Theorem cos2bnd 11771

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 9006	. . . . . 6 |- 7 e. CC
2		9cn 9010	. . . . . 6 |- 9 e. CC
3		9re 9009	. . . . . . 7 |- 9 e. RR
4		9pos 9026	. . . . . . 7 |- 0 < 9
5	3, 4	gt0ap0ii 8588	. . . . . 6 |- 9 # 0
6		divnegap 8666	. . . . . 6 |- ( ( 7 e. CC /\ 9 e. CC /\ 9 # 0 ) -> -u ( 7 / 9 ) = ( -u 7 / 9 ) )
7	1, 2, 5, 6	mp3an 1337	. . . . 5 |- -u ( 7 / 9 ) = ( -u 7 / 9 )
8		2cn 8993	. . . . . . 7 |- 2 e. CC
9	2, 5	pm3.2i 272	. . . . . . 7 |- ( 9 e. CC /\ 9 # 0 )
10		divsubdirap 8668	. . . . . . 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 1337	. . . . . 6 |- ( ( 2 - 9 ) / 9 ) = ( ( 2 / 9 ) - ( 9 / 9 ) )
12	2, 8	negsubdi2i 8246	. . . . . . . 8 |- -u ( 9 - 2 ) = ( 2 - 9 )
13		7p2e9 9073	. . . . . . . . . 10 |- ( 7 + 2 ) = 9
14	2, 8, 1	subadd2i 8248	. . . . . . . . . 10 |- ( ( 9 - 2 ) = 7 <-> ( 7 + 2 ) = 9 )
15	13, 14	mpbir 146	. . . . . . . . 9 |- ( 9 - 2 ) = 7
16	15	negeqi 8154	. . . . . . . 8 |- -u ( 9 - 2 ) = -u 7
17	12, 16	eqtr3i 2200	. . . . . . 7 |- ( 2 - 9 ) = -u 7
18	17	oveq1i 5888	. . . . . 6 |- ( ( 2 - 9 ) / 9 ) = ( -u 7 / 9 )
19	11, 18	eqtr3i 2200	. . . . 5 |- ( ( 2 / 9 ) - ( 9 / 9 ) ) = ( -u 7 / 9 )
20	2, 5	dividapi 8705	. . . . . 6 |- ( 9 / 9 ) = 1
21	20	oveq2i 5889	. . . . 5 |- ( ( 2 / 9 ) - ( 9 / 9 ) ) = ( ( 2 / 9 ) - 1 )
22	7, 19, 21	3eqtr2ri 2205	. . . 4 |- ( ( 2 / 9 ) - 1 ) = -u ( 7 / 9 )
23		ax-1cn 7907	. . . . . . . 8 |- 1 e. CC
24	8, 23, 2, 5	divassapi 8728	. . . . . . 7 |- ( ( 2 x. 1 ) / 9 ) = ( 2 x. ( 1 / 9 ) )
25		2t1e2 9075	. . . . . . . 8 |- ( 2 x. 1 ) = 2
26	25	oveq1i 5888	. . . . . . 7 |- ( ( 2 x. 1 ) / 9 ) = ( 2 / 9 )
27	24, 26	eqtr3i 2200	. . . . . 6 |- ( 2 x. ( 1 / 9 ) ) = ( 2 / 9 )
28		3cn 8997	. . . . . . . . . 10 |- 3 e. CC
29		3ap0 9018	. . . . . . . . . 10 |- 3 # 0
30	23, 28, 29	sqdivapi 10607	. . . . . . . . 9 |- ( ( 1 / 3 ) ^ 2 ) = ( ( 1 ^ 2 ) / ( 3 ^ 2 ) )
31		sq1 10617	. . . . . . . . . 10 |- ( 1 ^ 2 ) = 1
32		sq3 10620	. . . . . . . . . 10 |- ( 3 ^ 2 ) = 9
33	31, 32	oveq12i 5890	. . . . . . . . 9 |- ( ( 1 ^ 2 ) / ( 3 ^ 2 ) ) = ( 1 / 9 )
34	30, 33	eqtri 2198	. . . . . . . 8 |- ( ( 1 / 3 ) ^ 2 ) = ( 1 / 9 )
35		cos1bnd 11770	. . . . . . . . . 10 |- ( ( 1 / 3 ) < ( cos ` 1 ) /\ ( cos ` 1 ) < ( 2 / 3 ) )
36	35	simpli 111	. . . . . . . . 9 |- ( 1 / 3 ) < ( cos ` 1 )
37		0le1 8441	. . . . . . . . . . 11 |- 0 <_ 1
38		3pos 9016	. . . . . . . . . . 11 |- 0 < 3
39		1re 7959	. . . . . . . . . . . 12 |- 1 e. RR
40		3re 8996	. . . . . . . . . . . 12 |- 3 e. RR
41	39, 40	divge0i 8871	. . . . . . . . . . 11 |- ( ( 0 <_ 1 /\ 0 < 3 ) -> 0 <_ ( 1 / 3 ) )
42	37, 38, 41	mp2an 426	. . . . . . . . . 10 |- 0 <_ ( 1 / 3 )
43		0re 7960	. . . . . . . . . . 11 |- 0 e. RR
44		recoscl 11732	. . . . . . . . . . . 12 |- ( 1 e. RR -> ( cos ` 1 ) e. RR )
45	39, 44	ax-mp 5	. . . . . . . . . . 11 |- ( cos ` 1 ) e. RR
46	40, 29	rerecclapi 8737	. . . . . . . . . . . . 13 |- ( 1 / 3 ) e. RR
47	43, 46, 45	lelttri 8066	. . . . . . . . . . . 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 8063	. . . . . . . . . 10 |- 0 <_ ( cos ` 1 )
50	46, 45	lt2sqi 10611	. . . . . . . . . 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 4028	. . . . . . 7 |- ( 1 / 9 ) < ( ( cos ` 1 ) ^ 2 )
54		2pos 9013	. . . . . . . 8 |- 0 < 2
55	3, 5	rerecclapi 8737	. . . . . . . . 9 |- ( 1 / 9 ) e. RR
56	45	resqcli 10608	. . . . . . . . 9 |- ( ( cos ` 1 ) ^ 2 ) e. RR
57		2re 8992	. . . . . . . . 9 |- 2 e. RR
58	55, 56, 57	ltmul2i 8883	. . . . . . . 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 4028	. . . . 5 |- ( 2 / 9 ) < ( 2 x. ( ( cos ` 1 ) ^ 2 ) )
62	57, 3, 5	redivclapi 8739	. . . . . 6 |- ( 2 / 9 ) e. RR
63	57, 56	remulcli 7974	. . . . . 6 |- ( 2 x. ( ( cos ` 1 ) ^ 2 ) ) e. RR
64		ltsub1 8418	. . . . . 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 1337	. . . . 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 4028	. . 3 |- -u ( 7 / 9 ) < ( ( 2 x. ( ( cos ` 1 ) ^ 2 ) ) - 1 )
68	25	fveq2i 5520	. . . 4 |- ( cos ` ( 2 x. 1 ) ) = ( cos ` 2 )
69		cos2t 11761	. . . . 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 2200	. . 3 |- ( cos ` 2 ) = ( ( 2 x. ( ( cos ` 1 ) ^ 2 ) ) - 1 )
72	67, 71	breqtrri 4032	. 2 |- -u ( 7 / 9 ) < ( cos ` 2 )
73	35	simpri 113	. . . . . . . . 9 |- ( cos ` 1 ) < ( 2 / 3 )
74		0le2 9012	. . . . . . . . . . 11 |- 0 <_ 2
75	57, 40	divge0i 8871	. . . . . . . . . . 11 |- ( ( 0 <_ 2 /\ 0 < 3 ) -> 0 <_ ( 2 / 3 ) )
76	74, 38, 75	mp2an 426	. . . . . . . . . 10 |- 0 <_ ( 2 / 3 )
77	57, 40, 29	redivclapi 8739	. . . . . . . . . . 11 |- ( 2 / 3 ) e. RR
78	45, 77	lt2sqi 10611	. . . . . . . . . 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 10607	. . . . . . . . 9 |- ( ( 2 / 3 ) ^ 2 ) = ( ( 2 ^ 2 ) / ( 3 ^ 2 ) )
82		sq2 10619	. . . . . . . . . 10 |- ( 2 ^ 2 ) = 4
83	82, 32	oveq12i 5890	. . . . . . . . 9 |- ( ( 2 ^ 2 ) / ( 3 ^ 2 ) ) = ( 4 / 9 )
84	81, 83	eqtri 2198	. . . . . . . 8 |- ( ( 2 / 3 ) ^ 2 ) = ( 4 / 9 )
85	80, 84	breqtri 4030	. . . . . . 7 |- ( ( cos ` 1 ) ^ 2 ) < ( 4 / 9 )
86		4re 8999	. . . . . . . . . 10 |- 4 e. RR
87	86, 3, 5	redivclapi 8739	. . . . . . . . 9 |- ( 4 / 9 ) e. RR
88	56, 87, 57	ltmul2i 8883	. . . . . . . 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 9000	. . . . . . . 8 |- 4 e. CC
92	8, 91, 2, 5	divassapi 8728	. . . . . . 7 |- ( ( 2 x. 4 ) / 9 ) = ( 2 x. ( 4 / 9 ) )
93		4t2e8 9080	. . . . . . . . 9 |- ( 4 x. 2 ) = 8
94	91, 8, 93	mulcomli 7967	. . . . . . . 8 |- ( 2 x. 4 ) = 8
95	94	oveq1i 5888	. . . . . . 7 |- ( ( 2 x. 4 ) / 9 ) = ( 8 / 9 )
96	92, 95	eqtr3i 2200	. . . . . 6 |- ( 2 x. ( 4 / 9 ) ) = ( 8 / 9 )
97	90, 96	breqtri 4030	. . . . 5 |- ( 2 x. ( ( cos ` 1 ) ^ 2 ) ) < ( 8 / 9 )
98		8re 9007	. . . . . . 7 |- 8 e. RR
99	98, 3, 5	redivclapi 8739	. . . . . 6 |- ( 8 / 9 ) e. RR
100		ltsub1 8418	. . . . . 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 1337	. . . . 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 5889	. . . . 5 |- ( ( 8 / 9 ) - ( 9 / 9 ) ) = ( ( 8 / 9 ) - 1 )
104		divnegap 8666	. . . . . . 7 |- ( ( 1 e. CC /\ 9 e. CC /\ 9 # 0 ) -> -u ( 1 / 9 ) = ( -u 1 / 9 ) )
105	23, 2, 5, 104	mp3an 1337	. . . . . 6 |- -u ( 1 / 9 ) = ( -u 1 / 9 )
106		8cn 9008	. . . . . . . . 9 |- 8 e. CC
107	2, 106	negsubdi2i 8246	. . . . . . . 8 |- -u ( 9 - 8 ) = ( 8 - 9 )
108		8p1e9 9062	. . . . . . . . . 10 |- ( 8 + 1 ) = 9
109	2, 106, 23, 108	subaddrii 8249	. . . . . . . . 9 |- ( 9 - 8 ) = 1
110	109	negeqi 8154	. . . . . . . 8 |- -u ( 9 - 8 ) = -u 1
111	107, 110	eqtr3i 2200	. . . . . . 7 |- ( 8 - 9 ) = -u 1
112	111	oveq1i 5888	. . . . . 6 |- ( ( 8 - 9 ) / 9 ) = ( -u 1 / 9 )
113		divsubdirap 8668	. . . . . . 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 1337	. . . . . 6 |- ( ( 8 - 9 ) / 9 ) = ( ( 8 / 9 ) - ( 9 / 9 ) )
115	105, 112, 114	3eqtr2ri 2205	. . . . 5 |- ( ( 8 / 9 ) - ( 9 / 9 ) ) = -u ( 1 / 9 )
116	103, 115	eqtr3i 2200	. . . 4 |- ( ( 8 / 9 ) - 1 ) = -u ( 1 / 9 )
117	102, 116	breqtri 4030	. . 3 |- ( ( 2 x. ( ( cos ` 1 ) ^ 2 ) ) - 1 ) < -u ( 1 / 9 )
118	71, 117	eqbrtri 4026	. 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 1353   e. wcel 2148   class class class wbr 4005   ` cfv 5218  (class class class)co 5878   CCcc 7812   RRcr 7813   0cc0 7814   1c1 7815   + caddc 7817   x. cmul 7819   < clt 7995   <_ cle 7996   - cmin 8131   -ucneg 8132   # cap 8541   / cdiv 8632   2c2 8973   3c3 8974   4c4 8975   7c7 8978   8c8 8979   9c9 8980   ^cexp 10522   cosccos 11656

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589  ax-cnex 7905  ax-resscn 7906  ax-1cn 7907  ax-1re 7908  ax-icn 7909  ax-addcl 7910  ax-addrcl 7911  ax-mulcl 7912  ax-mulrcl 7913  ax-addcom 7914  ax-mulcom 7915  ax-addass 7916  ax-mulass 7917  ax-distr 7918  ax-i2m1 7919  ax-0lt1 7920  ax-1rid 7921  ax-0id 7922  ax-rnegex 7923  ax-precex 7924  ax-cnre 7925  ax-pre-ltirr 7926  ax-pre-ltwlin 7927  ax-pre-lttrn 7928  ax-pre-apti 7929  ax-pre-ltadd 7930  ax-pre-mulgt0 7931  ax-pre-mulext 7932  ax-arch 7933  ax-caucvg 7934

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-if 3537  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-disj 3983  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-ilim 4371  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-isom 5227  df-riota 5834  df-ov 5881  df-oprab 5882  df-mpo 5883  df-1st 6144  df-2nd 6145  df-recs 6309  df-irdg 6374  df-frec 6395  df-1o 6420  df-oadd 6424  df-er 6538  df-en 6744  df-dom 6745  df-fin 6746  df-sup 6986  df-pnf 7997  df-mnf 7998  df-xr 7999  df-ltxr 8000  df-le 8001  df-sub 8133  df-neg 8134  df-reap 8535  df-ap 8542  df-div 8633  df-inn 8923  df-2 8981  df-3 8982  df-4 8983  df-5 8984  df-6 8985  df-7 8986  df-8 8987  df-9 8988  df-n0 9180  df-z 9257  df-uz 9532  df-q 9623  df-rp 9657  df-ioc 9896  df-ico 9897  df-fz 10012  df-fzo 10146  df-seqfrec 10449  df-exp 10523  df-fac 10709  df-bc 10731  df-ihash 10759  df-shft 10827  df-cj 10854  df-re 10855  df-im 10856  df-rsqrt 11010  df-abs 11011  df-clim 11290  df-sumdc 11365  df-ef 11659  df-sin 11661  df-cos 11662

This theorem is referenced by:  sincos2sgn  11776