Theorem cos2bnd 12320

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 9226	. . . . . 6 |- 7 e. CC
2		9cn 9230	. . . . . 6 |- 9 e. CC
3		9re 9229	. . . . . . 7 |- 9 e. RR
4		9pos 9246	. . . . . . 7 |- 0 < 9
5	3, 4	gt0ap0ii 8807	. . . . . 6 |- 9 # 0
6		divnegap 8885	. . . . . 6 |- ( ( 7 e. CC /\ 9 e. CC /\ 9 # 0 ) -> -u ( 7 / 9 ) = ( -u 7 / 9 ) )
7	1, 2, 5, 6	mp3an 1373	. . . . 5 |- -u ( 7 / 9 ) = ( -u 7 / 9 )
8		2cn 9213	. . . . . . 7 |- 2 e. CC
9	2, 5	pm3.2i 272	. . . . . . 7 |- ( 9 e. CC /\ 9 # 0 )
10		divsubdirap 8887	. . . . . . 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 1373	. . . . . 6 |- ( ( 2 - 9 ) / 9 ) = ( ( 2 / 9 ) - ( 9 / 9 ) )
12	2, 8	negsubdi2i 8464	. . . . . . . 8 |- -u ( 9 - 2 ) = ( 2 - 9 )
13		7p2e9 9294	. . . . . . . . . 10 |- ( 7 + 2 ) = 9
14	2, 8, 1	subadd2i 8466	. . . . . . . . . 10 |- ( ( 9 - 2 ) = 7 <-> ( 7 + 2 ) = 9 )
15	13, 14	mpbir 146	. . . . . . . . 9 |- ( 9 - 2 ) = 7
16	15	negeqi 8372	. . . . . . . 8 |- -u ( 9 - 2 ) = -u 7
17	12, 16	eqtr3i 2254	. . . . . . 7 |- ( 2 - 9 ) = -u 7
18	17	oveq1i 6027	. . . . . 6 |- ( ( 2 - 9 ) / 9 ) = ( -u 7 / 9 )
19	11, 18	eqtr3i 2254	. . . . 5 |- ( ( 2 / 9 ) - ( 9 / 9 ) ) = ( -u 7 / 9 )
20	2, 5	dividapi 8924	. . . . . 6 |- ( 9 / 9 ) = 1
21	20	oveq2i 6028	. . . . 5 |- ( ( 2 / 9 ) - ( 9 / 9 ) ) = ( ( 2 / 9 ) - 1 )
22	7, 19, 21	3eqtr2ri 2259	. . . 4 |- ( ( 2 / 9 ) - 1 ) = -u ( 7 / 9 )
23		ax-1cn 8124	. . . . . . . 8 |- 1 e. CC
24	8, 23, 2, 5	divassapi 8947	. . . . . . 7 |- ( ( 2 x. 1 ) / 9 ) = ( 2 x. ( 1 / 9 ) )
25		2t1e2 9296	. . . . . . . 8 |- ( 2 x. 1 ) = 2
26	25	oveq1i 6027	. . . . . . 7 |- ( ( 2 x. 1 ) / 9 ) = ( 2 / 9 )
27	24, 26	eqtr3i 2254	. . . . . 6 |- ( 2 x. ( 1 / 9 ) ) = ( 2 / 9 )
28		3cn 9217	. . . . . . . . . 10 |- 3 e. CC
29		3ap0 9238	. . . . . . . . . 10 |- 3 # 0
30	23, 28, 29	sqdivapi 10884	. . . . . . . . 9 |- ( ( 1 / 3 ) ^ 2 ) = ( ( 1 ^ 2 ) / ( 3 ^ 2 ) )
31		sq1 10894	. . . . . . . . . 10 |- ( 1 ^ 2 ) = 1
32		sq3 10897	. . . . . . . . . 10 |- ( 3 ^ 2 ) = 9
33	31, 32	oveq12i 6029	. . . . . . . . 9 |- ( ( 1 ^ 2 ) / ( 3 ^ 2 ) ) = ( 1 / 9 )
34	30, 33	eqtri 2252	. . . . . . . 8 |- ( ( 1 / 3 ) ^ 2 ) = ( 1 / 9 )
35		cos1bnd 12319	. . . . . . . . . 10 |- ( ( 1 / 3 ) < ( cos ` 1 ) /\ ( cos ` 1 ) < ( 2 / 3 ) )
36	35	simpli 111	. . . . . . . . 9 |- ( 1 / 3 ) < ( cos ` 1 )
37		0le1 8660	. . . . . . . . . . 11 |- 0 <_ 1
38		3pos 9236	. . . . . . . . . . 11 |- 0 < 3
39		1re 8177	. . . . . . . . . . . 12 |- 1 e. RR
40		3re 9216	. . . . . . . . . . . 12 |- 3 e. RR
41	39, 40	divge0i 9090	. . . . . . . . . . 11 |- ( ( 0 <_ 1 /\ 0 < 3 ) -> 0 <_ ( 1 / 3 ) )
42	37, 38, 41	mp2an 426	. . . . . . . . . 10 |- 0 <_ ( 1 / 3 )
43		0re 8178	. . . . . . . . . . 11 |- 0 e. RR
44		recoscl 12281	. . . . . . . . . . . 12 |- ( 1 e. RR -> ( cos ` 1 ) e. RR )
45	39, 44	ax-mp 5	. . . . . . . . . . 11 |- ( cos ` 1 ) e. RR
46	40, 29	rerecclapi 8956	. . . . . . . . . . . . 13 |- ( 1 / 3 ) e. RR
47	43, 46, 45	lelttri 8284	. . . . . . . . . . . 12 |- ( ( 0 <_ ( 1 / 3 ) /\ ( 1 / 3 ) < ( cos ` 1 ) ) -> 0 < ( cos ` 1 ) )
48	42, 36, 47	mp2an 426	. . . . . . . . . . 11 |- 0 < ( cos ` 1 )
49	43, 45, 48	ltleii 8281	. . . . . . . . . 10 |- 0 <_ ( cos ` 1 )
50	46, 45	lt2sqi 10888	. . . . . . . . . 10 |- ( ( 0 <_ ( 1 / 3 ) /\ 0 <_ ( cos ` 1 ) ) -> ( ( 1 / 3 ) < ( cos ` 1 ) <-> ( ( 1 / 3 ) ^ 2 ) < ( ( cos ` 1 ) ^ 2 ) ) )
51	42, 49, 50	mp2an 426	. . . . . . . . 9 |- ( ( 1 / 3 ) < ( cos ` 1 ) <-> ( ( 1 / 3 ) ^ 2 ) < ( ( cos ` 1 ) ^ 2 ) )
52	36, 51	mpbi 145	. . . . . . . 8 |- ( ( 1 / 3 ) ^ 2 ) < ( ( cos ` 1 ) ^ 2 )
53	34, 52	eqbrtrri 4111	. . . . . . 7 |- ( 1 / 9 ) < ( ( cos ` 1 ) ^ 2 )
54		2pos 9233	. . . . . . . 8 |- 0 < 2
55	3, 5	rerecclapi 8956	. . . . . . . . 9 |- ( 1 / 9 ) e. RR
56	45	resqcli 10885	. . . . . . . . 9 |- ( ( cos ` 1 ) ^ 2 ) e. RR
57		2re 9212	. . . . . . . . 9 |- 2 e. RR
58	55, 56, 57	ltmul2i 9102	. . . . . . . 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 145	. . . . . 6 |- ( 2 x. ( 1 / 9 ) ) < ( 2 x. ( ( cos ` 1 ) ^ 2 ) )
61	27, 60	eqbrtrri 4111	. . . . 5 |- ( 2 / 9 ) < ( 2 x. ( ( cos ` 1 ) ^ 2 ) )
62	57, 3, 5	redivclapi 8958	. . . . . 6 |- ( 2 / 9 ) e. RR
63	57, 56	remulcli 8192	. . . . . 6 |- ( 2 x. ( ( cos ` 1 ) ^ 2 ) ) e. RR
64		ltsub1 8637	. . . . . 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 1373	. . . . 5 |- ( ( 2 / 9 ) < ( 2 x. ( ( cos ` 1 ) ^ 2 ) ) <-> ( ( 2 / 9 ) - 1 ) < ( ( 2 x. ( ( cos ` 1 ) ^ 2 ) ) - 1 ) )
66	61, 65	mpbi 145	. . . 4 |- ( ( 2 / 9 ) - 1 ) < ( ( 2 x. ( ( cos ` 1 ) ^ 2 ) ) - 1 )
67	22, 66	eqbrtrri 4111	. . 3 |- -u ( 7 / 9 ) < ( ( 2 x. ( ( cos ` 1 ) ^ 2 ) ) - 1 )
68	25	fveq2i 5642	. . . 4 |- ( cos ` ( 2 x. 1 ) ) = ( cos ` 2 )
69		cos2t 12310	. . . . 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 2254	. . 3 |- ( cos ` 2 ) = ( ( 2 x. ( ( cos ` 1 ) ^ 2 ) ) - 1 )
72	67, 71	breqtrri 4115	. 2 |- -u ( 7 / 9 ) < ( cos ` 2 )
73	35	simpri 113	. . . . . . . . 9 |- ( cos ` 1 ) < ( 2 / 3 )
74		0le2 9232	. . . . . . . . . . 11 |- 0 <_ 2
75	57, 40	divge0i 9090	. . . . . . . . . . 11 |- ( ( 0 <_ 2 /\ 0 < 3 ) -> 0 <_ ( 2 / 3 ) )
76	74, 38, 75	mp2an 426	. . . . . . . . . 10 |- 0 <_ ( 2 / 3 )
77	57, 40, 29	redivclapi 8958	. . . . . . . . . . 11 |- ( 2 / 3 ) e. RR
78	45, 77	lt2sqi 10888	. . . . . . . . . 10 |- ( ( 0 <_ ( cos ` 1 ) /\ 0 <_ ( 2 / 3 ) ) -> ( ( cos ` 1 ) < ( 2 / 3 ) <-> ( ( cos ` 1 ) ^ 2 ) < ( ( 2 / 3 ) ^ 2 ) ) )
79	49, 76, 78	mp2an 426	. . . . . . . . 9 |- ( ( cos ` 1 ) < ( 2 / 3 ) <-> ( ( cos ` 1 ) ^ 2 ) < ( ( 2 / 3 ) ^ 2 ) )
80	73, 79	mpbi 145	. . . . . . . 8 |- ( ( cos ` 1 ) ^ 2 ) < ( ( 2 / 3 ) ^ 2 )
81	8, 28, 29	sqdivapi 10884	. . . . . . . . 9 |- ( ( 2 / 3 ) ^ 2 ) = ( ( 2 ^ 2 ) / ( 3 ^ 2 ) )
82		sq2 10896	. . . . . . . . . 10 |- ( 2 ^ 2 ) = 4
83	82, 32	oveq12i 6029	. . . . . . . . 9 |- ( ( 2 ^ 2 ) / ( 3 ^ 2 ) ) = ( 4 / 9 )
84	81, 83	eqtri 2252	. . . . . . . 8 |- ( ( 2 / 3 ) ^ 2 ) = ( 4 / 9 )
85	80, 84	breqtri 4113	. . . . . . 7 |- ( ( cos ` 1 ) ^ 2 ) < ( 4 / 9 )
86		4re 9219	. . . . . . . . . 10 |- 4 e. RR
87	86, 3, 5	redivclapi 8958	. . . . . . . . 9 |- ( 4 / 9 ) e. RR
88	56, 87, 57	ltmul2i 9102	. . . . . . . 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 145	. . . . . 6 |- ( 2 x. ( ( cos ` 1 ) ^ 2 ) ) < ( 2 x. ( 4 / 9 ) )
91		4cn 9220	. . . . . . . 8 |- 4 e. CC
92	8, 91, 2, 5	divassapi 8947	. . . . . . 7 |- ( ( 2 x. 4 ) / 9 ) = ( 2 x. ( 4 / 9 ) )
93		4t2e8 9301	. . . . . . . . 9 |- ( 4 x. 2 ) = 8
94	91, 8, 93	mulcomli 8185	. . . . . . . 8 |- ( 2 x. 4 ) = 8
95	94	oveq1i 6027	. . . . . . 7 |- ( ( 2 x. 4 ) / 9 ) = ( 8 / 9 )
96	92, 95	eqtr3i 2254	. . . . . 6 |- ( 2 x. ( 4 / 9 ) ) = ( 8 / 9 )
97	90, 96	breqtri 4113	. . . . 5 |- ( 2 x. ( ( cos ` 1 ) ^ 2 ) ) < ( 8 / 9 )
98		8re 9227	. . . . . . 7 |- 8 e. RR
99	98, 3, 5	redivclapi 8958	. . . . . 6 |- ( 8 / 9 ) e. RR
100		ltsub1 8637	. . . . . 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 1373	. . . . 5 |- ( ( 2 x. ( ( cos ` 1 ) ^ 2 ) ) < ( 8 / 9 ) <-> ( ( 2 x. ( ( cos ` 1 ) ^ 2 ) ) - 1 ) < ( ( 8 / 9 ) - 1 ) )
102	97, 101	mpbi 145	. . . 4 |- ( ( 2 x. ( ( cos ` 1 ) ^ 2 ) ) - 1 ) < ( ( 8 / 9 ) - 1 )
103	20	oveq2i 6028	. . . . 5 |- ( ( 8 / 9 ) - ( 9 / 9 ) ) = ( ( 8 / 9 ) - 1 )
104		divnegap 8885	. . . . . . 7 |- ( ( 1 e. CC /\ 9 e. CC /\ 9 # 0 ) -> -u ( 1 / 9 ) = ( -u 1 / 9 ) )
105	23, 2, 5, 104	mp3an 1373	. . . . . 6 |- -u ( 1 / 9 ) = ( -u 1 / 9 )
106		8cn 9228	. . . . . . . . 9 |- 8 e. CC
107	2, 106	negsubdi2i 8464	. . . . . . . 8 |- -u ( 9 - 8 ) = ( 8 - 9 )
108		8p1e9 9283	. . . . . . . . . 10 |- ( 8 + 1 ) = 9
109	2, 106, 23, 108	subaddrii 8467	. . . . . . . . 9 |- ( 9 - 8 ) = 1
110	109	negeqi 8372	. . . . . . . 8 |- -u ( 9 - 8 ) = -u 1
111	107, 110	eqtr3i 2254	. . . . . . 7 |- ( 8 - 9 ) = -u 1
112	111	oveq1i 6027	. . . . . 6 |- ( ( 8 - 9 ) / 9 ) = ( -u 1 / 9 )
113		divsubdirap 8887	. . . . . . 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 1373	. . . . . 6 |- ( ( 8 - 9 ) / 9 ) = ( ( 8 / 9 ) - ( 9 / 9 ) )
115	105, 112, 114	3eqtr2ri 2259	. . . . 5 |- ( ( 8 / 9 ) - ( 9 / 9 ) ) = -u ( 1 / 9 )
116	103, 115	eqtr3i 2254	. . . 4 |- ( ( 8 / 9 ) - 1 ) = -u ( 1 / 9 )
117	102, 116	breqtri 4113	. . 3 |- ( ( 2 x. ( ( cos ` 1 ) ^ 2 ) ) - 1 ) < -u ( 1 / 9 )
118	71, 117	eqbrtri 4109	. 2 |- ( cos ` 2 ) < -u ( 1 / 9 )
119	72, 118	pm3.2i 272	1 |- ( -u ( 7 / 9 ) < ( cos ` 2 ) /\ ( cos ` 2 ) < -u ( 1 / 9 ) )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1397   e. wcel 2202   class class class wbr 4088   ` cfv 5326  (class class class)co 6017   CCcc 8029   RRcr 8030   0cc0 8031   1c1 8032   + caddc 8034   x. cmul 8036   < clt 8213   <_ cle 8214   - cmin 8349   -ucneg 8350   # cap 8760   / cdiv 8851   2c2 9193   3c3 9194   4c4 9195   7c7 9198   8c8 9199   9c9 9200   ^cexp 10799   cosccos 12205

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

This theorem depends on definitions:  df-bi 117  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-1st 6302  df-2nd 6303  df-recs 6470  df-irdg 6535  df-frec 6556  df-1o 6581  df-oadd 6585  df-er 6701  df-en 6909  df-dom 6910  df-fin 6911  df-sup 7182  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-ioc 10127  df-ico 10128  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

This theorem is referenced by:  sincos2sgn  12326