Theorem cos2bnd 11815

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 9044	. . . . . 6 |- 7 e. CC
2		9cn 9048	. . . . . 6 |- 9 e. CC
3		9re 9047	. . . . . . 7 |- 9 e. RR
4		9pos 9064	. . . . . . 7 |- 0 < 9
5	3, 4	gt0ap0ii 8626	. . . . . 6 |- 9 # 0
6		divnegap 8704	. . . . . 6 |- ( ( 7 e. CC /\ 9 e. CC /\ 9 # 0 ) -> -u ( 7 / 9 ) = ( -u 7 / 9 ) )
7	1, 2, 5, 6	mp3an 1348	. . . . 5 |- -u ( 7 / 9 ) = ( -u 7 / 9 )
8		2cn 9031	. . . . . . 7 |- 2 e. CC
9	2, 5	pm3.2i 272	. . . . . . 7 |- ( 9 e. CC /\ 9 # 0 )
10		divsubdirap 8706	. . . . . . 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 1348	. . . . . 6 |- ( ( 2 - 9 ) / 9 ) = ( ( 2 / 9 ) - ( 9 / 9 ) )
12	2, 8	negsubdi2i 8284	. . . . . . . 8 |- -u ( 9 - 2 ) = ( 2 - 9 )
13		7p2e9 9111	. . . . . . . . . 10 |- ( 7 + 2 ) = 9
14	2, 8, 1	subadd2i 8286	. . . . . . . . . 10 |- ( ( 9 - 2 ) = 7 <-> ( 7 + 2 ) = 9 )
15	13, 14	mpbir 146	. . . . . . . . 9 |- ( 9 - 2 ) = 7
16	15	negeqi 8192	. . . . . . . 8 |- -u ( 9 - 2 ) = -u 7
17	12, 16	eqtr3i 2212	. . . . . . 7 |- ( 2 - 9 ) = -u 7
18	17	oveq1i 5914	. . . . . 6 |- ( ( 2 - 9 ) / 9 ) = ( -u 7 / 9 )
19	11, 18	eqtr3i 2212	. . . . 5 |- ( ( 2 / 9 ) - ( 9 / 9 ) ) = ( -u 7 / 9 )
20	2, 5	dividapi 8743	. . . . . 6 |- ( 9 / 9 ) = 1
21	20	oveq2i 5915	. . . . 5 |- ( ( 2 / 9 ) - ( 9 / 9 ) ) = ( ( 2 / 9 ) - 1 )
22	7, 19, 21	3eqtr2ri 2217	. . . 4 |- ( ( 2 / 9 ) - 1 ) = -u ( 7 / 9 )
23		ax-1cn 7944	. . . . . . . 8 |- 1 e. CC
24	8, 23, 2, 5	divassapi 8766	. . . . . . 7 |- ( ( 2 x. 1 ) / 9 ) = ( 2 x. ( 1 / 9 ) )
25		2t1e2 9113	. . . . . . . 8 |- ( 2 x. 1 ) = 2
26	25	oveq1i 5914	. . . . . . 7 |- ( ( 2 x. 1 ) / 9 ) = ( 2 / 9 )
27	24, 26	eqtr3i 2212	. . . . . 6 |- ( 2 x. ( 1 / 9 ) ) = ( 2 / 9 )
28		3cn 9035	. . . . . . . . . 10 |- 3 e. CC
29		3ap0 9056	. . . . . . . . . 10 |- 3 # 0
30	23, 28, 29	sqdivapi 10650	. . . . . . . . 9 |- ( ( 1 / 3 ) ^ 2 ) = ( ( 1 ^ 2 ) / ( 3 ^ 2 ) )
31		sq1 10660	. . . . . . . . . 10 |- ( 1 ^ 2 ) = 1
32		sq3 10663	. . . . . . . . . 10 |- ( 3 ^ 2 ) = 9
33	31, 32	oveq12i 5916	. . . . . . . . 9 |- ( ( 1 ^ 2 ) / ( 3 ^ 2 ) ) = ( 1 / 9 )
34	30, 33	eqtri 2210	. . . . . . . 8 |- ( ( 1 / 3 ) ^ 2 ) = ( 1 / 9 )
35		cos1bnd 11814	. . . . . . . . . 10 |- ( ( 1 / 3 ) < ( cos ` 1 ) /\ ( cos ` 1 ) < ( 2 / 3 ) )
36	35	simpli 111	. . . . . . . . 9 |- ( 1 / 3 ) < ( cos ` 1 )
37		0le1 8479	. . . . . . . . . . 11 |- 0 <_ 1
38		3pos 9054	. . . . . . . . . . 11 |- 0 < 3
39		1re 7997	. . . . . . . . . . . 12 |- 1 e. RR
40		3re 9034	. . . . . . . . . . . 12 |- 3 e. RR
41	39, 40	divge0i 8909	. . . . . . . . . . 11 |- ( ( 0 <_ 1 /\ 0 < 3 ) -> 0 <_ ( 1 / 3 ) )
42	37, 38, 41	mp2an 426	. . . . . . . . . 10 |- 0 <_ ( 1 / 3 )
43		0re 7998	. . . . . . . . . . 11 |- 0 e. RR
44		recoscl 11776	. . . . . . . . . . . 12 |- ( 1 e. RR -> ( cos ` 1 ) e. RR )
45	39, 44	ax-mp 5	. . . . . . . . . . 11 |- ( cos ` 1 ) e. RR
46	40, 29	rerecclapi 8775	. . . . . . . . . . . . 13 |- ( 1 / 3 ) e. RR
47	43, 46, 45	lelttri 8104	. . . . . . . . . . . 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 8101	. . . . . . . . . 10 |- 0 <_ ( cos ` 1 )
50	46, 45	lt2sqi 10654	. . . . . . . . . 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 4048	. . . . . . 7 |- ( 1 / 9 ) < ( ( cos ` 1 ) ^ 2 )
54		2pos 9051	. . . . . . . 8 |- 0 < 2
55	3, 5	rerecclapi 8775	. . . . . . . . 9 |- ( 1 / 9 ) e. RR
56	45	resqcli 10651	. . . . . . . . 9 |- ( ( cos ` 1 ) ^ 2 ) e. RR
57		2re 9030	. . . . . . . . 9 |- 2 e. RR
58	55, 56, 57	ltmul2i 8921	. . . . . . . 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 4048	. . . . 5 |- ( 2 / 9 ) < ( 2 x. ( ( cos ` 1 ) ^ 2 ) )
62	57, 3, 5	redivclapi 8777	. . . . . 6 |- ( 2 / 9 ) e. RR
63	57, 56	remulcli 8012	. . . . . 6 |- ( 2 x. ( ( cos ` 1 ) ^ 2 ) ) e. RR
64		ltsub1 8456	. . . . . 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 1348	. . . . 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 4048	. . 3 |- -u ( 7 / 9 ) < ( ( 2 x. ( ( cos ` 1 ) ^ 2 ) ) - 1 )
68	25	fveq2i 5544	. . . 4 |- ( cos ` ( 2 x. 1 ) ) = ( cos ` 2 )
69		cos2t 11805	. . . . 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 2212	. . 3 |- ( cos ` 2 ) = ( ( 2 x. ( ( cos ` 1 ) ^ 2 ) ) - 1 )
72	67, 71	breqtrri 4052	. 2 |- -u ( 7 / 9 ) < ( cos ` 2 )
73	35	simpri 113	. . . . . . . . 9 |- ( cos ` 1 ) < ( 2 / 3 )
74		0le2 9050	. . . . . . . . . . 11 |- 0 <_ 2
75	57, 40	divge0i 8909	. . . . . . . . . . 11 |- ( ( 0 <_ 2 /\ 0 < 3 ) -> 0 <_ ( 2 / 3 ) )
76	74, 38, 75	mp2an 426	. . . . . . . . . 10 |- 0 <_ ( 2 / 3 )
77	57, 40, 29	redivclapi 8777	. . . . . . . . . . 11 |- ( 2 / 3 ) e. RR
78	45, 77	lt2sqi 10654	. . . . . . . . . 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 10650	. . . . . . . . 9 |- ( ( 2 / 3 ) ^ 2 ) = ( ( 2 ^ 2 ) / ( 3 ^ 2 ) )
82		sq2 10662	. . . . . . . . . 10 |- ( 2 ^ 2 ) = 4
83	82, 32	oveq12i 5916	. . . . . . . . 9 |- ( ( 2 ^ 2 ) / ( 3 ^ 2 ) ) = ( 4 / 9 )
84	81, 83	eqtri 2210	. . . . . . . 8 |- ( ( 2 / 3 ) ^ 2 ) = ( 4 / 9 )
85	80, 84	breqtri 4050	. . . . . . 7 |- ( ( cos ` 1 ) ^ 2 ) < ( 4 / 9 )
86		4re 9037	. . . . . . . . . 10 |- 4 e. RR
87	86, 3, 5	redivclapi 8777	. . . . . . . . 9 |- ( 4 / 9 ) e. RR
88	56, 87, 57	ltmul2i 8921	. . . . . . . 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 9038	. . . . . . . 8 |- 4 e. CC
92	8, 91, 2, 5	divassapi 8766	. . . . . . 7 |- ( ( 2 x. 4 ) / 9 ) = ( 2 x. ( 4 / 9 ) )
93		4t2e8 9118	. . . . . . . . 9 |- ( 4 x. 2 ) = 8
94	91, 8, 93	mulcomli 8005	. . . . . . . 8 |- ( 2 x. 4 ) = 8
95	94	oveq1i 5914	. . . . . . 7 |- ( ( 2 x. 4 ) / 9 ) = ( 8 / 9 )
96	92, 95	eqtr3i 2212	. . . . . 6 |- ( 2 x. ( 4 / 9 ) ) = ( 8 / 9 )
97	90, 96	breqtri 4050	. . . . 5 |- ( 2 x. ( ( cos ` 1 ) ^ 2 ) ) < ( 8 / 9 )
98		8re 9045	. . . . . . 7 |- 8 e. RR
99	98, 3, 5	redivclapi 8777	. . . . . 6 |- ( 8 / 9 ) e. RR
100		ltsub1 8456	. . . . . 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 1348	. . . . 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 5915	. . . . 5 |- ( ( 8 / 9 ) - ( 9 / 9 ) ) = ( ( 8 / 9 ) - 1 )
104		divnegap 8704	. . . . . . 7 |- ( ( 1 e. CC /\ 9 e. CC /\ 9 # 0 ) -> -u ( 1 / 9 ) = ( -u 1 / 9 ) )
105	23, 2, 5, 104	mp3an 1348	. . . . . 6 |- -u ( 1 / 9 ) = ( -u 1 / 9 )
106		8cn 9046	. . . . . . . . 9 |- 8 e. CC
107	2, 106	negsubdi2i 8284	. . . . . . . 8 |- -u ( 9 - 8 ) = ( 8 - 9 )
108		8p1e9 9100	. . . . . . . . . 10 |- ( 8 + 1 ) = 9
109	2, 106, 23, 108	subaddrii 8287	. . . . . . . . 9 |- ( 9 - 8 ) = 1
110	109	negeqi 8192	. . . . . . . 8 |- -u ( 9 - 8 ) = -u 1
111	107, 110	eqtr3i 2212	. . . . . . 7 |- ( 8 - 9 ) = -u 1
112	111	oveq1i 5914	. . . . . 6 |- ( ( 8 - 9 ) / 9 ) = ( -u 1 / 9 )
113		divsubdirap 8706	. . . . . . 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 1348	. . . . . 6 |- ( ( 8 - 9 ) / 9 ) = ( ( 8 / 9 ) - ( 9 / 9 ) )
115	105, 112, 114	3eqtr2ri 2217	. . . . 5 |- ( ( 8 / 9 ) - ( 9 / 9 ) ) = -u ( 1 / 9 )
116	103, 115	eqtr3i 2212	. . . 4 |- ( ( 8 / 9 ) - 1 ) = -u ( 1 / 9 )
117	102, 116	breqtri 4050	. . 3 |- ( ( 2 x. ( ( cos ` 1 ) ^ 2 ) ) - 1 ) < -u ( 1 / 9 )
118	71, 117	eqbrtri 4046	. 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 1364   e. wcel 2160   class class class wbr 4025   ` cfv 5242  (class class class)co 5904   CCcc 7849   RRcr 7850   0cc0 7851   1c1 7852   + caddc 7854   x. cmul 7856   < clt 8033   <_ cle 8034   - cmin 8169   -ucneg 8170   # cap 8579   / cdiv 8670   2c2 9011   3c3 9012   4c4 9013   7c7 9016   8c8 9017   9c9 9018   ^cexp 10565   cosccos 11700

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4140  ax-sep 4143  ax-nul 4151  ax-pow 4199  ax-pr 4234  ax-un 4458  ax-setind 4561  ax-iinf 4612  ax-cnex 7942  ax-resscn 7943  ax-1cn 7944  ax-1re 7945  ax-icn 7946  ax-addcl 7947  ax-addrcl 7948  ax-mulcl 7949  ax-mulrcl 7950  ax-addcom 7951  ax-mulcom 7952  ax-addass 7953  ax-mulass 7954  ax-distr 7955  ax-i2m1 7956  ax-0lt1 7957  ax-1rid 7958  ax-0id 7959  ax-rnegex 7960  ax-precex 7961  ax-cnre 7962  ax-pre-ltirr 7963  ax-pre-ltwlin 7964  ax-pre-lttrn 7965  ax-pre-apti 7966  ax-pre-ltadd 7967  ax-pre-mulgt0 7968  ax-pre-mulext 7969  ax-arch 7970  ax-caucvg 7971

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2758  df-sbc 2982  df-csb 3077  df-dif 3150  df-un 3152  df-in 3154  df-ss 3161  df-nul 3442  df-if 3554  df-pw 3599  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3832  df-int 3867  df-iun 3910  df-disj 4003  df-br 4026  df-opab 4087  df-mpt 4088  df-tr 4124  df-id 4318  df-po 4321  df-iso 4322  df-iord 4391  df-on 4393  df-ilim 4394  df-suc 4396  df-iom 4615  df-xp 4657  df-rel 4658  df-cnv 4659  df-co 4660  df-dm 4661  df-rn 4662  df-res 4663  df-ima 4664  df-iota 5203  df-fun 5244  df-fn 5245  df-f 5246  df-f1 5247  df-fo 5248  df-f1o 5249  df-fv 5250  df-isom 5251  df-riota 5859  df-ov 5907  df-oprab 5908  df-mpo 5909  df-1st 6173  df-2nd 6174  df-recs 6338  df-irdg 6403  df-frec 6424  df-1o 6449  df-oadd 6453  df-er 6567  df-en 6775  df-dom 6776  df-fin 6777  df-sup 7022  df-pnf 8035  df-mnf 8036  df-xr 8037  df-ltxr 8038  df-le 8039  df-sub 8171  df-neg 8172  df-reap 8573  df-ap 8580  df-div 8671  df-inn 8961  df-2 9019  df-3 9020  df-4 9021  df-5 9022  df-6 9023  df-7 9024  df-8 9025  df-9 9026  df-n0 9218  df-z 9295  df-uz 9570  df-q 9662  df-rp 9696  df-ioc 9935  df-ico 9936  df-fz 10051  df-fzo 10185  df-seqfrec 10491  df-exp 10566  df-fac 10753  df-bc 10775  df-ihash 10803  df-shft 10871  df-cj 10898  df-re 10899  df-im 10900  df-rsqrt 11054  df-abs 11055  df-clim 11334  df-sumdc 11409  df-ef 11703  df-sin 11705  df-cos 11706

This theorem is referenced by:  sincos2sgn  11820