. 2 |- ( ( A e. ( 0 (,) ( 2 x. pi ) ) /\ ( pi / 2 ) < A ) -> ( cos ` A ) < 1 )
58	4	rexri 8236	. . . . . 6 |- pi e. RR*
59		0re 8178	. . . . . . 7 |- 0 e. RR
60	59, 4, 46	ltleii 8281	. . . . . 6 |- 0 <_ pi
61		lbicc2 10218	. . . . . 6 |- ( ( 0 e. RR* /\ pi e. RR* /\ 0 <_ pi ) -> 0 e. ( 0 [,] pi ) )
62	8, 58, 60, 61	mp3an 1373	. . . . 5 |- 0 e. ( 0 [,] pi )
63	62	a1i 9	. . . 4 |- ( ( A e. ( 0 (,) ( 2 x. pi ) ) /\ A < pi ) -> 0 e. ( 0 [,] pi ) )
64	1	adantr 276	. . . . 5 |- ( ( A e. ( 0 (,) ( 2 x. pi ) ) /\ A < pi ) -> A e. RR )
65		0red 8179	. . . . . 6 |- ( ( A e. ( 0 (,) ( 2 x. pi ) ) /\ A < pi ) -> 0 e. RR )
66	13	simp2bi 1039	. . . . . . 7 |- ( A e. ( 0 (,) ( 2 x. pi ) ) -> 0 < A )
67	66	adantr 276	. . . . . 6 |- ( ( A e. ( 0 (,) ( 2 x. pi ) ) /\ A < pi ) -> 0 < A )
68	65, 64, 67	ltled 8297	. . . . 5 |- ( ( A e. ( 0 (,) ( 2 x. pi ) ) /\ A < pi ) -> 0 <_ A )
69	4	a1i 9	. . . . . 6 |- ( ( A e. ( 0 (,) ( 2 x. pi ) ) /\ A < pi ) -> pi e. RR )
70		simpr 110	. . . . . 6 |- ( ( A e. ( 0 (,) ( 2 x. pi ) ) /\ A < pi ) -> A < pi )
71	64, 69, 70	ltled 8297	. . . . 5 |- ( ( A e. ( 0 (,) ( 2 x. pi ) ) /\ A < pi ) -> A <_ pi )
72	59, 4	elicc2i 10173	. . . . 5 |- ( A e. ( 0 [,] pi ) <-> ( A e. RR /\ 0 <_ A /\ A <_ pi ) )
73	64, 68, 71, 72	syl3anbrc 1207	. . . 4 |- ( ( A e. ( 0 (,) ( 2 x. pi ) ) /\ A < pi ) -> A e. ( 0 [,] pi ) )
74	63, 73, 67	cosordlem 15572	. . 3 |- ( ( A e. ( 0 (,) ( 2 x. pi ) ) /\ A < pi ) -> ( cos ` A ) < ( cos ` 0 ) )
75		cos0 12290	. . 3 |- ( cos ` 0 ) = 1
76	74, 75	breqtrdi 4129	. 2 |- ( ( A e. ( 0 (,) ( 2 x. pi ) ) /\ A < pi ) -> ( cos ` A ) < 1 )
77		pirp 15512	. . . 4 |- pi e. RR+
78		rphalflt 9917	. . . 4 |- ( pi e. RR+ -> ( pi / 2 ) < pi )
79	77, 78	ax-mp 5	. . 3 |- ( pi / 2 ) < pi
80		axltwlin 8246	. . . 4 |- ( ( ( pi / 2 ) e. RR /\ pi e. RR /\ A e. RR ) -> ( ( pi / 2 ) < pi -> ( ( pi / 2 ) < A \/ A < pi ) ) )
81	24, 4, 1, 80	mp3an12i 1377	. . 3 |- ( A e. ( 0 (,) ( 2 x. pi ) ) -> ( ( pi / 2 ) < pi -> ( ( pi / 2 ) < A \/ A < pi ) ) )
82	79, 81	mpi 15	. 2 |- ( A e. ( 0 (,) ( 2 x. pi ) ) -> ( ( pi / 2 ) < A \/ A < pi ) )
83	57, 76, 82	mpjaodan 805	1 |- ( A e. ( 0 (,) ( 2 x. pi ) ) -> ( cos ` A ) < 1 )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 715   /\ w3a 1004   e. wcel 2202   class class class wbr 4088   ` cfv 5326  (class class class)co 6017   RRcr 8030   0cc0 8031   1c1 8032   x. cmul 8036   RR*cxr 8212   < clt 8213   <_ cle 8214   / cdiv 8851   2c2 9193   3c3 9194   RR+crp 9887   (,)cioo 10122   [,)cico 10124   [,]cicc 10125   cosccos 12205   picpi 12207

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-mulrcl 8130  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-precex 8141  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147  ax-pre-mulgt0 8148  ax-pre-mulext 8149  ax-arch 8150  ax-caucvg 8151  ax-pre-suploc 8152  ax-addf 8153  ax-mulf 8154

This theorem depends on definitions:  df-bi 117  df-stab 838  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-disj 4065  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-isom 5335  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-of 6234  df-1st 6302  df-2nd 6303  df-recs 6470  df-irdg 6535  df-frec 6556  df-1o 6581  df-oadd 6585  df-er 6701  df-map 6818  df-pm 6819  df-en 6909  df-dom 6910  df-fin 6911  df-sup 7182  df-inf 7183  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-reap 8754  df-ap 8761  df-div 8852  df-inn 9143  df-2 9201  df-3 9202  df-4 9203  df-5 9204  df-6 9205  df-7 9206  df-8 9207  df-9 9208  df-n0 9402  df-z 9479  df-uz 9755  df-q 9853  df-rp 9888  df-xneg 10006  df-xadd 10007  df-ioo 10126  df-ioc 10127  df-ico 10128  df-icc 10129  df-fz 10243  df-fzo 10377  df-seqfrec 10709  df-exp 10800  df-fac 10987  df-bc 11009  df-ihash 11037  df-shft 11375  df-cj 11402  df-re 11403  df-im 11404  df-rsqrt 11558  df-abs 11559  df-clim 11839  df-sumdc 11914  df-ef 12208  df-sin 12210  df-cos 12211  df-pi 12213  df-rest 13323  df-topgen 13342  df-psmet 14556  df-xmet 14557  df-met 14558  df-bl 14559  df-mopn 14560  df-top 14721  df-topon 14734  df-bases 14766  df-ntr 14819  df-cn 14911  df-cnp 14912  df-tx 14976  df-cncf 15294  df-limced 15379  df-dvap 15380

This theorem is referenced by:  cos0pilt1  15575  taupi  16677