Theorem sincos6thpi 15565

Description: The sine and cosine of pi / 6. (Contributed by Paul Chapman, 25-Jan-2008.) (Revised by Wolf Lammen, 24-Sep-2020.)

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 9213	. . . . 5 |- 2 e. CC
2	1	a1i 9	. . . 4 |- ( T. -> 2 e. CC )
3		pire 15509	. . . . . . . 8 |- pi e. RR
4		6re 9223	. . . . . . . 8 |- 6 e. RR
5		6pos 9243	. . . . . . . . 9 |- 0 < 6
6	4, 5	gt0ap0ii 8807	. . . . . . . 8 |- 6 # 0
7	3, 4, 6	redivclapi 8958	. . . . . . 7 |- ( pi / 6 ) e. RR
8	7	recni 8190	. . . . . 6 |- ( pi / 6 ) e. CC
9		sincl 12266	. . . . . 6 |- ( ( pi / 6 ) e. CC -> ( sin ` ( pi / 6 ) ) e. CC )
10	8, 9	ax-mp 5	. . . . 5 |- ( sin ` ( pi / 6 ) ) e. CC
11	10	a1i 9	. . . 4 |- ( T. -> ( sin ` ( pi / 6 ) ) e. CC )
12		2ap0 9235	. . . . 5 |- 2 # 0
13	12	a1i 9	. . . 4 |- ( T. -> 2 # 0 )
14		recoscl 12281	. . . . . . . . . . . 12 |- ( ( pi / 6 ) e. RR -> ( cos ` ( pi / 6 ) ) e. RR )
15	7, 14	ax-mp 5	. . . . . . . . . . 11 |- ( cos ` ( pi / 6 ) ) e. RR
16	15	recni 8190	. . . . . . . . . 10 |- ( cos ` ( pi / 6 ) ) e. CC
17	1, 10, 16	mulassi 8187	. . . . . . . . 9 |- ( ( 2 x. ( sin ` ( pi / 6 ) ) ) x. ( cos ` ( pi / 6 ) ) ) = ( 2 x. ( ( sin ` ( pi / 6 ) ) x. ( cos ` ( pi / 6 ) ) ) )
18		sin2t 12309	. . . . . . . . . 10 |- ( ( pi / 6 ) e. CC -> ( sin ` ( 2 x. ( pi / 6 ) ) ) = ( 2 x. ( ( sin ` ( pi / 6 ) ) x. ( cos ` ( pi / 6 ) ) ) ) )
19	8, 18	ax-mp 5	. . . . . . . . 9 |- ( sin ` ( 2 x. ( pi / 6 ) ) ) = ( 2 x. ( ( sin ` ( pi / 6 ) ) x. ( cos ` ( pi / 6 ) ) ) )
20	17, 19	eqtr4i 2255	. . . . . . . 8 |- ( ( 2 x. ( sin ` ( pi / 6 ) ) ) x. ( cos ` ( pi / 6 ) ) ) = ( sin ` ( 2 x. ( pi / 6 ) ) )
21		3cn 9217	. . . . . . . . . . . 12 |- 3 e. CC
22		3ap0 9238	. . . . . . . . . . . 12 |- 3 # 0
23	1, 21, 22	divclapi 8933	. . . . . . . . . . 11 |- ( 2 / 3 ) e. CC
24	21, 22	recclapi 8921	. . . . . . . . . . 11 |- ( 1 / 3 ) e. CC
25		df-3 9202	. . . . . . . . . . . . 13 |- 3 = ( 2 + 1 )
26	25	oveq1i 6027	. . . . . . . . . . . 12 |- ( 3 / 3 ) = ( ( 2 + 1 ) / 3 )
27	21, 22	dividapi 8924	. . . . . . . . . . . 12 |- ( 3 / 3 ) = 1
28		ax-1cn 8124	. . . . . . . . . . . . 13 |- 1 e. CC
29	1, 28, 21, 22	divdirapi 8948	. . . . . . . . . . . 12 |- ( ( 2 + 1 ) / 3 ) = ( ( 2 / 3 ) + ( 1 / 3 ) )
30	26, 27, 29	3eqtr3ri 2261	. . . . . . . . . . 11 |- ( ( 2 / 3 ) + ( 1 / 3 ) ) = 1
31		sincosq1eq 15562	. . . . . . . . . . 11 |- ( ( ( 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 ) ) ) )
32	23, 24, 30, 31	mp3an 1373	. . . . . . . . . 10 |- ( sin ` ( ( 2 / 3 ) x. ( pi / 2 ) ) ) = ( cos ` ( ( 1 / 3 ) x. ( pi / 2 ) ) )
33		picn 15510	. . . . . . . . . . . . 13 |- pi e. CC
34	1, 21, 33, 1, 22, 12	divmuldivapi 8951	. . . . . . . . . . . 12 |- ( ( 2 / 3 ) x. ( pi / 2 ) ) = ( ( 2 x. pi ) / ( 3 x. 2 ) )
35		3t2e6 9299	. . . . . . . . . . . . 13 |- ( 3 x. 2 ) = 6
36	35	oveq2i 6028	. . . . . . . . . . . 12 |- ( ( 2 x. pi ) / ( 3 x. 2 ) ) = ( ( 2 x. pi ) / 6 )
37		6cn 9224	. . . . . . . . . . . . 13 |- 6 e. CC
38	1, 33, 37, 6	divassapi 8947	. . . . . . . . . . . 12 |- ( ( 2 x. pi ) / 6 ) = ( 2 x. ( pi / 6 ) )
39	34, 36, 38	3eqtri 2256	. . . . . . . . . . 11 |- ( ( 2 / 3 ) x. ( pi / 2 ) ) = ( 2 x. ( pi / 6 ) )
40	39	fveq2i 5642	. . . . . . . . . 10 |- ( sin ` ( ( 2 / 3 ) x. ( pi / 2 ) ) ) = ( sin ` ( 2 x. ( pi / 6 ) ) )
41	32, 40	eqtr3i 2254	. . . . . . . . 9 |- ( cos ` ( ( 1 / 3 ) x. ( pi / 2 ) ) ) = ( sin ` ( 2 x. ( pi / 6 ) ) )
42	28, 21, 33, 1, 22, 12	divmuldivapi 8951	. . . . . . . . . . 11 |- ( ( 1 / 3 ) x. ( pi / 2 ) ) = ( ( 1 x. pi ) / ( 3 x. 2 ) )
43	33	mullidi 8181	. . . . . . . . . . . 12 |- ( 1 x. pi ) = pi
44	43, 35	oveq12i 6029	. . . . . . . . . . 11 |- ( ( 1 x. pi ) / ( 3 x. 2 ) ) = ( pi / 6 )
45	42, 44	eqtri 2252	. . . . . . . . . 10 |- ( ( 1 / 3 ) x. ( pi / 2 ) ) = ( pi / 6 )
46	45	fveq2i 5642	. . . . . . . . 9 |- ( cos ` ( ( 1 / 3 ) x. ( pi / 2 ) ) ) = ( cos ` ( pi / 6 ) )
47	41, 46	eqtr3i 2254	. . . . . . . 8 |- ( sin ` ( 2 x. ( pi / 6 ) ) ) = ( cos ` ( pi / 6 ) )
48	20, 47	eqtri 2252	. . . . . . 7 |- ( ( 2 x. ( sin ` ( pi / 6 ) ) ) x. ( cos ` ( pi / 6 ) ) ) = ( cos ` ( pi / 6 ) )
49	16	mullidi 8181	. . . . . . 7 |- ( 1 x. ( cos ` ( pi / 6 ) ) ) = ( cos ` ( pi / 6 ) )
50	48, 49	eqtr4i 2255	. . . . . 6 |- ( ( 2 x. ( sin ` ( pi / 6 ) ) ) x. ( cos ` ( pi / 6 ) ) ) = ( 1 x. ( cos ` ( pi / 6 ) ) )
51	1, 10	mulcli 8183	. . . . . . 7 |- ( 2 x. ( sin ` ( pi / 6 ) ) ) e. CC
52		pipos 15511	. . . . . . . . . . . . 13 |- 0 < pi
53	3, 4, 52, 5	divgt0ii 9098	. . . . . . . . . . . 12 |- 0 < ( pi / 6 )
54		2lt6 9325	. . . . . . . . . . . . 13 |- 2 < 6
55		2re 9212	. . . . . . . . . . . . . . 15 |- 2 e. RR
56		2pos 9233	. . . . . . . . . . . . . . 15 |- 0 < 2
57	55, 56	pm3.2i 272	. . . . . . . . . . . . . 14 |- ( 2 e. RR /\ 0 < 2 )
58	4, 5	pm3.2i 272	. . . . . . . . . . . . . 14 |- ( 6 e. RR /\ 0 < 6 )
59	3, 52	pm3.2i 272	. . . . . . . . . . . . . 14 |- ( pi e. RR /\ 0 < pi )
60		ltdiv2 9066	. . . . . . . . . . . . . 14 |- ( ( ( 2 e. RR /\ 0 < 2 ) /\ ( 6 e. RR /\ 0 < 6 ) /\ ( pi e. RR /\ 0 < pi ) ) -> ( 2 < 6 <-> ( pi / 6 ) < ( pi / 2 ) ) )
61	57, 58, 59, 60	mp3an 1373	. . . . . . . . . . . . 13 |- ( 2 < 6 <-> ( pi / 6 ) < ( pi / 2 ) )
62	54, 61	mpbi 145	. . . . . . . . . . . 12 |- ( pi / 6 ) < ( pi / 2 )
63		0re 8178	. . . . . . . . . . . . 13 |- 0 e. RR
64		halfpire 15515	. . . . . . . . . . . . 13 |- ( pi / 2 ) e. RR
65		rexr 8224	. . . . . . . . . . . . . 14 |- ( 0 e. RR -> 0 e. RR* )
66		rexr 8224	. . . . . . . . . . . . . 14 |- ( ( pi / 2 ) e. RR -> ( pi / 2 ) e. RR* )
67		elioo2 10155	. . . . . . . . . . . . . 14 |- ( ( 0 e. RR* /\ ( pi / 2 ) e. RR* ) -> ( ( pi / 6 ) e. ( 0 (,) ( pi / 2 ) ) <-> ( ( pi / 6 ) e. RR /\ 0 < ( pi / 6 ) /\ ( pi / 6 ) < ( pi / 2 ) ) ) )
68	65, 66, 67	syl2an 289	. . . . . . . . . . . . 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 ) ) ) )
69	63, 64, 68	mp2an 426	. . . . . . . . . . . 12 |- ( ( pi / 6 ) e. ( 0 (,) ( pi / 2 ) ) <-> ( ( pi / 6 ) e. RR /\ 0 < ( pi / 6 ) /\ ( pi / 6 ) < ( pi / 2 ) ) )
70	7, 53, 62, 69	mpbir3an 1205	. . . . . . . . . . 11 |- ( pi / 6 ) e. ( 0 (,) ( pi / 2 ) )
71		sincosq1sgn 15549	. . . . . . . . . . 11 |- ( ( pi / 6 ) e. ( 0 (,) ( pi / 2 ) ) -> ( 0 < ( sin ` ( pi / 6 ) ) /\ 0 < ( cos ` ( pi / 6 ) ) ) )
72	70, 71	ax-mp 5	. . . . . . . . . 10 |- ( 0 < ( sin ` ( pi / 6 ) ) /\ 0 < ( cos ` ( pi / 6 ) ) )
73	72	simpri 113	. . . . . . . . 9 |- 0 < ( cos ` ( pi / 6 ) )
74	15, 73	gt0ap0ii 8807	. . . . . . . 8 |- ( cos ` ( pi / 6 ) ) # 0
75	16, 74	pm3.2i 272	. . . . . . 7 |- ( ( cos ` ( pi / 6 ) ) e. CC /\ ( cos ` ( pi / 6 ) ) # 0 )
76		mulcanap2 8845	. . . . . . 7 |- ( ( ( 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 ) )
77	51, 28, 75, 76	mp3an 1373	. . . . . 6 |- ( ( ( 2 x. ( sin ` ( pi / 6 ) ) ) x. ( cos ` ( pi / 6 ) ) ) = ( 1 x. ( cos ` ( pi / 6 ) ) ) <-> ( 2 x. ( sin ` ( pi / 6 ) ) ) = 1 )
78	50, 77	mpbi 145	. . . . 5 |- ( 2 x. ( sin ` ( pi / 6 ) ) ) = 1
79	78	a1i 9	. . . 4 |- ( T. -> ( 2 x. ( sin ` ( pi / 6 ) ) ) = 1 )
80	2, 11, 13, 79	mvllmulapd 9021	. . 3 |- ( T. -> ( sin ` ( pi / 6 ) ) = ( 1 / 2 ) )
81	80	mptru 1406	. 2 |- ( sin ` ( pi / 6 ) ) = ( 1 / 2 )
82		3re 9216	. . . . . . . 8 |- 3 e. RR
83		3pos 9236	. . . . . . . 8 |- 0 < 3
84	82, 83	sqrtpclii 11690	. . . . . . 7 |- ( sqr ` 3 ) e. RR
85	84	recni 8190	. . . . . 6 |- ( sqr ` 3 ) e. CC
86	85, 1, 12	sqdivapi 10884	. . . . 5 |- ( ( ( sqr ` 3 ) / 2 ) ^ 2 ) = ( ( ( sqr ` 3 ) ^ 2 ) / ( 2 ^ 2 ) )
87	63, 82, 83	ltleii 8281	. . . . . . 7 |- 0 <_ 3
88	82	sqsqrti 11684	. . . . . . 7 |- ( 0 <_ 3 -> ( ( sqr ` 3 ) ^ 2 ) = 3 )
89	87, 88	ax-mp 5	. . . . . 6 |- ( ( sqr ` 3 ) ^ 2 ) = 3
90		sq2 10896	. . . . . 6 |- ( 2 ^ 2 ) = 4
91	89, 90	oveq12i 6029	. . . . 5 |- ( ( ( sqr ` 3 ) ^ 2 ) / ( 2 ^ 2 ) ) = ( 3 / 4 )
92	86, 91	eqtri 2252	. . . 4 |- ( ( ( sqr ` 3 ) / 2 ) ^ 2 ) = ( 3 / 4 )
93	92	fveq2i 5642	. . 3 |- ( sqr ` ( ( ( sqr ` 3 ) / 2 ) ^ 2 ) ) = ( sqr ` ( 3 / 4 ) )
94	82	sqrtge0i 11685	. . . . . 6 |- ( 0 <_ 3 -> 0 <_ ( sqr ` 3 ) )
95	87, 94	ax-mp 5	. . . . 5 |- 0 <_ ( sqr ` 3 )
96	84, 55	divge0i 9090	. . . . 5 |- ( ( 0 <_ ( sqr ` 3 ) /\ 0 < 2 ) -> 0 <_ ( ( sqr ` 3 ) / 2 ) )
97	95, 56, 96	mp2an 426	. . . 4 |- 0 <_ ( ( sqr ` 3 ) / 2 )
98	84, 55, 12	redivclapi 8958	. . . . 5 |- ( ( sqr ` 3 ) / 2 ) e. RR
99	98	sqrtsqi 11683	. . . 4 |- ( 0 <_ ( ( sqr ` 3 ) / 2 ) -> ( sqr ` ( ( ( sqr ` 3 ) / 2 ) ^ 2 ) ) = ( ( sqr ` 3 ) / 2 ) )
100	97, 99	ax-mp 5	. . 3 |- ( sqr ` ( ( ( sqr ` 3 ) / 2 ) ^ 2 ) ) = ( ( sqr ` 3 ) / 2 )
101		4cn 9220	. . . . . . . 8 |- 4 e. CC
102		4ap0 9241	. . . . . . . 8 |- 4 # 0
103	101, 102	dividapi 8924	. . . . . . 7 |- ( 4 / 4 ) = 1
104	103	oveq1i 6027	. . . . . 6 |- ( ( 4 / 4 ) - ( 1 / 4 ) ) = ( 1 - ( 1 / 4 ) )
105	101, 102	pm3.2i 272	. . . . . . . 8 |- ( 4 e. CC /\ 4 # 0 )
106		divsubdirap 8887	. . . . . . . 8 |- ( ( 4 e. CC /\ 1 e. CC /\ ( 4 e. CC /\ 4 # 0 ) ) -> ( ( 4 - 1 ) / 4 ) = ( ( 4 / 4 ) - ( 1 / 4 ) ) )
107	101, 28, 105, 106	mp3an 1373	. . . . . . 7 |- ( ( 4 - 1 ) / 4 ) = ( ( 4 / 4 ) - ( 1 / 4 ) )
108		4m1e3 9263	. . . . . . . 8 |- ( 4 - 1 ) = 3
109	108	oveq1i 6027	. . . . . . 7 |- ( ( 4 - 1 ) / 4 ) = ( 3 / 4 )
110	107, 109	eqtr3i 2254	. . . . . 6 |- ( ( 4 / 4 ) - ( 1 / 4 ) ) = ( 3 / 4 )
111	101, 102	recclapi 8921	. . . . . . 7 |- ( 1 / 4 ) e. CC
112	16	sqcli 10881	. . . . . . 7 |- ( ( cos ` ( pi / 6 ) ) ^ 2 ) e. CC
113	81	oveq1i 6027	. . . . . . . . . 10 |- ( ( sin ` ( pi / 6 ) ) ^ 2 ) = ( ( 1 / 2 ) ^ 2 )
114		2z 9506	. . . . . . . . . . 11 |- 2 e. ZZ
115		exprecap 10841	. . . . . . . . . . 11 |- ( ( 2 e. CC /\ 2 # 0 /\ 2 e. ZZ ) -> ( ( 1 / 2 ) ^ 2 ) = ( 1 / ( 2 ^ 2 ) ) )
116	1, 12, 114, 115	mp3an 1373	. . . . . . . . . 10 |- ( ( 1 / 2 ) ^ 2 ) = ( 1 / ( 2 ^ 2 ) )
117	90	oveq2i 6028	. . . . . . . . . 10 |- ( 1 / ( 2 ^ 2 ) ) = ( 1 / 4 )
118	113, 116, 117	3eqtri 2256	. . . . . . . . 9 |- ( ( sin ` ( pi / 6 ) ) ^ 2 ) = ( 1 / 4 )
119	118	oveq1i 6027	. . . . . . . 8 |- ( ( ( sin ` ( pi / 6 ) ) ^ 2 ) + ( ( cos ` ( pi / 6 ) ) ^ 2 ) ) = ( ( 1 / 4 ) + ( ( cos ` ( pi / 6 ) ) ^ 2 ) )
120		sincossq 12308	. . . . . . . . 9 |- ( ( pi / 6 ) e. CC -> ( ( ( sin ` ( pi / 6 ) ) ^ 2 ) + ( ( cos ` ( pi / 6 ) ) ^ 2 ) ) = 1 )
121	8, 120	ax-mp 5	. . . . . . . 8 |- ( ( ( sin ` ( pi / 6 ) ) ^ 2 ) + ( ( cos ` ( pi / 6 ) ) ^ 2 ) ) = 1
122	119, 121	eqtr3i 2254	. . . . . . 7 |- ( ( 1 / 4 ) + ( ( cos ` ( pi / 6 ) ) ^ 2 ) ) = 1
123	28, 111, 112, 122	subaddrii 8467	. . . . . 6 |- ( 1 - ( 1 / 4 ) ) = ( ( cos ` ( pi / 6 ) ) ^ 2 )
124	104, 110, 123	3eqtr3ri 2261	. . . . 5 |- ( ( cos ` ( pi / 6 ) ) ^ 2 ) = ( 3 / 4 )
125	124	fveq2i 5642	. . . 4 |- ( sqr ` ( ( cos ` ( pi / 6 ) ) ^ 2 ) ) = ( sqr ` ( 3 / 4 ) )
126	63, 15, 73	ltleii 8281	. . . . 5 |- 0 <_ ( cos ` ( pi / 6 ) )
127	15	sqrtsqi 11683	. . . . 5 |- ( 0 <_ ( cos ` ( pi / 6 ) ) -> ( sqr ` ( ( cos ` ( pi / 6 ) ) ^ 2 ) ) = ( cos ` ( pi / 6 ) ) )
128	126, 127	ax-mp 5	. . . 4 |- ( sqr ` ( ( cos ` ( pi / 6 ) ) ^ 2 ) ) = ( cos ` ( pi / 6 ) )
129	125, 128	eqtr3i 2254	. . 3 |- ( sqr ` ( 3 / 4 ) ) = ( cos ` ( pi / 6 ) )
130	93, 100, 129	3eqtr3ri 2261	. 2 |- ( cos ` ( pi / 6 ) ) = ( ( sqr ` 3 ) / 2 )
131	81, 130	pm3.2i 272	1 |- ( ( sin ` ( pi / 6 ) ) = ( 1 / 2 ) /\ ( cos ` ( pi / 6 ) ) = ( ( sqr ` 3 ) / 2 ) )

Syntax hints:   /\ wa 104   <-> wb 105   /\ w3a 1004   = wceq 1397   T. wtru 1398   e. wcel 2202   class class class wbr 4088   ` cfv 5326  (class class class)co 6017   CCcc 8029   RRcr 8030   0cc0 8031   1c1 8032   + caddc 8034   x. cmul 8036   RR*cxr 8212   < clt 8213   <_ cle 8214   - cmin 8349   # cap 8760   / cdiv 8851   2c2 9193   3c3 9194   4c4 9195   6c6 9197   ZZcz 9478   (,)cioo 10122   ^cexp 10799   sqrcsqrt 11556   sincsin 12204   cosccos 12205   picpi 12207

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-mulrcl 8130  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-precex 8141  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147  ax-pre-mulgt0 8148  ax-pre-mulext 8149  ax-arch 8150  ax-caucvg 8151  ax-pre-suploc 8152  ax-addf 8153  ax-mulf 8154

This theorem depends on definitions:  df-bi 117  df-stab 838  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-disj 4065  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-isom 5335  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-of 6234  df-1st 6302  df-2nd 6303  df-recs 6470  df-irdg 6535  df-frec 6556  df-1o 6581  df-oadd 6585  df-er 6701  df-map 6818  df-pm 6819  df-en 6909  df-dom 6910  df-fin 6911  df-sup 7182  df-inf 7183  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-reap 8754  df-ap 8761  df-div 8852  df-inn 9143  df-2 9201  df-3 9202  df-4 9203  df-5 9204  df-6 9205  df-7 9206  df-8 9207  df-9 9208  df-n0 9402  df-z 9479  df-uz 9755  df-q 9853  df-rp 9888  df-xneg 10006  df-xadd 10007  df-ioo 10126  df-ioc 10127  df-ico 10128  df-icc 10129  df-fz 10243  df-fzo 10377  df-seqfrec 10709  df-exp 10800  df-fac 10987  df-bc 11009  df-ihash 11037  df-shft 11375  df-cj 11402  df-re 11403  df-im 11404  df-rsqrt 11558  df-abs 11559  df-clim 11839  df-sumdc 11914  df-ef 12208  df-sin 12210  df-cos 12211  df-pi 12213  df-rest 13323  df-topgen 13342  df-psmet 14556  df-xmet 14557  df-met 14558  df-bl 14559  df-mopn 14560  df-top 14721  df-topon 14734  df-bases 14766  df-ntr 14819  df-cn 14911  df-cnp 14912  df-tx 14976  df-cncf 15294  df-limced 15379  df-dvap 15380

This theorem is referenced by:  sincos3rdpi  15566  pigt3  15567