Theorem sincos6thpi 15078

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 9061	. . . . 5 |- 2 e. CC
2	1	a1i 9	. . . 4 |- ( T. -> 2 e. CC )
3		pire 15022	. . . . . . . 8 |- pi e. RR
4		6re 9071	. . . . . . . 8 |- 6 e. RR
5		6pos 9091	. . . . . . . . 9 |- 0 < 6
6	4, 5	gt0ap0ii 8655	. . . . . . . 8 |- 6 # 0
7	3, 4, 6	redivclapi 8806	. . . . . . 7 |- ( pi / 6 ) e. RR
8	7	recni 8038	. . . . . 6 |- ( pi / 6 ) e. CC
9		sincl 11871	. . . . . 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 9083	. . . . 5 |- 2 # 0
13	12	a1i 9	. . . 4 |- ( T. -> 2 # 0 )
14		recoscl 11886	. . . . . . . . . . . 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 8038	. . . . . . . . . 10 |- ( cos ` ( pi / 6 ) ) e. CC
17	1, 10, 16	mulassi 8035	. . . . . . . . 9 |- ( ( 2 x. ( sin ` ( pi / 6 ) ) ) x. ( cos ` ( pi / 6 ) ) ) = ( 2 x. ( ( sin ` ( pi / 6 ) ) x. ( cos ` ( pi / 6 ) ) ) )
18		sin2t 11914	. . . . . . . . . 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 2220	. . . . . . . 8 |- ( ( 2 x. ( sin ` ( pi / 6 ) ) ) x. ( cos ` ( pi / 6 ) ) ) = ( sin ` ( 2 x. ( pi / 6 ) ) )
21		3cn 9065	. . . . . . . . . . . 12 |- 3 e. CC
22		3ap0 9086	. . . . . . . . . . . 12 |- 3 # 0
23	1, 21, 22	divclapi 8781	. . . . . . . . . . 11 |- ( 2 / 3 ) e. CC
24	21, 22	recclapi 8769	. . . . . . . . . . 11 |- ( 1 / 3 ) e. CC
25		df-3 9050	. . . . . . . . . . . . 13 |- 3 = ( 2 + 1 )
26	25	oveq1i 5932	. . . . . . . . . . . 12 |- ( 3 / 3 ) = ( ( 2 + 1 ) / 3 )
27	21, 22	dividapi 8772	. . . . . . . . . . . 12 |- ( 3 / 3 ) = 1
28		ax-1cn 7972	. . . . . . . . . . . . 13 |- 1 e. CC
29	1, 28, 21, 22	divdirapi 8796	. . . . . . . . . . . 12 |- ( ( 2 + 1 ) / 3 ) = ( ( 2 / 3 ) + ( 1 / 3 ) )
30	26, 27, 29	3eqtr3ri 2226	. . . . . . . . . . 11 |- ( ( 2 / 3 ) + ( 1 / 3 ) ) = 1
31		sincosq1eq 15075	. . . . . . . . . . 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 1348	. . . . . . . . . 10 |- ( sin ` ( ( 2 / 3 ) x. ( pi / 2 ) ) ) = ( cos ` ( ( 1 / 3 ) x. ( pi / 2 ) ) )
33		picn 15023	. . . . . . . . . . . . 13 |- pi e. CC
34	1, 21, 33, 1, 22, 12	divmuldivapi 8799	. . . . . . . . . . . 12 |- ( ( 2 / 3 ) x. ( pi / 2 ) ) = ( ( 2 x. pi ) / ( 3 x. 2 ) )
35		3t2e6 9147	. . . . . . . . . . . . 13 |- ( 3 x. 2 ) = 6
36	35	oveq2i 5933	. . . . . . . . . . . 12 |- ( ( 2 x. pi ) / ( 3 x. 2 ) ) = ( ( 2 x. pi ) / 6 )
37		6cn 9072	. . . . . . . . . . . . 13 |- 6 e. CC
38	1, 33, 37, 6	divassapi 8795	. . . . . . . . . . . 12 |- ( ( 2 x. pi ) / 6 ) = ( 2 x. ( pi / 6 ) )
39	34, 36, 38	3eqtri 2221	. . . . . . . . . . 11 |- ( ( 2 / 3 ) x. ( pi / 2 ) ) = ( 2 x. ( pi / 6 ) )
40	39	fveq2i 5561	. . . . . . . . . 10 |- ( sin ` ( ( 2 / 3 ) x. ( pi / 2 ) ) ) = ( sin ` ( 2 x. ( pi / 6 ) ) )
41	32, 40	eqtr3i 2219	. . . . . . . . 9 |- ( cos ` ( ( 1 / 3 ) x. ( pi / 2 ) ) ) = ( sin ` ( 2 x. ( pi / 6 ) ) )
42	28, 21, 33, 1, 22, 12	divmuldivapi 8799	. . . . . . . . . . 11 |- ( ( 1 / 3 ) x. ( pi / 2 ) ) = ( ( 1 x. pi ) / ( 3 x. 2 ) )
43	33	mullidi 8029	. . . . . . . . . . . 12 |- ( 1 x. pi ) = pi
44	43, 35	oveq12i 5934	. . . . . . . . . . 11 |- ( ( 1 x. pi ) / ( 3 x. 2 ) ) = ( pi / 6 )
45	42, 44	eqtri 2217	. . . . . . . . . 10 |- ( ( 1 / 3 ) x. ( pi / 2 ) ) = ( pi / 6 )
46	45	fveq2i 5561	. . . . . . . . 9 |- ( cos ` ( ( 1 / 3 ) x. ( pi / 2 ) ) ) = ( cos ` ( pi / 6 ) )
47	41, 46	eqtr3i 2219	. . . . . . . 8 |- ( sin ` ( 2 x. ( pi / 6 ) ) ) = ( cos ` ( pi / 6 ) )
48	20, 47	eqtri 2217	. . . . . . 7 |- ( ( 2 x. ( sin ` ( pi / 6 ) ) ) x. ( cos ` ( pi / 6 ) ) ) = ( cos ` ( pi / 6 ) )
49	16	mullidi 8029	. . . . . . 7 |- ( 1 x. ( cos ` ( pi / 6 ) ) ) = ( cos ` ( pi / 6 ) )
50	48, 49	eqtr4i 2220	. . . . . 6 |- ( ( 2 x. ( sin ` ( pi / 6 ) ) ) x. ( cos ` ( pi / 6 ) ) ) = ( 1 x. ( cos ` ( pi / 6 ) ) )
51	1, 10	mulcli 8031	. . . . . . 7 |- ( 2 x. ( sin ` ( pi / 6 ) ) ) e. CC
52		pipos 15024	. . . . . . . . . . . . 13 |- 0 < pi
53	3, 4, 52, 5	divgt0ii 8946	. . . . . . . . . . . 12 |- 0 < ( pi / 6 )
54		2lt6 9173	. . . . . . . . . . . . 13 |- 2 < 6
55		2re 9060	. . . . . . . . . . . . . . 15 |- 2 e. RR
56		2pos 9081	. . . . . . . . . . . . . . 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 8914	. . . . . . . . . . . . . 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 1348	. . . . . . . . . . . . 13 |- ( 2 < 6 <-> ( pi / 6 ) < ( pi / 2 ) )
62	54, 61	mpbi 145	. . . . . . . . . . . 12 |- ( pi / 6 ) < ( pi / 2 )
63		0re 8026	. . . . . . . . . . . . 13 |- 0 e. RR
64		halfpire 15028	. . . . . . . . . . . . 13 |- ( pi / 2 ) e. RR
65		rexr 8072	. . . . . . . . . . . . . 14 |- ( 0 e. RR -> 0 e. RR* )
66		rexr 8072	. . . . . . . . . . . . . 14 |- ( ( pi / 2 ) e. RR -> ( pi / 2 ) e. RR* )
67		elioo2 9996	. . . . . . . . . . . . . 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 1181	. . . . . . . . . . 11 |- ( pi / 6 ) e. ( 0 (,) ( pi / 2 ) )
71		sincosq1sgn 15062	. . . . . . . . . . 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 8655	. . . . . . . 8 |- ( cos ` ( pi / 6 ) ) # 0
75	16, 74	pm3.2i 272	. . . . . . 7 |- ( ( cos ` ( pi / 6 ) ) e. CC /\ ( cos ` ( pi / 6 ) ) # 0 )
76		mulcanap2 8693	. . . . . . 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 1348	. . . . . 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 8869	. . 3 |- ( T. -> ( sin ` ( pi / 6 ) ) = ( 1 / 2 ) )
81	80	mptru 1373	. 2 |- ( sin ` ( pi / 6 ) ) = ( 1 / 2 )
82		3re 9064	. . . . . . . 8 |- 3 e. RR
83		3pos 9084	. . . . . . . 8 |- 0 < 3
84	82, 83	sqrtpclii 11295	. . . . . . 7 |- ( sqr ` 3 ) e. RR
85	84	recni 8038	. . . . . 6 |- ( sqr ` 3 ) e. CC
86	85, 1, 12	sqdivapi 10715	. . . . 5 |- ( ( ( sqr ` 3 ) / 2 ) ^ 2 ) = ( ( ( sqr ` 3 ) ^ 2 ) / ( 2 ^ 2 ) )
87	63, 82, 83	ltleii 8129	. . . . . . 7 |- 0 <_ 3
88	82	sqsqrti 11289	. . . . . . 7 |- ( 0 <_ 3 -> ( ( sqr ` 3 ) ^ 2 ) = 3 )
89	87, 88	ax-mp 5	. . . . . 6 |- ( ( sqr ` 3 ) ^ 2 ) = 3
90		sq2 10727	. . . . . 6 |- ( 2 ^ 2 ) = 4
91	89, 90	oveq12i 5934	. . . . 5 |- ( ( ( sqr ` 3 ) ^ 2 ) / ( 2 ^ 2 ) ) = ( 3 / 4 )
92	86, 91	eqtri 2217	. . . 4 |- ( ( ( sqr ` 3 ) / 2 ) ^ 2 ) = ( 3 / 4 )
93	92	fveq2i 5561	. . 3 |- ( sqr ` ( ( ( sqr ` 3 ) / 2 ) ^ 2 ) ) = ( sqr ` ( 3 / 4 ) )
94	82	sqrtge0i 11290	. . . . . 6 |- ( 0 <_ 3 -> 0 <_ ( sqr ` 3 ) )
95	87, 94	ax-mp 5	. . . . 5 |- 0 <_ ( sqr ` 3 )
96	84, 55	divge0i 8938	. . . . 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 8806	. . . . 5 |- ( ( sqr ` 3 ) / 2 ) e. RR
99	98	sqrtsqi 11288	. . . 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 9068	. . . . . . . 8 |- 4 e. CC
102		4ap0 9089	. . . . . . . 8 |- 4 # 0
103	101, 102	dividapi 8772	. . . . . . 7 |- ( 4 / 4 ) = 1
104	103	oveq1i 5932	. . . . . 6 |- ( ( 4 / 4 ) - ( 1 / 4 ) ) = ( 1 - ( 1 / 4 ) )
105	101, 102	pm3.2i 272	. . . . . . . 8 |- ( 4 e. CC /\ 4 # 0 )
106		divsubdirap 8735	. . . . . . . 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 1348	. . . . . . 7 |- ( ( 4 - 1 ) / 4 ) = ( ( 4 / 4 ) - ( 1 / 4 ) )
108		4m1e3 9111	. . . . . . . 8 |- ( 4 - 1 ) = 3
109	108	oveq1i 5932	. . . . . . 7 |- ( ( 4 - 1 ) / 4 ) = ( 3 / 4 )
110	107, 109	eqtr3i 2219	. . . . . 6 |- ( ( 4 / 4 ) - ( 1 / 4 ) ) = ( 3 / 4 )
111	101, 102	recclapi 8769	. . . . . . 7 |- ( 1 / 4 ) e. CC
112	16	sqcli 10712	. . . . . . 7 |- ( ( cos ` ( pi / 6 ) ) ^ 2 ) e. CC
113	81	oveq1i 5932	. . . . . . . . . 10 |- ( ( sin ` ( pi / 6 ) ) ^ 2 ) = ( ( 1 / 2 ) ^ 2 )
114		2z 9354	. . . . . . . . . . 11 |- 2 e. ZZ
115		exprecap 10672	. . . . . . . . . . 11 |- ( ( 2 e. CC /\ 2 # 0 /\ 2 e. ZZ ) -> ( ( 1 / 2 ) ^ 2 ) = ( 1 / ( 2 ^ 2 ) ) )
116	1, 12, 114, 115	mp3an 1348	. . . . . . . . . 10 |- ( ( 1 / 2 ) ^ 2 ) = ( 1 / ( 2 ^ 2 ) )
117	90	oveq2i 5933	. . . . . . . . . 10 |- ( 1 / ( 2 ^ 2 ) ) = ( 1 / 4 )
118	113, 116, 117	3eqtri 2221	. . . . . . . . 9 |- ( ( sin ` ( pi / 6 ) ) ^ 2 ) = ( 1 / 4 )
119	118	oveq1i 5932	. . . . . . . 8 |- ( ( ( sin ` ( pi / 6 ) ) ^ 2 ) + ( ( cos ` ( pi / 6 ) ) ^ 2 ) ) = ( ( 1 / 4 ) + ( ( cos ` ( pi / 6 ) ) ^ 2 ) )
120		sincossq 11913	. . . . . . . . 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 2219	. . . . . . 7 |- ( ( 1 / 4 ) + ( ( cos ` ( pi / 6 ) ) ^ 2 ) ) = 1
123	28, 111, 112, 122	subaddrii 8315	. . . . . 6 |- ( 1 - ( 1 / 4 ) ) = ( ( cos ` ( pi / 6 ) ) ^ 2 )
124	104, 110, 123	3eqtr3ri 2226	. . . . 5 |- ( ( cos ` ( pi / 6 ) ) ^ 2 ) = ( 3 / 4 )
125	124	fveq2i 5561	. . . 4 |- ( sqr ` ( ( cos ` ( pi / 6 ) ) ^ 2 ) ) = ( sqr ` ( 3 / 4 ) )
126	63, 15, 73	ltleii 8129	. . . . 5 |- 0 <_ ( cos ` ( pi / 6 ) )
127	15	sqrtsqi 11288	. . . . 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 2219	. . 3 |- ( sqr ` ( 3 / 4 ) ) = ( cos ` ( pi / 6 ) )
130	93, 100, 129	3eqtr3ri 2226	. 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 980   = wceq 1364   T. wtru 1365   e. wcel 2167   class class class wbr 4033   ` cfv 5258  (class class class)co 5922   CCcc 7877   RRcr 7878   0cc0 7879   1c1 7880   + caddc 7882   x. cmul 7884   RR*cxr 8060   < clt 8061   <_ cle 8062   - cmin 8197   # cap 8608   / cdiv 8699   2c2 9041   3c3 9042   4c4 9043   6c6 9045   ZZcz 9326   (,)cioo 9963   ^cexp 10630   sqrcsqrt 11161   sincsin 11809   cosccos 11810   picpi 11812

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-mulrcl 7978  ax-addcom 7979  ax-mulcom 7980  ax-addass 7981  ax-mulass 7982  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-1rid 7986  ax-0id 7987  ax-rnegex 7988  ax-precex 7989  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-apti 7994  ax-pre-ltadd 7995  ax-pre-mulgt0 7996  ax-pre-mulext 7997  ax-arch 7998  ax-caucvg 7999  ax-pre-suploc 8000  ax-addf 8001  ax-mulf 8002

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-disj 4011  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-po 4331  df-iso 4332  df-iord 4401  df-on 4403  df-ilim 4404  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-isom 5267  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-of 6135  df-1st 6198  df-2nd 6199  df-recs 6363  df-irdg 6428  df-frec 6449  df-1o 6474  df-oadd 6478  df-er 6592  df-map 6709  df-pm 6710  df-en 6800  df-dom 6801  df-fin 6802  df-sup 7050  df-inf 7051  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-reap 8602  df-ap 8609  df-div 8700  df-inn 8991  df-2 9049  df-3 9050  df-4 9051  df-5 9052  df-6 9053  df-7 9054  df-8 9055  df-9 9056  df-n0 9250  df-z 9327  df-uz 9602  df-q 9694  df-rp 9729  df-xneg 9847  df-xadd 9848  df-ioo 9967  df-ioc 9968  df-ico 9969  df-icc 9970  df-fz 10084  df-fzo 10218  df-seqfrec 10540  df-exp 10631  df-fac 10818  df-bc 10840  df-ihash 10868  df-shft 10980  df-cj 11007  df-re 11008  df-im 11009  df-rsqrt 11163  df-abs 11164  df-clim 11444  df-sumdc 11519  df-ef 11813  df-sin 11815  df-cos 11816  df-pi 11818  df-rest 12912  df-topgen 12931  df-psmet 14099  df-xmet 14100  df-met 14101  df-bl 14102  df-mopn 14103  df-top 14234  df-topon 14247  df-bases 14279  df-ntr 14332  df-cn 14424  df-cnp 14425  df-tx 14489  df-cncf 14807  df-limced 14892  df-dvap 14893

This theorem is referenced by:  sincos3rdpi  15079  pigt3  15080