Theorem cos2bnd 13904

Description: Bounds on the cosine of 2. (Contributed by Paul Chapman, 19-Jan-2008.)

Assertion

Ref	Expression
cos2bnd	|- ( -u ( 7 / 9 ) < ( cos ` 2 ) /\ ( cos ` 2 ) < -u ( 1 / 9 ) )

Proof of Theorem cos2bnd

Step	Hyp	Ref	Expression
1		7cn 10626	. . . . . 6 |- 7 e. CC
2		9cn 10630	. . . . . 6 |- 9 e. CC
3		9re 10629	. . . . . . 7 |- 9 e. RR
4		9pos 10644	. . . . . . 7 |- 0 < 9
5	3, 4	gt0ne0ii 10096	. . . . . 6 |- 9 =/= 0
6		divneg 10246	. . . . . 6 |- ( ( 7 e. CC /\ 9 e. CC /\ 9 =/= 0 ) -> -u ( 7 / 9 ) = ( -u 7 / 9 ) )
7	1, 2, 5, 6	mp3an 1325	. . . . 5 |- -u ( 7 / 9 ) = ( -u 7 / 9 )
8		2cn 10613	. . . . . . 7 |- 2 e. CC
9	2, 5	pm3.2i 455	. . . . . . 7 |- ( 9 e. CC /\ 9 =/= 0 )
10		divsubdir 10247	. . . . . . 7 |- ( ( 2 e. CC /\ 9 e. CC /\ ( 9 e. CC /\ 9 =/= 0 ) ) -> ( ( 2 - 9 ) / 9 ) = ( ( 2 / 9 ) - ( 9 / 9 ) ) )
11	8, 2, 9, 10	mp3an 1325	. . . . . 6 |- ( ( 2 - 9 ) / 9 ) = ( ( 2 / 9 ) - ( 9 / 9 ) )
12	2, 8	negsubdi2i 9911	. . . . . . . 8 |- -u ( 9 - 2 ) = ( 2 - 9 )
13		7p2e9 10687	. . . . . . . . . 10 |- ( 7 + 2 ) = 9
14	2, 8, 1	subadd2i 9913	. . . . . . . . . 10 |- ( ( 9 - 2 ) = 7 <-> ( 7 + 2 ) = 9 )
15	13, 14	mpbir 209	. . . . . . . . 9 |- ( 9 - 2 ) = 7
16	15	negeqi 9818	. . . . . . . 8 |- -u ( 9 - 2 ) = -u 7
17	12, 16	eqtr3i 2474	. . . . . . 7 |- ( 2 - 9 ) = -u 7
18	17	oveq1i 6291	. . . . . 6 |- ( ( 2 - 9 ) / 9 ) = ( -u 7 / 9 )
19	11, 18	eqtr3i 2474	. . . . 5 |- ( ( 2 / 9 ) - ( 9 / 9 ) ) = ( -u 7 / 9 )
20	2, 5	dividi 10284	. . . . . 6 |- ( 9 / 9 ) = 1
21	20	oveq2i 6292	. . . . 5 |- ( ( 2 / 9 ) - ( 9 / 9 ) ) = ( ( 2 / 9 ) - 1 )
22	7, 19, 21	3eqtr2ri 2479	. . . 4 |- ( ( 2 / 9 ) - 1 ) = -u ( 7 / 9 )
23		ax-1cn 9553	. . . . . . . 8 |- 1 e. CC
24	8, 23, 2, 5	divassi 10307	. . . . . . 7 |- ( ( 2 x. 1 ) / 9 ) = ( 2 x. ( 1 / 9 ) )
25		2t1e2 10691	. . . . . . . 8 |- ( 2 x. 1 ) = 2
26	25	oveq1i 6291	. . . . . . 7 |- ( ( 2 x. 1 ) / 9 ) = ( 2 / 9 )
27	24, 26	eqtr3i 2474	. . . . . 6 |- ( 2 x. ( 1 / 9 ) ) = ( 2 / 9 )
28		3cn 10617	. . . . . . . . . 10 |- 3 e. CC
29		3ne0 10637	. . . . . . . . . 10 |- 3 =/= 0
30	23, 28, 29	sqdivi 12233	. . . . . . . . 9 |- ( ( 1 / 3 ) ^ 2 ) = ( ( 1 ^ 2 ) / ( 3 ^ 2 ) )
31		sq1 12243	. . . . . . . . . 10 |- ( 1 ^ 2 ) = 1
32		sq3 12246	. . . . . . . . . 10 |- ( 3 ^ 2 ) = 9
33	31, 32	oveq12i 6293	. . . . . . . . 9 |- ( ( 1 ^ 2 ) / ( 3 ^ 2 ) ) = ( 1 / 9 )
34	30, 33	eqtri 2472	. . . . . . . 8 |- ( ( 1 / 3 ) ^ 2 ) = ( 1 / 9 )
35		cos1bnd 13903	. . . . . . . . . 10 |- ( ( 1 / 3 ) < ( cos ` 1 ) /\ ( cos ` 1 ) < ( 2 / 3 ) )
36	35	simpli 458	. . . . . . . . 9 |- ( 1 / 3 ) < ( cos ` 1 )
37		0le1 10083	. . . . . . . . . . 11 |- 0 <_ 1
38		3pos 10636	. . . . . . . . . . 11 |- 0 < 3
39		1re 9598	. . . . . . . . . . . 12 |- 1 e. RR
40		3re 10616	. . . . . . . . . . . 12 |- 3 e. RR
41	39, 40	divge0i 10462	. . . . . . . . . . 11 |- ( ( 0 <_ 1 /\ 0 < 3 ) -> 0 <_ ( 1 / 3 ) )
42	37, 38, 41	mp2an 672	. . . . . . . . . 10 |- 0 <_ ( 1 / 3 )
43		0re 9599	. . . . . . . . . . 11 |- 0 e. RR
44		recoscl 13857	. . . . . . . . . . . 12 |- ( 1 e. RR -> ( cos ` 1 ) e. RR )
45	39, 44	ax-mp 5	. . . . . . . . . . 11 |- ( cos ` 1 ) e. RR
46	40, 29	rereccli 10316	. . . . . . . . . . . . 13 |- ( 1 / 3 ) e. RR
47	43, 46, 45	lelttri 9714	. . . . . . . . . . . 12 |- ( ( 0 <_ ( 1 / 3 ) /\ ( 1 / 3 ) < ( cos ` 1 ) ) -> 0 < ( cos ` 1 ) )
48	42, 36, 47	mp2an 672	. . . . . . . . . . 11 |- 0 < ( cos ` 1 )
49	43, 45, 48	ltleii 9710	. . . . . . . . . 10 |- 0 <_ ( cos ` 1 )
50	46, 45	lt2sqi 12237	. . . . . . . . . 10 |- ( ( 0 <_ ( 1 / 3 ) /\ 0 <_ ( cos ` 1 ) ) -> ( ( 1 / 3 ) < ( cos ` 1 ) <-> ( ( 1 / 3 ) ^ 2 ) < ( ( cos ` 1 ) ^ 2 ) ) )
51	42, 49, 50	mp2an 672	. . . . . . . . 9 |- ( ( 1 / 3 ) < ( cos ` 1 ) <-> ( ( 1 / 3 ) ^ 2 ) < ( ( cos ` 1 ) ^ 2 ) )
52	36, 51	mpbi 208	. . . . . . . 8 |- ( ( 1 / 3 ) ^ 2 ) < ( ( cos ` 1 ) ^ 2 )
53	34, 52	eqbrtrri 4458	. . . . . . 7 |- ( 1 / 9 ) < ( ( cos ` 1 ) ^ 2 )
54		2pos 10634	. . . . . . . 8 |- 0 < 2
55	3, 5	rereccli 10316	. . . . . . . . 9 |- ( 1 / 9 ) e. RR
56	45	resqcli 12234	. . . . . . . . 9 |- ( ( cos ` 1 ) ^ 2 ) e. RR
57		2re 10612	. . . . . . . . 9 |- 2 e. RR
58	55, 56, 57	ltmul2i 10474	. . . . . . . 8 |- ( 0 < 2 -> ( ( 1 / 9 ) < ( ( cos ` 1 ) ^ 2 ) <-> ( 2 x. ( 1 / 9 ) ) < ( 2 x. ( ( cos ` 1 ) ^ 2 ) ) ) )
59	54, 58	ax-mp 5	. . . . . . 7 |- ( ( 1 / 9 ) < ( ( cos ` 1 ) ^ 2 ) <-> ( 2 x. ( 1 / 9 ) ) < ( 2 x. ( ( cos ` 1 ) ^ 2 ) ) )
60	53, 59	mpbi 208	. . . . . 6 |- ( 2 x. ( 1 / 9 ) ) < ( 2 x. ( ( cos ` 1 ) ^ 2 ) )
61	27, 60	eqbrtrri 4458	. . . . 5 |- ( 2 / 9 ) < ( 2 x. ( ( cos ` 1 ) ^ 2 ) )
62	57, 3, 5	redivcli 10318	. . . . . 6 |- ( 2 / 9 ) e. RR
63	57, 56	remulcli 9613	. . . . . 6 |- ( 2 x. ( ( cos ` 1 ) ^ 2 ) ) e. RR
64		ltsub1 10055	. . . . . 6 |- ( ( ( 2 / 9 ) e. RR /\ ( 2 x. ( ( cos ` 1 ) ^ 2 ) ) e. RR /\ 1 e. RR ) -> ( ( 2 / 9 ) < ( 2 x. ( ( cos ` 1 ) ^ 2 ) ) <-> ( ( 2 / 9 ) - 1 ) < ( ( 2 x. ( ( cos ` 1 ) ^ 2 ) ) - 1 ) ) )
65	62, 63, 39, 64	mp3an 1325	. . . . 5 |- ( ( 2 / 9 ) < ( 2 x. ( ( cos ` 1 ) ^ 2 ) ) <-> ( ( 2 / 9 ) - 1 ) < ( ( 2 x. ( ( cos ` 1 ) ^ 2 ) ) - 1 ) )
66	61, 65	mpbi 208	. . . 4 |- ( ( 2 / 9 ) - 1 ) < ( ( 2 x. ( ( cos ` 1 ) ^ 2 ) ) - 1 )
67	22, 66	eqbrtrri 4458	. . 3 |- -u ( 7 / 9 ) < ( ( 2 x. ( ( cos ` 1 ) ^ 2 ) ) - 1 )
68	25	fveq2i 5859	. . . 4 |- ( cos ` ( 2 x. 1 ) ) = ( cos ` 2 )
69		cos2t 13894	. . . . 5 |- ( 1 e. CC -> ( cos ` ( 2 x. 1 ) ) = ( ( 2 x. ( ( cos ` 1 ) ^ 2 ) ) - 1 ) )
70	23, 69	ax-mp 5	. . . 4 |- ( cos ` ( 2 x. 1 ) ) = ( ( 2 x. ( ( cos ` 1 ) ^ 2 ) ) - 1 )
71	68, 70	eqtr3i 2474	. . 3 |- ( cos ` 2 ) = ( ( 2 x. ( ( cos ` 1 ) ^ 2 ) ) - 1 )
72	67, 71	breqtrri 4462	. 2 |- -u ( 7 / 9 ) < ( cos ` 2 )
73	35	simpri 462	. . . . . . . . 9 |- ( cos ` 1 ) < ( 2 / 3 )
74		0le2 10633	. . . . . . . . . . 11 |- 0 <_ 2
75	57, 40	divge0i 10462	. . . . . . . . . . 11 |- ( ( 0 <_ 2 /\ 0 < 3 ) -> 0 <_ ( 2 / 3 ) )
76	74, 38, 75	mp2an 672	. . . . . . . . . 10 |- 0 <_ ( 2 / 3 )
77	57, 40, 29	redivcli 10318	. . . . . . . . . . 11 |- ( 2 / 3 ) e. RR
78	45, 77	lt2sqi 12237	. . . . . . . . . 10 |- ( ( 0 <_ ( cos ` 1 ) /\ 0 <_ ( 2 / 3 ) ) -> ( ( cos ` 1 ) < ( 2 / 3 ) <-> ( ( cos ` 1 ) ^ 2 ) < ( ( 2 / 3 ) ^ 2 ) ) )
79	49, 76, 78	mp2an 672	. . . . . . . . 9 |- ( ( cos ` 1 ) < ( 2 / 3 ) <-> ( ( cos ` 1 ) ^ 2 ) < ( ( 2 / 3 ) ^ 2 ) )
80	73, 79	mpbi 208	. . . . . . . 8 |- ( ( cos ` 1 ) ^ 2 ) < ( ( 2 / 3 ) ^ 2 )
81	8, 28, 29	sqdivi 12233	. . . . . . . . 9 |- ( ( 2 / 3 ) ^ 2 ) = ( ( 2 ^ 2 ) / ( 3 ^ 2 ) )
82		sq2 12245	. . . . . . . . . 10 |- ( 2 ^ 2 ) = 4
83	82, 32	oveq12i 6293	. . . . . . . . 9 |- ( ( 2 ^ 2 ) / ( 3 ^ 2 ) ) = ( 4 / 9 )
84	81, 83	eqtri 2472	. . . . . . . 8 |- ( ( 2 / 3 ) ^ 2 ) = ( 4 / 9 )
85	80, 84	breqtri 4460	. . . . . . 7 |- ( ( cos ` 1 ) ^ 2 ) < ( 4 / 9 )
86		4re 10619	. . . . . . . . . 10 |- 4 e. RR
87	86, 3, 5	redivcli 10318	. . . . . . . . 9 |- ( 4 / 9 ) e. RR
88	56, 87, 57	ltmul2i 10474	. . . . . . . 8 |- ( 0 < 2 -> ( ( ( cos ` 1 ) ^ 2 ) < ( 4 / 9 ) <-> ( 2 x. ( ( cos ` 1 ) ^ 2 ) ) < ( 2 x. ( 4 / 9 ) ) ) )
89	54, 88	ax-mp 5	. . . . . . 7 |- ( ( ( cos ` 1 ) ^ 2 ) < ( 4 / 9 ) <-> ( 2 x. ( ( cos ` 1 ) ^ 2 ) ) < ( 2 x. ( 4 / 9 ) ) )
90	85, 89	mpbi 208	. . . . . 6 |- ( 2 x. ( ( cos ` 1 ) ^ 2 ) ) < ( 2 x. ( 4 / 9 ) )
91		4cn 10620	. . . . . . . 8 |- 4 e. CC
92	8, 91, 2, 5	divassi 10307	. . . . . . 7 |- ( ( 2 x. 4 ) / 9 ) = ( 2 x. ( 4 / 9 ) )
93		4t2e8 10696	. . . . . . . . 9 |- ( 4 x. 2 ) = 8
94	91, 8, 93	mulcomli 9606	. . . . . . . 8 |- ( 2 x. 4 ) = 8
95	94	oveq1i 6291	. . . . . . 7 |- ( ( 2 x. 4 ) / 9 ) = ( 8 / 9 )
96	92, 95	eqtr3i 2474	. . . . . 6 |- ( 2 x. ( 4 / 9 ) ) = ( 8 / 9 )
97	90, 96	breqtri 4460	. . . . 5 |- ( 2 x. ( ( cos ` 1 ) ^ 2 ) ) < ( 8 / 9 )
98		8re 10627	. . . . . . 7 |- 8 e. RR
99	98, 3, 5	redivcli 10318	. . . . . 6 |- ( 8 / 9 ) e. RR
100		ltsub1 10055	. . . . . 6 |- ( ( ( 2 x. ( ( cos ` 1 ) ^ 2 ) ) e. RR /\ ( 8 / 9 ) e. RR /\ 1 e. RR ) -> ( ( 2 x. ( ( cos ` 1 ) ^ 2 ) ) < ( 8 / 9 ) <-> ( ( 2 x. ( ( cos ` 1 ) ^ 2 ) ) - 1 ) < ( ( 8 / 9 ) - 1 ) ) )
101	63, 99, 39, 100	mp3an 1325	. . . . 5 |- ( ( 2 x. ( ( cos ` 1 ) ^ 2 ) ) < ( 8 / 9 ) <-> ( ( 2 x. ( ( cos ` 1 ) ^ 2 ) ) - 1 ) < ( ( 8 / 9 ) - 1 ) )
102	97, 101	mpbi 208	. . . 4 |- ( ( 2 x. ( ( cos ` 1 ) ^ 2 ) ) - 1 ) < ( ( 8 / 9 ) - 1 )
103	20	oveq2i 6292	. . . . 5 |- ( ( 8 / 9 ) - ( 9 / 9 ) ) = ( ( 8 / 9 ) - 1 )
104		divneg 10246	. . . . . . 7 |- ( ( 1 e. CC /\ 9 e. CC /\ 9 =/= 0 ) -> -u ( 1 / 9 ) = ( -u 1 / 9 ) )
105	23, 2, 5, 104	mp3an 1325	. . . . . 6 |- -u ( 1 / 9 ) = ( -u 1 / 9 )
106		8cn 10628	. . . . . . . . 9 |- 8 e. CC
107	2, 106	negsubdi2i 9911	. . . . . . . 8 |- -u ( 9 - 8 ) = ( 8 - 9 )
108		8p1e9 10673	. . . . . . . . . 10 |- ( 8 + 1 ) = 9
109	2, 106, 23, 108	subaddrii 9914	. . . . . . . . 9 |- ( 9 - 8 ) = 1
110	109	negeqi 9818	. . . . . . . 8 |- -u ( 9 - 8 ) = -u 1
111	107, 110	eqtr3i 2474	. . . . . . 7 |- ( 8 - 9 ) = -u 1
112	111	oveq1i 6291	. . . . . 6 |- ( ( 8 - 9 ) / 9 ) = ( -u 1 / 9 )
113		divsubdir 10247	. . . . . . 7 |- ( ( 8 e. CC /\ 9 e. CC /\ ( 9 e. CC /\ 9 =/= 0 ) ) -> ( ( 8 - 9 ) / 9 ) = ( ( 8 / 9 ) - ( 9 / 9 ) ) )
114	106, 2, 9, 113	mp3an 1325	. . . . . 6 |- ( ( 8 - 9 ) / 9 ) = ( ( 8 / 9 ) - ( 9 / 9 ) )
115	105, 112, 114	3eqtr2ri 2479	. . . . 5 |- ( ( 8 / 9 ) - ( 9 / 9 ) ) = -u ( 1 / 9 )
116	103, 115	eqtr3i 2474	. . . 4 |- ( ( 8 / 9 ) - 1 ) = -u ( 1 / 9 )
117	102, 116	breqtri 4460	. . 3 |- ( ( 2 x. ( ( cos ` 1 ) ^ 2 ) ) - 1 ) < -u ( 1 / 9 )
118	71, 117	eqbrtri 4456	. 2 |- ( cos ` 2 ) < -u ( 1 / 9 )
119	72, 118	pm3.2i 455	1 |- ( -u ( 7 / 9 ) < ( cos ` 2 ) /\ ( cos ` 2 ) < -u ( 1 / 9 ) )

Syntax hints:   <-> wb 184   /\ wa 369   = wceq 1383   e. wcel 1804   =/= wne 2638   class class class wbr 4437   ` cfv 5578  (class class class)co 6281   CCcc 9493   RRcr 9494   0cc0 9495   1c1 9496   + caddc 9498   x. cmul 9500   < clt 9631   <_ cle 9632   - cmin 9810   -ucneg 9811   / cdiv 10213   2c2 10592   3c3 10593   4c4 10594   7c7 10597   8c8 10598   9c9 10599   ^cexp 12147   cosccos 13781

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-pre-sup 9573  ax-addf 9574  ax-mulf 9575

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-fal 1389  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-oadd 7136  df-er 7313  df-pm 7425  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-sup 7903  df-oi 7938  df-card 8323  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10214  df-nn 10544  df-2 10601  df-3 10602  df-4 10603  df-5 10604  df-6 10605  df-7 10606  df-8 10607  df-9 10608  df-n0 10803  df-z 10872  df-uz 11092  df-rp 11231  df-ioc 11544  df-ico 11545  df-fz 11683  df-fzo 11806  df-fl 11910  df-seq 12089  df-exp 12148  df-fac 12335  df-bc 12362  df-hash 12387  df-shft 12881  df-cj 12913  df-re 12914  df-im 12915  df-sqrt 13049  df-abs 13050  df-limsup 13275  df-clim 13292  df-rlim 13293  df-sum 13490  df-ef 13784  df-sin 13786  df-cos 13787

This theorem is referenced by:  sincos2sgn  13910