Theorem sincos6thpi 10061

Description: The sine and cosine of pi / 6. (Contributed by Paul Chapman, 25-Jan-2008.)

Assertion

Ref	Expression
sincos6thpi	|- ((sin` (pi / 6)) = (1 / 2) /\ (cos` (pi / 6)) = ((sqr` 3) / 2))

Proof of Theorem sincos6thpi

Step	Hyp	Ref	Expression
1		2cn 7164	. . . . . . . 8 |- 2 e. CC
2		pire 10026	. . . . . . . . . . 11 |- pi e. RR
3		6re 7168	. . . . . . . . . . 11 |- 6 e. RR
4		6pos 7178	. . . . . . . . . . . 12 |- 0 < 6
5	3, 4	gt0ne0ii 6799	. . . . . . . . . . 11 |- 6 =/= 0
6	2, 3, 5	redivcli 6976	. . . . . . . . . 10 |- (pi / 6) e. RR
7	6	recni 6467	. . . . . . . . 9 |- (pi / 6) e. CC
8		sincl 8696	. . . . . . . . 9 |- ((pi / 6) e. CC -> (sin` (pi / 6)) e. CC)
9	7, 8	ax-mp 7	. . . . . . . 8 |- (sin` (pi / 6)) e. CC
10		recoscl 8704	. . . . . . . . . 10 |- ((pi / 6) e. RR -> (cos` (pi / 6)) e. RR)
11	6, 10	ax-mp 7	. . . . . . . . 9 |- (cos` (pi / 6)) e. RR
12	11	recni 6467	. . . . . . . 8 |- (cos` (pi / 6)) e. CC
13	1, 9, 12	mulassi 6478	. . . . . . 7 |- ((2 x. (sin` (pi / 6))) x. (cos` (pi / 6))) = (2 x. ((sin` (pi / 6)) x. (cos` (pi / 6))))
14		sin2t 8727	. . . . . . . 8 |- ((pi / 6) e. CC -> (sin` (2 x. (pi / 6))) = (2 x. ((sin` (pi / 6)) x. (cos` (pi / 6)))))
15	7, 14	ax-mp 7	. . . . . . 7 |- (sin` (2 x. (pi / 6))) = (2 x. ((sin` (pi / 6)) x. (cos` (pi / 6))))
16		3re 7165	. . . . . . . . . . 11 |- 3 e. RR
17	16	recni 6467	. . . . . . . . . 10 |- 3 e. CC
18		3pos 7175	. . . . . . . . . . 11 |- 0 < 3
19	16, 18	gt0ne0ii 6799	. . . . . . . . . 10 |- 3 =/= 0
20	1, 17, 19	divcli 6899	. . . . . . . . 9 |- (2 / 3) e. CC
21	17, 19	reccli 6902	. . . . . . . . 9 |- (1 / 3) e. CC
22		df-3 7155	. . . . . . . . . . 11 |- 3 = (2 + 1)
23	22	opreq1i 4892	. . . . . . . . . 10 |- (3 / 3) = ((2 + 1) / 3)
24	17, 19	dividi 6946	. . . . . . . . . 10 |- (3 / 3) = 1
25		ax1cn 6422	. . . . . . . . . . 11 |- 1 e. CC
26	1, 25, 17, 19	divdiri 6930	. . . . . . . . . 10 |- ((2 + 1) / 3) = ((2 / 3) + (1 / 3))
27	23, 24, 26	3eqtr3ri 1920	. . . . . . . . 9 |- ((2 / 3) + (1 / 3)) = 1
28		sincosq1eq 10059	. . . . . . . . 9 |- (((2 / 3) e. CC /\ (1 / 3) e. CC /\ ((2 / 3) + (1 / 3)) = 1) -> (sin` ((2 / 3) x. (pi / 2))) = (cos` ((1 / 3) x. (pi / 2))))
29	20, 21, 27, 28	mp3an 1191	. . . . . . . 8 |- (sin` ((2 / 3) x. (pi / 2))) = (cos` ((1 / 3) x. (pi / 2)))
30	2	recni 6467	. . . . . . . . . . 11 |- pi e. CC
31		2ne0 7174	. . . . . . . . . . 11 |- 2 =/= 0
32	1, 17, 30, 1, 19, 31	divmuldivi 6963	. . . . . . . . . 10 |- ((2 / 3) x. (pi / 2)) = ((2 x. pi) / (3 x. 2))
33		3t2e6 7207	. . . . . . . . . . 11 |- (3 x. 2) = 6
34	33	opreq2i 4893	. . . . . . . . . 10 |- ((2 x. pi) / (3 x. 2)) = ((2 x. pi) / 6)
35	3	recni 6467	. . . . . . . . . . 11 |- 6 e. CC
36	1, 30, 35, 5	divassi 6929	. . . . . . . . . 10 |- ((2 x. pi) / 6) = (2 x. (pi / 6))
37	32, 34, 36	3eqtri 1912	. . . . . . . . 9 |- ((2 / 3) x. (pi / 2)) = (2 x. (pi / 6))
38	37	fveq2i 4684	. . . . . . . 8 |- (sin` ((2 / 3) x. (pi / 2))) = (sin` (2 x. (pi / 6)))
39	25, 17, 30, 1, 19, 31	divmuldivi 6963	. . . . . . . . . 10 |- ((1 / 3) x. (pi / 2)) = ((1 x. pi) / (3 x. 2))
40	30	mulid2i 6486	. . . . . . . . . . 11 |- (1 x. pi) = pi
41	40, 33	opreq12i 4894	. . . . . . . . . 10 |- ((1 x. pi) / (3 x. 2)) = (pi / 6)
42	39, 41	eqtri 1908	. . . . . . . . 9 |- ((1 / 3) x. (pi / 2)) = (pi / 6)
43	42	fveq2i 4684	. . . . . . . 8 |- (cos` ((1 / 3) x. (pi / 2))) = (cos` (pi / 6))
44	29, 38, 43	3eqtr3i 1918	. . . . . . 7 |- (sin` (2 x. (pi / 6))) = (cos` (pi / 6))
45	13, 15, 44	3eqtr2i 1915	. . . . . 6 |- ((2 x. (sin` (pi / 6))) x. (cos` (pi / 6))) = (cos` (pi / 6))
46	12	mulid2i 6486	. . . . . 6 |- (1 x. (cos` (pi / 6))) = (cos` (pi / 6))
47	45, 46	eqtr4i 1911	. . . . 5 |- ((2 x. (sin` (pi / 6))) x. (cos` (pi / 6))) = (1 x. (cos` (pi / 6)))
48	1, 9	mulcli 6474	. . . . . 6 |- (2 x. (sin` (pi / 6))) e. CC
49		0re 6603	. . . . . . . . . . . 12 |- 0 e. RR
50		2re 7163	. . . . . . . . . . . . 13 |- 2 e. RR
51	2, 50, 31	redivcli 6976	. . . . . . . . . . . 12 |- (pi / 2) e. RR
52		elioo2 7546	. . . . . . . . . . . . 13 |- ((0 e. RR* /\ (pi / 2) e. RR*) -> ((pi / 6) e. (0(,)(pi / 2)) <-> ((pi / 6) e. RR /\ 0 < (pi / 6) /\ (pi / 6) < (pi / 2))))
53		rexr 6668	. . . . . . . . . . . . 13 |- (0 e. RR -> 0 e. RR*)
54		rexr 6668	. . . . . . . . . . . . 13 |- ((pi / 2) e. RR -> (pi / 2) e. RR*)
55	52, 53, 54	syl2an 503	. . . . . . . . . . . 12 |- ((0 e. RR /\ (pi / 2) e. RR) -> ((pi / 6) e. (0(,)(pi / 2)) <-> ((pi / 6) e. RR /\ 0 < (pi / 6) /\ (pi / 6) < (pi / 2))))
56	49, 51, 55	mp2an 761	. . . . . . . . . . 11 |- ((pi / 6) e. (0(,)(pi / 2)) <-> ((pi / 6) e. RR /\ 0 < (pi / 6) /\ (pi / 6) < (pi / 2)))
57		pipos 10027	. . . . . . . . . . . 12 |- 0 < pi
58	2, 3, 57, 4	divgt0ii 7042	. . . . . . . . . . 11 |- 0 < (pi / 6)
59	25	addid2i 6484	. . . . . . . . . . . . . . . 16 |- (0 + 1) = 1
60		2pos 7173	. . . . . . . . . . . . . . . . 17 |- 0 < 2
61		1re 6598	. . . . . . . . . . . . . . . . . 18 |- 1 e. RR
62	49, 50, 61	ltadd1i 6766	. . . . . . . . . . . . . . . . 17 |- (0 < 2 <-> (0 + 1) < (2 + 1))
63	60, 62	mpbi 206	. . . . . . . . . . . . . . . 16 |- (0 + 1) < (2 + 1)
64	59, 63	eqbrtrri 3358	. . . . . . . . . . . . . . 15 |- 1 < (2 + 1)
65	64, 22	breqtrri 3362	. . . . . . . . . . . . . 14 |- 1 < 3
66	61, 16, 50, 60	ltmul1ii 6999	. . . . . . . . . . . . . 14 |- (1 < 3 <-> (1 x. 2) < (3 x. 2))
67	65, 66	mpbi 206	. . . . . . . . . . . . 13 |- (1 x. 2) < (3 x. 2)
68	1	mulid2i 6486	. . . . . . . . . . . . 13 |- (1 x. 2) = 2
69	67, 68, 33	3brtr3i 3364	. . . . . . . . . . . 12 |- 2 < 6
70	50, 3, 2	3pm3.2i 1048	. . . . . . . . . . . . . 14 |- (2 e. RR /\ 6 e. RR /\ pi e. RR)
71		ltdiv2 7070	. . . . . . . . . . . . . 14 |- (((2 e. RR /\ 6 e. RR /\ pi e. RR) /\ (0 < 2 /\ 0 < 6 /\ 0 < pi)) -> (2 < 6 <-> (pi / 6) < (pi / 2)))
72	70, 71	mpan 759	. . . . . . . . . . . . 13 |- ((0 < 2 /\ 0 < 6 /\ 0 < pi) -> (2 < 6 <-> (pi / 6) < (pi / 2)))
73	60, 4, 57, 72	mp3an 1191	. . . . . . . . . . . 12 |- (2 < 6 <-> (pi / 6) < (pi / 2))
74	69, 73	mpbi 206	. . . . . . . . . . 11 |- (pi / 6) < (pi / 2)
75	56, 6, 58, 74	mpbir3an 1052	. . . . . . . . . 10 |- (pi / 6) e. (0(,)(pi / 2))
76		sincosq1sgn 10053	. . . . . . . . . 10 |- ((pi / 6) e. (0(,)(pi / 2)) -> (0 < (sin` (pi / 6)) /\ 0 < (cos` (pi / 6))))
77	75, 76	ax-mp 7	. . . . . . . . 9 |- (0 < (sin` (pi / 6)) /\ 0 < (cos` (pi / 6)))
78	77	simpri 351	. . . . . . . 8 |- 0 < (cos` (pi / 6))
79	11, 78	gt0ne0ii 6799	. . . . . . 7 |- (cos` (pi / 6)) =/= 0
80	12, 79	pm3.2i 307	. . . . . 6 |- ((cos` (pi / 6)) e. CC /\ (cos` (pi / 6)) =/= 0)
81		mulcan2 6881	. . . . . 6 |- (((2 x. (sin` (pi / 6))) e. CC /\ 1 e. CC /\ ((cos` (pi / 6)) e. CC /\ (cos` (pi / 6)) =/= 0)) -> (((2 x. (sin` (pi / 6))) x. (cos` (pi / 6))) = (1 x. (cos` (pi / 6))) <-> (2 x. (sin` (pi / 6))) = 1))
82	48, 25, 80, 81	mp3an 1191	. . . . 5 |- (((2 x. (sin` (pi / 6))) x. (cos` (pi / 6))) = (1 x. (cos` (pi / 6))) <-> (2 x. (sin` (pi / 6))) = 1)
83	47, 82	mpbi 206	. . . 4 |- (2 x. (sin` (pi / 6))) = 1
84	25, 1, 9, 31	divmuli 6894	. . . 4 |- ((1 / 2) = (sin` (pi / 6)) <-> (2 x. (sin` (pi / 6))) = 1)
85	83, 84	mpbir 207	. . 3 |- (1 / 2) = (sin` (pi / 6))
86	85	eqcomi 1888	. 2 |- (sin` (pi / 6)) = (1 / 2)
87		4re 7166	. . . . . . . 8 |- 4 e. RR
88	87	recni 6467	. . . . . . 7 |- 4 e. CC
89		4pos 7176	. . . . . . . 8 |- 0 < 4
90	87, 89	gt0ne0ii 6799	. . . . . . 7 |- 4 =/= 0
91	88, 90	reccli 6902	. . . . . 6 |- (1 / 4) e. CC
92	12	sqcli 7860	. . . . . 6 |- ((cos` (pi / 6))^2) e. CC
93	86	opreq1i 4892	. . . . . . . . 9 |- ((sin` (pi / 6))^2) = ((1 / 2)^2)
94	1, 31	sqrecii 7864	. . . . . . . . 9 |- ((1 / 2)^2) = (1 / (2^2))
95		sq2 7883	. . . . . . . . . 10 |- (2^2) = 4
96	95	opreq2i 4893	. . . . . . . . 9 |- (1 / (2^2)) = (1 / 4)
97	93, 94, 96	3eqtri 1912	. . . . . . . 8 |- ((sin` (pi / 6))^2) = (1 / 4)
98	97	opreq1i 4892	. . . . . . 7 |- (((sin` (pi / 6))^2) + ((cos` (pi / 6))^2)) = ((1 / 4) + ((cos` (pi / 6))^2))
99		sincossq 8726	. . . . . . . 8 |- ((pi / 6) e. CC -> (((sin` (pi / 6))^2) + ((cos` (pi / 6))^2)) = 1)
100	7, 99	ax-mp 7	. . . . . . 7 |- (((sin` (pi / 6))^2) + ((cos` (pi / 6))^2)) = 1
101	98, 100	eqtr3i 1910	. . . . . 6 |- ((1 / 4) + ((cos` (pi / 6))^2)) = 1
102	25, 91, 92, 101	subaddrii 6529	. . . . 5 |- (1 - (1 / 4)) = ((cos` (pi / 6))^2)
103	88, 90	pm3.2i 307	. . . . . . 7 |- (4 e. CC /\ 4 =/= 0)
104		divsubdir 6951	. . . . . . 7 |- ((4 e. CC /\ 1 e. CC /\ (4 e. CC /\ 4 =/= 0)) -> ((4 - 1) / 4) = ((4 / 4) - (1 / 4)))
105	88, 25, 103, 104	mp3an 1191	. . . . . 6 |- ((4 - 1) / 4) = ((4 / 4) - (1 / 4))
106		df-4 7156	. . . . . . . . 9 |- 4 = (3 + 1)
107	106	eqcomi 1888	. . . . . . . 8 |- (3 + 1) = 4
108	88, 25, 17	subadd2i 6530	. . . . . . . 8 |- ((4 - 1) = 3 <-> (3 + 1) = 4)
109	107, 108	mpbir 207	. . . . . . 7 |- (4 - 1) = 3
110	109	opreq1i 4892	. . . . . 6 |- ((4 - 1) / 4) = (3 / 4)
111	88, 90	dividi 6946	. . . . . . 7 |- (4 / 4) = 1
112	111	opreq1i 4892	. . . . . 6 |- ((4 / 4) - (1 / 4)) = (1 - (1 / 4))
113	105, 110, 112	3eqtr3ri 1920	. . . . 5 |- (1 - (1 / 4)) = (3 / 4)
114	102, 113	eqtr3i 1910	. . . 4 |- ((cos` (pi / 6))^2) = (3 / 4)
115	114	fveq2i 4684	. . 3 |- (sqr` ((cos` (pi / 6))^2)) = (sqr` (3 / 4))
116	49, 11, 78	ltleii 6756	. . . 4 |- 0 <_ (cos` (pi / 6))
117	11	sqrsqi 7970	. . . 4 |- (0 <_ (cos` (pi / 6)) -> (sqr` ((cos` (pi / 6))^2)) = (cos` (pi / 6)))
118	116, 117	ax-mp 7	. . 3 |- (sqr` ((cos` (pi / 6))^2)) = (cos` (pi / 6))
119	16, 18	sqrlem24 7946	. . . . . . . 8 |- (sqr` 3) e. RR
120	119	recni 6467	. . . . . . 7 |- (sqr` 3) e. CC
121	120, 1, 31	sqdivi 7863	. . . . . 6 |- (((sqr` 3) / 2)^2) = (((sqr` 3)^2) / (2^2))
122	49, 16, 18	ltleii 6756	. . . . . . . 8 |- 0 <_ 3
123	16	sqsqri 7971	. . . . . . . 8 |- (0 <_ 3 -> ((sqr` 3)^2) = 3)
124	122, 123	ax-mp 7	. . . . . . 7 |- ((sqr` 3)^2) = 3
125	124, 95	opreq12i 4894	. . . . . 6 |- (((sqr` 3)^2) / (2^2)) = (3 / 4)
126	121, 125	eqtri 1908	. . . . 5 |- (((sqr` 3) / 2)^2) = (3 / 4)
127	126	fveq2i 4684	. . . 4 |- (sqr` (((sqr` 3) / 2)^2)) = (sqr` (3 / 4))
128	16	sqrge0i 7952	. . . . . . 7 |- (0 <_ 3 -> 0 <_ (sqr` 3))
129	122, 128	ax-mp 7	. . . . . 6 |- 0 <_ (sqr` 3)
130	119, 50	divge0i 7040	. . . . . 6 |- ((0 <_ (sqr` 3) /\ 0 < 2) -> 0 <_ ((sqr` 3) / 2))
131	129, 60, 130	mp2an 761	. . . . 5 |- 0 <_ ((sqr` 3) / 2)
132	119, 50, 31	redivcli 6976	. . . . . 6 |- ((sqr` 3) / 2) e. RR
133	132	sqrsqi 7970	. . . . 5 |- (0 <_ ((sqr` 3) / 2) -> (sqr` (((sqr` 3) / 2)^2)) = ((sqr` 3) / 2))
134	131, 133	ax-mp 7	. . . 4 |- (sqr` (((sqr` 3) / 2)^2)) = ((sqr` 3) / 2)
135	127, 134	eqtr3i 1910	. . 3 |- (sqr` (3 / 4)) = ((sqr` 3) / 2)
136	115, 118, 135	3eqtr3i 1918	. 2 |- (cos` (pi / 6)) = ((sqr` 3) / 2)
137	86, 136	pm3.2i 307	1 |- ((sin` (pi / 6)) = (1 / 2) /\ (cos` (pi / 6)) = ((sqr` 3) / 2))

Syntax hints:   <-> wb 163   /\ wa 240   /\ w3a 858   = 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   / cdiv 6447   <_ cle 6448  RR*cxr 6652   < clt 6653  2c2 7145  3c3 7146  4c4 7147  6c6 7149  (,)cioo 7524  ^cexp 7811  sqrcsqr 7919  sincsin 8557  cosccos 8558  picpi 8559

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-reg 5695  ax-inf2 5731  ax-ac 5906

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-iin 3258  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-map 5383  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-sup 5664  df-r1 5750  df-rank 5751  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-rp 7232  df-n0 7309  df-z 7345  df-q 7436  df-fl 7463  df-ioo 7528  df-ioc 7529  df-ico 7530  df-icc 7531  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-cncf 8525  df-ef 8560  df-sin 8562  df-cos 8563  df-pi 8564  df-top 8861  df-cn 9030  df-cnp 9031  df-met 9070  df-bl 9072  df-opn 9073  df-lm 9200

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