Theorem cos2bnd 13493

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 10426	. . . . . 6 |- 7 e. CC
2		9cn 10430	. . . . . 6 |- 9 e. CC
3		9re 10429	. . . . . . 7 |- 9 e. RR
4		9pos 10444	. . . . . . 7 |- 0 < 9
5	3, 4	gt0ne0ii 9897	. . . . . 6 |- 9 =/= 0
6		divneg 10047	. . . . . 6 |- ( ( 7 e. CC /\ 9 e. CC /\ 9 =/= 0 ) -> -u ( 7 / 9 ) = ( -u 7 / 9 ) )
7	1, 2, 5, 6	mp3an 1314	. . . . 5 |- -u ( 7 / 9 ) = ( -u 7 / 9 )
8		2cn 10413	. . . . . . 7 |- 2 e. CC
9	2, 5	pm3.2i 455	. . . . . . 7 |- ( 9 e. CC /\ 9 =/= 0 )
10		divsubdir 10048	. . . . . . 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 1314	. . . . . 6 |- ( ( 2 - 9 ) / 9 ) = ( ( 2 / 9 ) - ( 9 / 9 ) )
12	2, 8	negsubdi2i 9715	. . . . . . . 8 |- -u ( 9 - 2 ) = ( 2 - 9 )
13		7p2e9 10487	. . . . . . . . . 10 |- ( 7 + 2 ) = 9
14	2, 8, 1	subadd2i 9717	. . . . . . . . . 10 |- ( ( 9 - 2 ) = 7 <-> ( 7 + 2 ) = 9 )
15	13, 14	mpbir 209	. . . . . . . . 9 |- ( 9 - 2 ) = 7
16	15	negeqi 9624	. . . . . . . 8 |- -u ( 9 - 2 ) = -u 7
17	12, 16	eqtr3i 2465	. . . . . . 7 |- ( 2 - 9 ) = -u 7
18	17	oveq1i 6122	. . . . . 6 |- ( ( 2 - 9 ) / 9 ) = ( -u 7 / 9 )
19	11, 18	eqtr3i 2465	. . . . 5 |- ( ( 2 / 9 ) - ( 9 / 9 ) ) = ( -u 7 / 9 )
20	2, 5	dividi 10085	. . . . . 6 |- ( 9 / 9 ) = 1
21	20	oveq2i 6123	. . . . 5 |- ( ( 2 / 9 ) - ( 9 / 9 ) ) = ( ( 2 / 9 ) - 1 )
22	7, 19, 21	3eqtr2ri 2470	. . . 4 |- ( ( 2 / 9 ) - 1 ) = -u ( 7 / 9 )
23		ax-1cn 9361	. . . . . . . 8 |- 1 e. CC
24	8, 23, 2, 5	divassi 10108	. . . . . . 7 |- ( ( 2 x. 1 ) / 9 ) = ( 2 x. ( 1 / 9 ) )
25		2t1e2 10491	. . . . . . . 8 |- ( 2 x. 1 ) = 2
26	25	oveq1i 6122	. . . . . . 7 |- ( ( 2 x. 1 ) / 9 ) = ( 2 / 9 )
27	24, 26	eqtr3i 2465	. . . . . 6 |- ( 2 x. ( 1 / 9 ) ) = ( 2 / 9 )
28		3cn 10417	. . . . . . . . . 10 |- 3 e. CC
29		3ne0 10437	. . . . . . . . . 10 |- 3 =/= 0
30	23, 28, 29	sqdivi 11971	. . . . . . . . 9 |- ( ( 1 / 3 ) ^ 2 ) = ( ( 1 ^ 2 ) / ( 3 ^ 2 ) )
31		sq1 11981	. . . . . . . . . 10 |- ( 1 ^ 2 ) = 1
32		sq3 11984	. . . . . . . . . 10 |- ( 3 ^ 2 ) = 9
33	31, 32	oveq12i 6124	. . . . . . . . 9 |- ( ( 1 ^ 2 ) / ( 3 ^ 2 ) ) = ( 1 / 9 )
34	30, 33	eqtri 2463	. . . . . . . 8 |- ( ( 1 / 3 ) ^ 2 ) = ( 1 / 9 )
35		cos1bnd 13492	. . . . . . . . . 10 |- ( ( 1 / 3 ) < ( cos ` 1 ) /\ ( cos ` 1 ) < ( 2 / 3 ) )
36	35	simpli 458	. . . . . . . . 9 |- ( 1 / 3 ) < ( cos ` 1 )
37		0le1 9884	. . . . . . . . . . 11 |- 0 <_ 1
38		3pos 10436	. . . . . . . . . . 11 |- 0 < 3
39		1re 9406	. . . . . . . . . . . 12 |- 1 e. RR
40		3re 10416	. . . . . . . . . . . 12 |- 3 e. RR
41	39, 40	divge0i 10263	. . . . . . . . . . 11 |- ( ( 0 <_ 1 /\ 0 < 3 ) -> 0 <_ ( 1 / 3 ) )
42	37, 38, 41	mp2an 672	. . . . . . . . . 10 |- 0 <_ ( 1 / 3 )
43		0re 9407	. . . . . . . . . . 11 |- 0 e. RR
44		recoscl 13446	. . . . . . . . . . . 12 |- ( 1 e. RR -> ( cos ` 1 ) e. RR )
45	39, 44	ax-mp 5	. . . . . . . . . . 11 |- ( cos ` 1 ) e. RR
46	40, 29	rereccli 10117	. . . . . . . . . . . . 13 |- ( 1 / 3 ) e. RR
47	43, 46, 45	lelttri 9522	. . . . . . . . . . . 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 9518	. . . . . . . . . 10 |- 0 <_ ( cos ` 1 )
50	46, 45	lt2sqi 11975	. . . . . . . . . 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 4334	. . . . . . 7 |- ( 1 / 9 ) < ( ( cos ` 1 ) ^ 2 )
54		2pos 10434	. . . . . . . 8 |- 0 < 2
55	3, 5	rereccli 10117	. . . . . . . . 9 |- ( 1 / 9 ) e. RR
56	45	resqcli 11972	. . . . . . . . 9 |- ( ( cos ` 1 ) ^ 2 ) e. RR
57		2re 10412	. . . . . . . . 9 |- 2 e. RR
58	55, 56, 57	ltmul2i 10275	. . . . . . . 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 4334	. . . . 5 |- ( 2 / 9 ) < ( 2 x. ( ( cos ` 1 ) ^ 2 ) )
62	57, 3, 5	redivcli 10119	. . . . . 6 |- ( 2 / 9 ) e. RR
63	57, 56	remulcli 9421	. . . . . 6 |- ( 2 x. ( ( cos ` 1 ) ^ 2 ) ) e. RR
64		ltsub1 9856	. . . . . 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 1314	. . . . 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 4334	. . 3 |- -u ( 7 / 9 ) < ( ( 2 x. ( ( cos ` 1 ) ^ 2 ) ) - 1 )
68	25	fveq2i 5715	. . . 4 |- ( cos ` ( 2 x. 1 ) ) = ( cos ` 2 )
69		cos2t 13483	. . . . 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 2465	. . 3 |- ( cos ` 2 ) = ( ( 2 x. ( ( cos ` 1 ) ^ 2 ) ) - 1 )
72	67, 71	breqtrri 4338	. 2 |- -u ( 7 / 9 ) < ( cos ` 2 )
73	35	simpri 462	. . . . . . . . 9 |- ( cos ` 1 ) < ( 2 / 3 )
74		0le2 10433	. . . . . . . . . . 11 |- 0 <_ 2
75	57, 40	divge0i 10263	. . . . . . . . . . 11 |- ( ( 0 <_ 2 /\ 0 < 3 ) -> 0 <_ ( 2 / 3 ) )
76	74, 38, 75	mp2an 672	. . . . . . . . . 10 |- 0 <_ ( 2 / 3 )
77	57, 40, 29	redivcli 10119	. . . . . . . . . . 11 |- ( 2 / 3 ) e. RR
78	45, 77	lt2sqi 11975	. . . . . . . . . 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 11971	. . . . . . . . 9 |- ( ( 2 / 3 ) ^ 2 ) = ( ( 2 ^ 2 ) / ( 3 ^ 2 ) )
82		sq2 11983	. . . . . . . . . 10 |- ( 2 ^ 2 ) = 4
83	82, 32	oveq12i 6124	. . . . . . . . 9 |- ( ( 2 ^ 2 ) / ( 3 ^ 2 ) ) = ( 4 / 9 )
84	81, 83	eqtri 2463	. . . . . . . 8 |- ( ( 2 / 3 ) ^ 2 ) = ( 4 / 9 )
85	80, 84	breqtri 4336	. . . . . . 7 |- ( ( cos ` 1 ) ^ 2 ) < ( 4 / 9 )
86		4re 10419	. . . . . . . . . 10 |- 4 e. RR
87	86, 3, 5	redivcli 10119	. . . . . . . . 9 |- ( 4 / 9 ) e. RR
88	56, 87, 57	ltmul2i 10275	. . . . . . . 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 10420	. . . . . . . 8 |- 4 e. CC
92	8, 91, 2, 5	divassi 10108	. . . . . . 7 |- ( ( 2 x. 4 ) / 9 ) = ( 2 x. ( 4 / 9 ) )
93		4t2e8 10496	. . . . . . . . 9 |- ( 4 x. 2 ) = 8
94	91, 8, 93	mulcomli 9414	. . . . . . . 8 |- ( 2 x. 4 ) = 8
95	94	oveq1i 6122	. . . . . . 7 |- ( ( 2 x. 4 ) / 9 ) = ( 8 / 9 )
96	92, 95	eqtr3i 2465	. . . . . 6 |- ( 2 x. ( 4 / 9 ) ) = ( 8 / 9 )
97	90, 96	breqtri 4336	. . . . 5 |- ( 2 x. ( ( cos ` 1 ) ^ 2 ) ) < ( 8 / 9 )
98		8re 10427	. . . . . . 7 |- 8 e. RR
99	98, 3, 5	redivcli 10119	. . . . . 6 |- ( 8 / 9 ) e. RR
100		ltsub1 9856	. . . . . 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 1314	. . . . 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 6123	. . . . 5 |- ( ( 8 / 9 ) - ( 9 / 9 ) ) = ( ( 8 / 9 ) - 1 )
104		divneg 10047	. . . . . . 7 |- ( ( 1 e. CC /\ 9 e. CC /\ 9 =/= 0 ) -> -u ( 1 / 9 ) = ( -u 1 / 9 ) )
105	23, 2, 5, 104	mp3an 1314	. . . . . 6 |- -u ( 1 / 9 ) = ( -u 1 / 9 )
106		8cn 10428	. . . . . . . . 9 |- 8 e. CC
107	2, 106	negsubdi2i 9715	. . . . . . . 8 |- -u ( 9 - 8 ) = ( 8 - 9 )
108		8p1e9 10473	. . . . . . . . . 10 |- ( 8 + 1 ) = 9
109	2, 106, 23, 108	subaddrii 9718	. . . . . . . . 9 |- ( 9 - 8 ) = 1
110	109	negeqi 9624	. . . . . . . 8 |- -u ( 9 - 8 ) = -u 1
111	107, 110	eqtr3i 2465	. . . . . . 7 |- ( 8 - 9 ) = -u 1
112	111	oveq1i 6122	. . . . . 6 |- ( ( 8 - 9 ) / 9 ) = ( -u 1 / 9 )
113		divsubdir 10048	. . . . . . 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 1314	. . . . . 6 |- ( ( 8 - 9 ) / 9 ) = ( ( 8 / 9 ) - ( 9 / 9 ) )
115	105, 112, 114	3eqtr2ri 2470	. . . . 5 |- ( ( 8 / 9 ) - ( 9 / 9 ) ) = -u ( 1 / 9 )
116	103, 115	eqtr3i 2465	. . . 4 |- ( ( 8 / 9 ) - 1 ) = -u ( 1 / 9 )
117	102, 116	breqtri 4336	. . 3 |- ( ( 2 x. ( ( cos ` 1 ) ^ 2 ) ) - 1 ) < -u ( 1 / 9 )
118	71, 117	eqbrtri 4332	. 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 1369   e. wcel 1756   =/= wne 2620   class class class wbr 4313   ` cfv 5439  (class class class)co 6112   CCcc 9301   RRcr 9302   0cc0 9303   1c1 9304   + caddc 9306   x. cmul 9308   < clt 9439   <_ cle 9440   - cmin 9616   -ucneg 9617   / cdiv 10014   2c2 10392   3c3 10393   4c4 10394   7c7 10397   8c8 10398   9c9 10399   ^cexp 11886   cosccos 13371

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-inf2 7868  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380  ax-pre-sup 9381  ax-addf 9382  ax-mulf 9383

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-int 4150  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-se 4701  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-isom 5448  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-om 6498  df-1st 6598  df-2nd 6599  df-recs 6853  df-rdg 6887  df-1o 6941  df-oadd 6945  df-er 7122  df-pm 7238  df-en 7332  df-dom 7333  df-sdom 7334  df-fin 7335  df-sup 7712  df-oi 7745  df-card 8130  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-div 10015  df-nn 10344  df-2 10401  df-3 10402  df-4 10403  df-5 10404  df-6 10405  df-7 10406  df-8 10407  df-9 10408  df-n0 10601  df-z 10668  df-uz 10883  df-rp 11013  df-ioc 11326  df-ico 11327  df-fz 11459  df-fzo 11570  df-fl 11663  df-seq 11828  df-exp 11887  df-fac 12073  df-bc 12100  df-hash 12125  df-shft 12577  df-cj 12609  df-re 12610  df-im 12611  df-sqr 12745  df-abs 12746  df-limsup 12970  df-clim 12987  df-rlim 12988  df-sum 13185  df-ef 13374  df-sin 13376  df-cos 13377

This theorem is referenced by:  sincos2sgn  13499