Theorem cos2bnd 8741

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		2cn 7164	. . . . . . 7 |- 2 e. CC
2		9re 7171	. . . . . . . 8 |- 9 e. RR
3	2	recni 6467	. . . . . . 7 |- 9 e. CC
4		9pos 7181	. . . . . . . . 9 |- 0 < 9
5	2, 4	gt0ne0ii 6799	. . . . . . . 8 |- 9 =/= 0
6	3, 5	pm3.2i 307	. . . . . . 7 |- (9 e. CC /\ 9 =/= 0)
7		divsubdir 6951	. . . . . . 7 |- ((2 e. CC /\ 9 e. CC /\ (9 e. CC /\ 9 =/= 0)) -> ((2 - 9) / 9) = ((2 / 9) - (9 / 9)))
8	1, 3, 6, 7	mp3an 1191	. . . . . 6 |- ((2 - 9) / 9) = ((2 / 9) - (9 / 9))
9	3, 1	negsubdi2i 6614	. . . . . . . 8 |- -u(9 - 2) = (2 - 9)
10		7re 7169	. . . . . . . . . . . . . 14 |- 7 e. RR
11	10	recni 6467	. . . . . . . . . . . . 13 |- 7 e. CC
12		ax1cn 6422	. . . . . . . . . . . . 13 |- 1 e. CC
13	11, 12, 12	addassi 6477	. . . . . . . . . . . 12 |- ((7 + 1) + 1) = (7 + (1 + 1))
14		df-8 7160	. . . . . . . . . . . . 13 |- 8 = (7 + 1)
15	14	opreq1i 4892	. . . . . . . . . . . 12 |- (8 + 1) = ((7 + 1) + 1)
16		df-2 7154	. . . . . . . . . . . . 13 |- 2 = (1 + 1)
17	16	opreq2i 4893	. . . . . . . . . . . 12 |- (7 + 2) = (7 + (1 + 1))
18	13, 15, 17	3eqtr4ri 1923	. . . . . . . . . . 11 |- (7 + 2) = (8 + 1)
19		df-9 7161	. . . . . . . . . . 11 |- 9 = (8 + 1)
20	18, 19	eqtr4i 1911	. . . . . . . . . 10 |- (7 + 2) = 9
21	3, 1, 11	subadd2i 6530	. . . . . . . . . 10 |- ((9 - 2) = 7 <-> (7 + 2) = 9)
22	20, 21	mpbir 207	. . . . . . . . 9 |- (9 - 2) = 7
23	22	negeqi 6515	. . . . . . . 8 |- -u(9 - 2) = -u7
24	9, 23	eqtr3i 1910	. . . . . . 7 |- (2 - 9) = -u7
25	24	opreq1i 4892	. . . . . 6 |- ((2 - 9) / 9) = (-u7 / 9)
26	3, 5	dividi 6946	. . . . . . 7 |- (9 / 9) = 1
27	26	opreq2i 4893	. . . . . 6 |- ((2 / 9) - (9 / 9)) = ((2 / 9) - 1)
28	8, 25, 27	3eqtr3ri 1920	. . . . 5 |- ((2 / 9) - 1) = (-u7 / 9)
29		divneg 6950	. . . . . 6 |- ((7 e. CC /\ 9 e. CC /\ 9 =/= 0) -> -u(7 / 9) = (-u7 / 9))
30	11, 3, 5, 29	mp3an 1191	. . . . 5 |- -u(7 / 9) = (-u7 / 9)
31	28, 30	eqtr4i 1911	. . . 4 |- ((2 / 9) - 1) = -u(7 / 9)
32	1, 12, 3, 5	divassi 6929	. . . . . . 7 |- ((2 x. 1) / 9) = (2 x. (1 / 9))
33	1	mulid1i 6485	. . . . . . . 8 |- (2 x. 1) = 2
34	33	opreq1i 4892	. . . . . . 7 |- ((2 x. 1) / 9) = (2 / 9)
35	32, 34	eqtr3i 1910	. . . . . 6 |- (2 x. (1 / 9)) = (2 / 9)
36		3re 7165	. . . . . . . . . . 11 |- 3 e. RR
37	36	recni 6467	. . . . . . . . . 10 |- 3 e. CC
38		3nn 7184	. . . . . . . . . . 11 |- 3 e. NN
39	38	nnne0i 7134	. . . . . . . . . 10 |- 3 =/= 0
40	12, 37, 39	sqdivi 7863	. . . . . . . . 9 |- ((1 / 3)^2) = ((1^2) / (3^2))
41		sq1 7882	. . . . . . . . . 10 |- (1^2) = 1
42		sq3 7884	. . . . . . . . . 10 |- (3^2) = 9
43	41, 42	opreq12i 4894	. . . . . . . . 9 |- ((1^2) / (3^2)) = (1 / 9)
44	40, 43	eqtri 1908	. . . . . . . 8 |- ((1 / 3)^2) = (1 / 9)
45		cos1bnd 8740	. . . . . . . . . 10 |- ((1 / 3) < (cos` 1) /\ (cos` 1) < (2 / 3))
46	45	simpli 347	. . . . . . . . 9 |- (1 / 3) < (cos` 1)
47		1nn0 7323	. . . . . . . . . . . 12 |- 1 e. NN0
48	47	nn0ge0i 7327	. . . . . . . . . . 11 |- 0 <_ 1
49	38	nngt0i 7133	. . . . . . . . . . 11 |- 0 < 3
50		1re 6598	. . . . . . . . . . . 12 |- 1 e. RR
51	50, 36	divge0i 7040	. . . . . . . . . . 11 |- ((0 <_ 1 /\ 0 < 3) -> 0 <_ (1 / 3))
52	48, 49, 51	mp2an 761	. . . . . . . . . 10 |- 0 <_ (1 / 3)
53		0re 6603	. . . . . . . . . . 11 |- 0 e. RR
54		recoscl 8704	. . . . . . . . . . . 12 |- (1 e. RR -> (cos` 1) e. RR)
55	50, 54	ax-mp 7	. . . . . . . . . . 11 |- (cos` 1) e. RR
56	36, 39	rereccli 6979	. . . . . . . . . . . . 13 |- (1 / 3) e. RR
57	53, 56, 55	lelttri 6761	. . . . . . . . . . . 12 |- ((0 <_ (1 / 3) /\ (1 / 3) < (cos` 1)) -> 0 < (cos` 1))
58	52, 46, 57	mp2an 761	. . . . . . . . . . 11 |- 0 < (cos` 1)
59	53, 55, 58	ltleii 6756	. . . . . . . . . 10 |- 0 <_ (cos` 1)
60	56, 55	lt2sqi 7869	. . . . . . . . . 10 |- ((0 <_ (1 / 3) /\ 0 <_ (cos` 1)) -> ((1 / 3) < (cos` 1) <-> ((1 / 3)^2) < ((cos` 1)^2)))
61	52, 59, 60	mp2an 761	. . . . . . . . 9 |- ((1 / 3) < (cos` 1) <-> ((1 / 3)^2) < ((cos` 1)^2))
62	46, 61	mpbi 206	. . . . . . . 8 |- ((1 / 3)^2) < ((cos` 1)^2)
63	44, 62	eqbrtrri 3358	. . . . . . 7 |- (1 / 9) < ((cos` 1)^2)
64		2pos 7173	. . . . . . . 8 |- 0 < 2
65	2, 5	rereccli 6979	. . . . . . . . 9 |- (1 / 9) e. RR
66	55	resqcli 7868	. . . . . . . . 9 |- ((cos` 1)^2) e. RR
67		2re 7163	. . . . . . . . 9 |- 2 e. RR
68	65, 66, 67	ltmul2i 7015	. . . . . . . 8 |- (0 < 2 -> ((1 / 9) < ((cos` 1)^2) <-> (2 x. (1 / 9)) < (2 x. ((cos` 1)^2))))
69	64, 68	ax-mp 7	. . . . . . 7 |- ((1 / 9) < ((cos` 1)^2) <-> (2 x. (1 / 9)) < (2 x. ((cos` 1)^2)))
70	63, 69	mpbi 206	. . . . . 6 |- (2 x. (1 / 9)) < (2 x. ((cos` 1)^2))
71	35, 70	eqbrtrri 3358	. . . . 5 |- (2 / 9) < (2 x. ((cos` 1)^2))
72	67, 2, 5	redivcli 6976	. . . . . 6 |- (2 / 9) e. RR
73	67, 66	remulcli 6488	. . . . . 6 |- (2 x. ((cos` 1)^2)) e. RR
74		ltsub1 6851	. . . . . 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)))
75	72, 73, 50, 74	mp3an 1191	. . . . 5 |- ((2 / 9) < (2 x. ((cos` 1)^2)) <-> ((2 / 9) - 1) < ((2 x. ((cos` 1)^2)) - 1))
76	71, 75	mpbi 206	. . . 4 |- ((2 / 9) - 1) < ((2 x. ((cos` 1)^2)) - 1)
77	31, 76	eqbrtrri 3358	. . 3 |- -u(7 / 9) < ((2 x. ((cos` 1)^2)) - 1)
78	33	fveq2i 4684	. . . 4 |- (cos` (2 x. 1)) = (cos` 2)
79	12	cos2OLD 8730	. . . 4 |- (cos` (2 x. 1)) = ((2 x. ((cos` 1)^2)) - 1)
80	78, 79	eqtr3i 1910	. . 3 |- (cos` 2) = ((2 x. ((cos` 1)^2)) - 1)
81	77, 80	breqtrri 3362	. 2 |- -u(7 / 9) < (cos` 2)
82	45	simpri 351	. . . . . . . . 9 |- (cos` 1) < (2 / 3)
83		2nn0 7324	. . . . . . . . . . . 12 |- 2 e. NN0
84	83	nn0ge0i 7327	. . . . . . . . . . 11 |- 0 <_ 2
85	67, 36	divge0i 7040	. . . . . . . . . . 11 |- ((0 <_ 2 /\ 0 < 3) -> 0 <_ (2 / 3))
86	84, 49, 85	mp2an 761	. . . . . . . . . 10 |- 0 <_ (2 / 3)
87	67, 36, 39	redivcli 6976	. . . . . . . . . . 11 |- (2 / 3) e. RR
88	55, 87	lt2sqi 7869	. . . . . . . . . 10 |- ((0 <_ (cos` 1) /\ 0 <_ (2 / 3)) -> ((cos` 1) < (2 / 3) <-> ((cos` 1)^2) < ((2 / 3)^2)))
89	59, 86, 88	mp2an 761	. . . . . . . . 9 |- ((cos` 1) < (2 / 3) <-> ((cos` 1)^2) < ((2 / 3)^2))
90	82, 89	mpbi 206	. . . . . . . 8 |- ((cos` 1)^2) < ((2 / 3)^2)
91	1, 37, 39	sqdivi 7863	. . . . . . . . 9 |- ((2 / 3)^2) = ((2^2) / (3^2))
92		sq2 7883	. . . . . . . . . 10 |- (2^2) = 4
93	92, 42	opreq12i 4894	. . . . . . . . 9 |- ((2^2) / (3^2)) = (4 / 9)
94	91, 93	eqtri 1908	. . . . . . . 8 |- ((2 / 3)^2) = (4 / 9)
95	90, 94	breqtri 3360	. . . . . . 7 |- ((cos` 1)^2) < (4 / 9)
96		4re 7166	. . . . . . . . . 10 |- 4 e. RR
97	96, 2, 5	redivcli 6976	. . . . . . . . 9 |- (4 / 9) e. RR
98	66, 97, 67	ltmul2i 7015	. . . . . . . 8 |- (0 < 2 -> (((cos` 1)^2) < (4 / 9) <-> (2 x. ((cos` 1)^2)) < (2 x. (4 / 9))))
99	64, 98	ax-mp 7	. . . . . . 7 |- (((cos` 1)^2) < (4 / 9) <-> (2 x. ((cos` 1)^2)) < (2 x. (4 / 9)))
100	95, 99	mpbi 206	. . . . . 6 |- (2 x. ((cos` 1)^2)) < (2 x. (4 / 9))
101	96	recni 6467	. . . . . . . 8 |- 4 e. CC
102	1, 101, 3, 5	divassi 6929	. . . . . . 7 |- ((2 x. 4) / 9) = (2 x. (4 / 9))
103	1, 101	mulcomi 6476	. . . . . . . . 9 |- (2 x. 4) = (4 x. 2)
104		4t2e8 7209	. . . . . . . . 9 |- (4 x. 2) = 8
105	103, 104	eqtri 1908	. . . . . . . 8 |- (2 x. 4) = 8
106	105	opreq1i 4892	. . . . . . 7 |- ((2 x. 4) / 9) = (8 / 9)
107	102, 106	eqtr3i 1910	. . . . . 6 |- (2 x. (4 / 9)) = (8 / 9)
108	100, 107	breqtri 3360	. . . . 5 |- (2 x. ((cos` 1)^2)) < (8 / 9)
109		8re 7170	. . . . . . 7 |- 8 e. RR
110	109, 2, 5	redivcli 6976	. . . . . 6 |- (8 / 9) e. RR
111		ltsub1 6851	. . . . . 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)))
112	73, 110, 50, 111	mp3an 1191	. . . . 5 |- ((2 x. ((cos` 1)^2)) < (8 / 9) <-> ((2 x. ((cos` 1)^2)) - 1) < ((8 / 9) - 1))
113	108, 112	mpbi 206	. . . 4 |- ((2 x. ((cos` 1)^2)) - 1) < ((8 / 9) - 1)
114	26	opreq2i 4893	. . . . 5 |- ((8 / 9) - (9 / 9)) = ((8 / 9) - 1)
115		divneg 6950	. . . . . . 7 |- ((1 e. CC /\ 9 e. CC /\ 9 =/= 0) -> -u(1 / 9) = (-u1 / 9))
116	12, 3, 5, 115	mp3an 1191	. . . . . 6 |- -u(1 / 9) = (-u1 / 9)
117	109	recni 6467	. . . . . . . . 9 |- 8 e. CC
118	3, 117	negsubdi2i 6614	. . . . . . . 8 |- -u(9 - 8) = (8 - 9)
119	19	eqcomi 1888	. . . . . . . . . 10 |- (8 + 1) = 9
120	3, 117, 12	subaddi 6528	. . . . . . . . . 10 |- ((9 - 8) = 1 <-> (8 + 1) = 9)
121	119, 120	mpbir 207	. . . . . . . . 9 |- (9 - 8) = 1
122	121	negeqi 6515	. . . . . . . 8 |- -u(9 - 8) = -u1
123	118, 122	eqtr3i 1910	. . . . . . 7 |- (8 - 9) = -u1
124	123	opreq1i 4892	. . . . . 6 |- ((8 - 9) / 9) = (-u1 / 9)
125		divsubdir 6951	. . . . . . 7 |- ((8 e. CC /\ 9 e. CC /\ (9 e. CC /\ 9 =/= 0)) -> ((8 - 9) / 9) = ((8 / 9) - (9 / 9)))
126	117, 3, 6, 125	mp3an 1191	. . . . . 6 |- ((8 - 9) / 9) = ((8 / 9) - (9 / 9))
127	116, 124, 126	3eqtr2ri 1916	. . . . 5 |- ((8 / 9) - (9 / 9)) = -u(1 / 9)
128	114, 127	eqtr3i 1910	. . . 4 |- ((8 / 9) - 1) = -u(1 / 9)
129	113, 128	breqtri 3360	. . 3 |- ((2 x. ((cos` 1)^2)) - 1) < -u(1 / 9)
130	80, 129	eqbrtri 3356	. 2 |- (cos` 2) < -u(1 / 9)
131	81, 130	pm3.2i 307	1 |- (-u(7 / 9) < (cos` 2) /\ (cos` 2) < -u(1 / 9))

Syntax hints:   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300   =/= wne 2017   class class class wbr 3338  ` cfv 3998  (class class class)co 4884  CCcc 6384  RRcr 6385  0cc0 6386  1c1 6387   + caddc 6389   x. cmul 6391   - cmin 6445  -ucneg 6446   / cdiv 6447   <_ cle 6448   < clt 6653  2c2 7145  3c3 7146  4c4 7147  7c7 7150  8c8 7151  9c9 7152  ^cexp 7811  cosccos 8558

This theorem is referenced by:  sincos2sgn 8746

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-5 1302  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-sup 5664  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-div 6892  df-n 7108  df-2 7154  df-3 7155  df-4 7156  df-5 7157  df-6 7158  df-7 7159  df-8 7160  df-9 7161  df-n0 7309  df-z 7345  df-fl 7463  df-ioc 7529  df-uz 7587  df-fz 7638  df-seq1 7721  df-shft 7754  df-seqz 7776  df-seq0 7777  df-exp 7812  df-sqr 7920  df-re 8001  df-im 8002  df-cj 8003  df-abs 8004  df-fac 8184  df-bc 8209  df-clim 8235  df-sum 8240  df-ef 8560  df-sin 8562  df-cos 8563

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