Theorem cos2bnd 12552

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 9397	. . . . . 6 |- 7 e. CC
2		9cn 9401	. . . . . 6 |- 9 e. CC
3		9re 9400	. . . . . . 7 |- 9 e. RR
4		9pos 9417	. . . . . . 7 |- 0 < 9
5	3, 4	gt0ap0ii 8964	. . . . . 6 |- 9 # 0
6		divnegap 9044	. . . . . 6 |- ( ( 7 e. CC /\ 9 e. CC /\ 9 # 0 ) -> -u ( 7 / 9 ) = ( -u 7 / 9 ) )
7	1, 2, 5, 6	mp3an 1378	. . . . 5 |- -u ( 7 / 9 ) = ( -u 7 / 9 )
8		2cn 9384	. . . . . . 7 |- 2 e. CC
9	2, 5	pm3.2i 272	. . . . . . 7 |- ( 9 e. CC /\ 9 # 0 )
10		divsubdirap 9046	. . . . . . 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 1378	. . . . . 6 |- ( ( 2 - 9 ) / 9 ) = ( ( 2 / 9 ) - ( 9 / 9 ) )
12	2, 8	negsubdi2i 8619	. . . . . . . 8 |- -u ( 9 - 2 ) = ( 2 - 9 )
13		7p2e9 9465	. . . . . . . . . 10 |- ( 7 + 2 ) = 9
14	2, 8, 1	subadd2i 8621	. . . . . . . . . 10 |- ( ( 9 - 2 ) = 7 <-> ( 7 + 2 ) = 9 )
15	13, 14	mpbir 146	. . . . . . . . 9 |- ( 9 - 2 ) = 7
16	15	negeqi 8527	. . . . . . . 8 |- -u ( 9 - 2 ) = -u 7
17	12, 16	eqtr3i 2261	. . . . . . 7 |- ( 2 - 9 ) = -u 7
18	17	oveq1i 6098	. . . . . 6 |- ( ( 2 - 9 ) / 9 ) = ( -u 7 / 9 )
19	11, 18	eqtr3i 2261	. . . . 5 |- ( ( 2 / 9 ) - ( 9 / 9 ) ) = ( -u 7 / 9 )
20	2, 5	dividapi 9083	. . . . . 6 |- ( 9 / 9 ) = 1
21	20	oveq2i 6099	. . . . 5 |- ( ( 2 / 9 ) - ( 9 / 9 ) ) = ( ( 2 / 9 ) - 1 )
22	7, 19, 21	3eqtr2ri 2266	. . . 4 |- ( ( 2 / 9 ) - 1 ) = -u ( 7 / 9 )
23		ax-1cn 8278	. . . . . . . 8 |- 1 e. CC
24	8, 23, 2, 5	divassapi 9106	. . . . . . 7 |- ( ( 2 x. 1 ) / 9 ) = ( 2 x. ( 1 / 9 ) )
25		2t1e2 9467	. . . . . . . 8 |- ( 2 x. 1 ) = 2
26	25	oveq1i 6098	. . . . . . 7 |- ( ( 2 x. 1 ) / 9 ) = ( 2 / 9 )
27	24, 26	eqtr3i 2261	. . . . . 6 |- ( 2 x. ( 1 / 9 ) ) = ( 2 / 9 )
28		3cn 9388	. . . . . . . . . 10 |- 3 e. CC
29		3ap0 9409	. . . . . . . . . 10 |- 3 # 0
30	23, 28, 29	sqdivapi 11081	. . . . . . . . 9 |- ( ( 1 / 3 ) ^ 2 ) = ( ( 1 ^ 2 ) / ( 3 ^ 2 ) )
31		sq1 11091	. . . . . . . . . 10 |- ( 1 ^ 2 ) = 1
32		sq3 11094	. . . . . . . . . 10 |- ( 3 ^ 2 ) = 9
33	31, 32	oveq12i 6100	. . . . . . . . 9 |- ( ( 1 ^ 2 ) / ( 3 ^ 2 ) ) = ( 1 / 9 )
34	30, 33	eqtri 2259	. . . . . . . 8 |- ( ( 1 / 3 ) ^ 2 ) = ( 1 / 9 )
35		cos1bnd 12551	. . . . . . . . . 10 |- ( ( 1 / 3 ) < ( cos ` 1 ) /\ ( cos ` 1 ) < ( 2 / 3 ) )
36	35	simpli 111	. . . . . . . . 9 |- ( 1 / 3 ) < ( cos ` 1 )
37		0le1 8816	. . . . . . . . . . 11 |- 0 <_ 1
38		3pos 9407	. . . . . . . . . . 11 |- 0 < 3
39		1re 8331	. . . . . . . . . . . 12 |- 1 e. RR
40		3re 9387	. . . . . . . . . . . 12 |- 3 e. RR
41	39, 40	divge0i 9249	. . . . . . . . . . 11 |- ( ( 0 <_ 1 /\ 0 < 3 ) -> 0 <_ ( 1 / 3 ) )
42	37, 38, 41	mp2an 430	. . . . . . . . . 10 |- 0 <_ ( 1 / 3 )
43		0re 8332	. . . . . . . . . . 11 |- 0 e. RR
44		recoscl 12513	. . . . . . . . . . . 12 |- ( 1 e. RR -> ( cos ` 1 ) e. RR )
45	39, 44	ax-mp 5	. . . . . . . . . . 11 |- ( cos ` 1 ) e. RR
46	40, 29	rerecclapi 9115	. . . . . . . . . . . . 13 |- ( 1 / 3 ) e. RR
47	43, 46, 45	lelttri 8438	. . . . . . . . . . . 12 |- ( ( 0 <_ ( 1 / 3 ) /\ ( 1 / 3 ) < ( cos ` 1 ) ) -> 0 < ( cos ` 1 ) )
48	42, 36, 47	mp2an 430	. . . . . . . . . . 11 |- 0 < ( cos ` 1 )
49	43, 45, 48	ltleii 8435	. . . . . . . . . 10 |- 0 <_ ( cos ` 1 )
50	46, 45	lt2sqi 11085	. . . . . . . . . 10 |- ( ( 0 <_ ( 1 / 3 ) /\ 0 <_ ( cos ` 1 ) ) -> ( ( 1 / 3 ) < ( cos ` 1 ) <-> ( ( 1 / 3 ) ^ 2 ) < ( ( cos ` 1 ) ^ 2 ) ) )
51	42, 49, 50	mp2an 430	. . . . . . . . 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 4154	. . . . . . 7 |- ( 1 / 9 ) < ( ( cos ` 1 ) ^ 2 )
54		2pos 9404	. . . . . . . 8 |- 0 < 2
55	3, 5	rerecclapi 9115	. . . . . . . . 9 |- ( 1 / 9 ) e. RR
56	45	resqcli 11082	. . . . . . . . 9 |- ( ( cos ` 1 ) ^ 2 ) e. RR
57		2re 9383	. . . . . . . . 9 |- 2 e. RR
58	55, 56, 57	ltmul2i 9261	. . . . . . . 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 4154	. . . . 5 |- ( 2 / 9 ) < ( 2 x. ( ( cos ` 1 ) ^ 2 ) )
62	57, 3, 5	redivclapi 9117	. . . . . 6 |- ( 2 / 9 ) e. RR
63	57, 56	remulcli 8346	. . . . . 6 |- ( 2 x. ( ( cos ` 1 ) ^ 2 ) ) e. RR
64		ltsub1 8793	. . . . . 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 1378	. . . . 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 4154	. . 3 |- -u ( 7 / 9 ) < ( ( 2 x. ( ( cos ` 1 ) ^ 2 ) ) - 1 )
68	25	fveq2i 5701	. . . 4 |- ( cos ` ( 2 x. 1 ) ) = ( cos ` 2 )
69		cos2t 12542	. . . . 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 2261	. . 3 |- ( cos ` 2 ) = ( ( 2 x. ( ( cos ` 1 ) ^ 2 ) ) - 1 )
72	67, 71	breqtrri 4158	. 2 |- -u ( 7 / 9 ) < ( cos ` 2 )
73	35	simpri 113	. . . . . . . . 9 |- ( cos ` 1 ) < ( 2 / 3 )
74		0le2 9403	. . . . . . . . . . 11 |- 0 <_ 2
75	57, 40	divge0i 9249	. . . . . . . . . . 11 |- ( ( 0 <_ 2 /\ 0 < 3 ) -> 0 <_ ( 2 / 3 ) )
76	74, 38, 75	mp2an 430	. . . . . . . . . 10 |- 0 <_ ( 2 / 3 )
77	57, 40, 29	redivclapi 9117	. . . . . . . . . . 11 |- ( 2 / 3 ) e. RR
78	45, 77	lt2sqi 11085	. . . . . . . . . 10 |- ( ( 0 <_ ( cos ` 1 ) /\ 0 <_ ( 2 / 3 ) ) -> ( ( cos ` 1 ) < ( 2 / 3 ) <-> ( ( cos ` 1 ) ^ 2 ) < ( ( 2 / 3 ) ^ 2 ) ) )
79	49, 76, 78	mp2an 430	. . . . . . . . 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 11081	. . . . . . . . 9 |- ( ( 2 / 3 ) ^ 2 ) = ( ( 2 ^ 2 ) / ( 3 ^ 2 ) )
82		sq2 11093	. . . . . . . . . 10 |- ( 2 ^ 2 ) = 4
83	82, 32	oveq12i 6100	. . . . . . . . 9 |- ( ( 2 ^ 2 ) / ( 3 ^ 2 ) ) = ( 4 / 9 )
84	81, 83	eqtri 2259	. . . . . . . 8 |- ( ( 2 / 3 ) ^ 2 ) = ( 4 / 9 )
85	80, 84	breqtri 4156	. . . . . . 7 |- ( ( cos ` 1 ) ^ 2 ) < ( 4 / 9 )
86		4re 9390	. . . . . . . . . 10 |- 4 e. RR
87	86, 3, 5	redivclapi 9117	. . . . . . . . 9 |- ( 4 / 9 ) e. RR
88	56, 87, 57	ltmul2i 9261	. . . . . . . 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 9391	. . . . . . . 8 |- 4 e. CC
92	8, 91, 2, 5	divassapi 9106	. . . . . . 7 |- ( ( 2 x. 4 ) / 9 ) = ( 2 x. ( 4 / 9 ) )
93		4t2e8 9473	. . . . . . . . 9 |- ( 4 x. 2 ) = 8
94	91, 8, 93	mulcomli 8339	. . . . . . . 8 |- ( 2 x. 4 ) = 8
95	94	oveq1i 6098	. . . . . . 7 |- ( ( 2 x. 4 ) / 9 ) = ( 8 / 9 )
96	92, 95	eqtr3i 2261	. . . . . 6 |- ( 2 x. ( 4 / 9 ) ) = ( 8 / 9 )
97	90, 96	breqtri 4156	. . . . 5 |- ( 2 x. ( ( cos ` 1 ) ^ 2 ) ) < ( 8 / 9 )
98		8re 9398	. . . . . . 7 |- 8 e. RR
99	98, 3, 5	redivclapi 9117	. . . . . 6 |- ( 8 / 9 ) e. RR
100		ltsub1 8793	. . . . . 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 1378	. . . . 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 6099	. . . . 5 |- ( ( 8 / 9 ) - ( 9 / 9 ) ) = ( ( 8 / 9 ) - 1 )
104		divnegap 9044	. . . . . . 7 |- ( ( 1 e. CC /\ 9 e. CC /\ 9 # 0 ) -> -u ( 1 / 9 ) = ( -u 1 / 9 ) )
105	23, 2, 5, 104	mp3an 1378	. . . . . 6 |- -u ( 1 / 9 ) = ( -u 1 / 9 )
106		8cn 9399	. . . . . . . . 9 |- 8 e. CC
107	2, 106	negsubdi2i 8619	. . . . . . . 8 |- -u ( 9 - 8 ) = ( 8 - 9 )
108		8p1e9 9454	. . . . . . . . . 10 |- ( 8 + 1 ) = 9
109	2, 106, 23, 108	subaddrii 8622	. . . . . . . . 9 |- ( 9 - 8 ) = 1
110	109	negeqi 8527	. . . . . . . 8 |- -u ( 9 - 8 ) = -u 1
111	107, 110	eqtr3i 2261	. . . . . . 7 |- ( 8 - 9 ) = -u 1
112	111	oveq1i 6098	. . . . . 6 |- ( ( 8 - 9 ) / 9 ) = ( -u 1 / 9 )
113		divsubdirap 9046	. . . . . . 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 1378	. . . . . 6 |- ( ( 8 - 9 ) / 9 ) = ( ( 8 / 9 ) - ( 9 / 9 ) )
115	105, 112, 114	3eqtr2ri 2266	. . . . 5 |- ( ( 8 / 9 ) - ( 9 / 9 ) ) = -u ( 1 / 9 )
116	103, 115	eqtr3i 2261	. . . 4 |- ( ( 8 / 9 ) - 1 ) = -u ( 1 / 9 )
117	102, 116	breqtri 4156	. . 3 |- ( ( 2 x. ( ( cos ` 1 ) ^ 2 ) ) - 1 ) < -u ( 1 / 9 )
118	71, 117	eqbrtri 4152	. 2 |- ( cos ` 2 ) < -u ( 1 / 9 )
119	72, 118	pm3.2i 272	1 |- ( -u ( 7 / 9 ) < ( cos ` 2 ) /\ ( cos ` 2 ) < -u ( 1 / 9 ) )

This proof depends on syntax axioms:   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   class class class wbr 4131   ` cfv 5380  (class class class)co 6088   CCcc 8183   RRcr 8184   0cc0 8185   1c1 8186   + caddc 8188   x. cmul 8190   < clt 8366   <_ cle 8367   - cmin 8504   -ucneg 8505   # cap 8917   / cdiv 9010   2c2 9364   3c3 9365   4c4 9366   7c7 9369   8c8 9370   9c9 9371   ^cexp 10996   cosccos 12437

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-mulrcl 8284  ax-addcom 8285  ax-mulcom 8286  ax-addass 8287  ax-mulass 8288  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-1rid 8292  ax-0id 8293  ax-rnegex 8294  ax-precex 8295  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-apti 8300  ax-pre-ltadd 8301  ax-pre-mulgt0 8302  ax-pre-mulext 8303  ax-arch 8304  ax-caucvg 8305

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-disj 4107  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-po 4443  df-iso 4444  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-isom 5389  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-irdg 6645  df-frec 6666  df-1o 6691  df-oadd 6695  df-er 6811  df-en 7027  df-dom 7028  df-fin 7029  df-sup 7330  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-reap 8911  df-ap 8918  df-div 9011  df-inn 9314  df-2 9372  df-3 9373  df-4 9374  df-5 9375  df-6 9376  df-7 9377  df-8 9378  df-9 9379  df-n0 9575  df-z 9656  df-uz 9938  df-q 10036  df-rp 10072  df-ioc 10312  df-ico 10313  df-fz 10429  df-fzo 10567  df-seqfrec 10906  df-exp 10997  df-fac 11186  df-bc 11208  df-ihash 11237  df-shft 11602  df-cj 11629  df-re 11630  df-im 11631  df-rsqrt 11786  df-abs 11787  df-clim 12070  df-sumdc 12145  df-ef 12440  df-sin 12442  df-cos 12443

This theorem is used by:  sincos2sgn  12558