Theorem sincos6thpi 12971

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 8815	. . . . 5 |- 2 e. CC
2	1	a1i 9	. . . 4 |- ( T. -> 2 e. CC )
3		pire 12915	. . . . . . . 8 |- pi e. RR
4		6re 8825	. . . . . . . 8 |- 6 e. RR
5		6pos 8845	. . . . . . . . 9 |- 0 < 6
6	4, 5	gt0ap0ii 8414	. . . . . . . 8 |- 6 # 0
7	3, 4, 6	redivclapi 8563	. . . . . . 7 |- ( pi / 6 ) e. RR
8	7	recni 7802	. . . . . 6 |- ( pi / 6 ) e. CC
9		sincl 11449	. . . . . 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 8837	. . . . 5 |- 2 # 0
13	12	a1i 9	. . . 4 |- ( T. -> 2 # 0 )
14		recoscl 11464	. . . . . . . . . . . 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 7802	. . . . . . . . . 10 |- ( cos ` ( pi / 6 ) ) e. CC
17	1, 10, 16	mulassi 7799	. . . . . . . . 9 |- ( ( 2 x. ( sin ` ( pi / 6 ) ) ) x. ( cos ` ( pi / 6 ) ) ) = ( 2 x. ( ( sin ` ( pi / 6 ) ) x. ( cos ` ( pi / 6 ) ) ) )
18		sin2t 11492	. . . . . . . . . 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 2164	. . . . . . . 8 |- ( ( 2 x. ( sin ` ( pi / 6 ) ) ) x. ( cos ` ( pi / 6 ) ) ) = ( sin ` ( 2 x. ( pi / 6 ) ) )
21		3cn 8819	. . . . . . . . . . . 12 |- 3 e. CC
22		3ap0 8840	. . . . . . . . . . . 12 |- 3 # 0
23	1, 21, 22	divclapi 8538	. . . . . . . . . . 11 |- ( 2 / 3 ) e. CC
24	21, 22	recclapi 8526	. . . . . . . . . . 11 |- ( 1 / 3 ) e. CC
25		df-3 8804	. . . . . . . . . . . . 13 |- 3 = ( 2 + 1 )
26	25	oveq1i 5792	. . . . . . . . . . . 12 |- ( 3 / 3 ) = ( ( 2 + 1 ) / 3 )
27	21, 22	dividapi 8529	. . . . . . . . . . . 12 |- ( 3 / 3 ) = 1
28		ax-1cn 7737	. . . . . . . . . . . . 13 |- 1 e. CC
29	1, 28, 21, 22	divdirapi 8553	. . . . . . . . . . . 12 |- ( ( 2 + 1 ) / 3 ) = ( ( 2 / 3 ) + ( 1 / 3 ) )
30	26, 27, 29	3eqtr3ri 2170	. . . . . . . . . . 11 |- ( ( 2 / 3 ) + ( 1 / 3 ) ) = 1
31		sincosq1eq 12968	. . . . . . . . . . 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 1316	. . . . . . . . . 10 |- ( sin ` ( ( 2 / 3 ) x. ( pi / 2 ) ) ) = ( cos ` ( ( 1 / 3 ) x. ( pi / 2 ) ) )
33		picn 12916	. . . . . . . . . . . . 13 |- pi e. CC
34	1, 21, 33, 1, 22, 12	divmuldivapi 8556	. . . . . . . . . . . 12 |- ( ( 2 / 3 ) x. ( pi / 2 ) ) = ( ( 2 x. pi ) / ( 3 x. 2 ) )
35		3t2e6 8900	. . . . . . . . . . . . 13 |- ( 3 x. 2 ) = 6
36	35	oveq2i 5793	. . . . . . . . . . . 12 |- ( ( 2 x. pi ) / ( 3 x. 2 ) ) = ( ( 2 x. pi ) / 6 )
37		6cn 8826	. . . . . . . . . . . . 13 |- 6 e. CC
38	1, 33, 37, 6	divassapi 8552	. . . . . . . . . . . 12 |- ( ( 2 x. pi ) / 6 ) = ( 2 x. ( pi / 6 ) )
39	34, 36, 38	3eqtri 2165	. . . . . . . . . . 11 |- ( ( 2 / 3 ) x. ( pi / 2 ) ) = ( 2 x. ( pi / 6 ) )
40	39	fveq2i 5432	. . . . . . . . . 10 |- ( sin ` ( ( 2 / 3 ) x. ( pi / 2 ) ) ) = ( sin ` ( 2 x. ( pi / 6 ) ) )
41	32, 40	eqtr3i 2163	. . . . . . . . 9 |- ( cos ` ( ( 1 / 3 ) x. ( pi / 2 ) ) ) = ( sin ` ( 2 x. ( pi / 6 ) ) )
42	28, 21, 33, 1, 22, 12	divmuldivapi 8556	. . . . . . . . . . 11 |- ( ( 1 / 3 ) x. ( pi / 2 ) ) = ( ( 1 x. pi ) / ( 3 x. 2 ) )
43	33	mulid2i 7793	. . . . . . . . . . . 12 |- ( 1 x. pi ) = pi
44	43, 35	oveq12i 5794	. . . . . . . . . . 11 |- ( ( 1 x. pi ) / ( 3 x. 2 ) ) = ( pi / 6 )
45	42, 44	eqtri 2161	. . . . . . . . . 10 |- ( ( 1 / 3 ) x. ( pi / 2 ) ) = ( pi / 6 )
46	45	fveq2i 5432	. . . . . . . . 9 |- ( cos ` ( ( 1 / 3 ) x. ( pi / 2 ) ) ) = ( cos ` ( pi / 6 ) )
47	41, 46	eqtr3i 2163	. . . . . . . 8 |- ( sin ` ( 2 x. ( pi / 6 ) ) ) = ( cos ` ( pi / 6 ) )
48	20, 47	eqtri 2161	. . . . . . 7 |- ( ( 2 x. ( sin ` ( pi / 6 ) ) ) x. ( cos ` ( pi / 6 ) ) ) = ( cos ` ( pi / 6 ) )
49	16	mulid2i 7793	. . . . . . 7 |- ( 1 x. ( cos ` ( pi / 6 ) ) ) = ( cos ` ( pi / 6 ) )
50	48, 49	eqtr4i 2164	. . . . . 6 |- ( ( 2 x. ( sin ` ( pi / 6 ) ) ) x. ( cos ` ( pi / 6 ) ) ) = ( 1 x. ( cos ` ( pi / 6 ) ) )
51	1, 10	mulcli 7795	. . . . . . 7 |- ( 2 x. ( sin ` ( pi / 6 ) ) ) e. CC
52		pipos 12917	. . . . . . . . . . . . 13 |- 0 < pi
53	3, 4, 52, 5	divgt0ii 8701	. . . . . . . . . . . 12 |- 0 < ( pi / 6 )
54		2lt6 8926	. . . . . . . . . . . . 13 |- 2 < 6
55		2re 8814	. . . . . . . . . . . . . . 15 |- 2 e. RR
56		2pos 8835	. . . . . . . . . . . . . . 15 |- 0 < 2
57	55, 56	pm3.2i 270	. . . . . . . . . . . . . 14 |- ( 2 e. RR /\ 0 < 2 )
58	4, 5	pm3.2i 270	. . . . . . . . . . . . . 14 |- ( 6 e. RR /\ 0 < 6 )
59	3, 52	pm3.2i 270	. . . . . . . . . . . . . 14 |- ( pi e. RR /\ 0 < pi )
60		ltdiv2 8669	. . . . . . . . . . . . . 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 1316	. . . . . . . . . . . . 13 |- ( 2 < 6 <-> ( pi / 6 ) < ( pi / 2 ) )
62	54, 61	mpbi 144	. . . . . . . . . . . 12 |- ( pi / 6 ) < ( pi / 2 )
63		0re 7790	. . . . . . . . . . . . 13 |- 0 e. RR
64		halfpire 12921	. . . . . . . . . . . . 13 |- ( pi / 2 ) e. RR
65		rexr 7835	. . . . . . . . . . . . . 14 |- ( 0 e. RR -> 0 e. RR* )
66		rexr 7835	. . . . . . . . . . . . . 14 |- ( ( pi / 2 ) e. RR -> ( pi / 2 ) e. RR* )
67		elioo2 9734	. . . . . . . . . . . . . 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 287	. . . . . . . . . . . . 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 423	. . . . . . . . . . . 12 |- ( ( pi / 6 ) e. ( 0 (,) ( pi / 2 ) ) <-> ( ( pi / 6 ) e. RR /\ 0 < ( pi / 6 ) /\ ( pi / 6 ) < ( pi / 2 ) ) )
70	7, 53, 62, 69	mpbir3an 1164	. . . . . . . . . . 11 |- ( pi / 6 ) e. ( 0 (,) ( pi / 2 ) )
71		sincosq1sgn 12955	. . . . . . . . . . 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 112	. . . . . . . . 9 |- 0 < ( cos ` ( pi / 6 ) )
74	15, 73	gt0ap0ii 8414	. . . . . . . 8 |- ( cos ` ( pi / 6 ) ) # 0
75	16, 74	pm3.2i 270	. . . . . . 7 |- ( ( cos ` ( pi / 6 ) ) e. CC /\ ( cos ` ( pi / 6 ) ) # 0 )
76		mulcanap2 8451	. . . . . . 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 1316	. . . . . 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 144	. . . . 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 8625	. . 3 |- ( T. -> ( sin ` ( pi / 6 ) ) = ( 1 / 2 ) )
81	80	mptru 1341	. 2 |- ( sin ` ( pi / 6 ) ) = ( 1 / 2 )
82		3re 8818	. . . . . . . 8 |- 3 e. RR
83		3pos 8838	. . . . . . . 8 |- 0 < 3
84	82, 83	sqrtpclii 10934	. . . . . . 7 |- ( sqr ` 3 ) e. RR
85	84	recni 7802	. . . . . 6 |- ( sqr ` 3 ) e. CC
86	85, 1, 12	sqdivapi 10407	. . . . 5 |- ( ( ( sqr ` 3 ) / 2 ) ^ 2 ) = ( ( ( sqr ` 3 ) ^ 2 ) / ( 2 ^ 2 ) )
87	63, 82, 83	ltleii 7890	. . . . . . 7 |- 0 <_ 3
88	82	sqsqrti 10928	. . . . . . 7 |- ( 0 <_ 3 -> ( ( sqr ` 3 ) ^ 2 ) = 3 )
89	87, 88	ax-mp 5	. . . . . 6 |- ( ( sqr ` 3 ) ^ 2 ) = 3
90		sq2 10419	. . . . . 6 |- ( 2 ^ 2 ) = 4
91	89, 90	oveq12i 5794	. . . . 5 |- ( ( ( sqr ` 3 ) ^ 2 ) / ( 2 ^ 2 ) ) = ( 3 / 4 )
92	86, 91	eqtri 2161	. . . 4 |- ( ( ( sqr ` 3 ) / 2 ) ^ 2 ) = ( 3 / 4 )
93	92	fveq2i 5432	. . 3 |- ( sqr ` ( ( ( sqr ` 3 ) / 2 ) ^ 2 ) ) = ( sqr ` ( 3 / 4 ) )
94	82	sqrtge0i 10929	. . . . . 6 |- ( 0 <_ 3 -> 0 <_ ( sqr ` 3 ) )
95	87, 94	ax-mp 5	. . . . 5 |- 0 <_ ( sqr ` 3 )
96	84, 55	divge0i 8693	. . . . 5 |- ( ( 0 <_ ( sqr ` 3 ) /\ 0 < 2 ) -> 0 <_ ( ( sqr ` 3 ) / 2 ) )
97	95, 56, 96	mp2an 423	. . . 4 |- 0 <_ ( ( sqr ` 3 ) / 2 )
98	84, 55, 12	redivclapi 8563	. . . . 5 |- ( ( sqr ` 3 ) / 2 ) e. RR
99	98	sqrtsqi 10927	. . . 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 8822	. . . . . . . 8 |- 4 e. CC
102		4ap0 8843	. . . . . . . 8 |- 4 # 0
103	101, 102	dividapi 8529	. . . . . . 7 |- ( 4 / 4 ) = 1
104	103	oveq1i 5792	. . . . . 6 |- ( ( 4 / 4 ) - ( 1 / 4 ) ) = ( 1 - ( 1 / 4 ) )
105	101, 102	pm3.2i 270	. . . . . . . 8 |- ( 4 e. CC /\ 4 # 0 )
106		divsubdirap 8492	. . . . . . . 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 1316	. . . . . . 7 |- ( ( 4 - 1 ) / 4 ) = ( ( 4 / 4 ) - ( 1 / 4 ) )
108		4m1e3 8865	. . . . . . . 8 |- ( 4 - 1 ) = 3
109	108	oveq1i 5792	. . . . . . 7 |- ( ( 4 - 1 ) / 4 ) = ( 3 / 4 )
110	107, 109	eqtr3i 2163	. . . . . 6 |- ( ( 4 / 4 ) - ( 1 / 4 ) ) = ( 3 / 4 )
111	101, 102	recclapi 8526	. . . . . . 7 |- ( 1 / 4 ) e. CC
112	16	sqcli 10404	. . . . . . 7 |- ( ( cos ` ( pi / 6 ) ) ^ 2 ) e. CC
113	81	oveq1i 5792	. . . . . . . . . 10 |- ( ( sin ` ( pi / 6 ) ) ^ 2 ) = ( ( 1 / 2 ) ^ 2 )
114		2z 9106	. . . . . . . . . . 11 |- 2 e. ZZ
115		exprecap 10365	. . . . . . . . . . 11 |- ( ( 2 e. CC /\ 2 # 0 /\ 2 e. ZZ ) -> ( ( 1 / 2 ) ^ 2 ) = ( 1 / ( 2 ^ 2 ) ) )
116	1, 12, 114, 115	mp3an 1316	. . . . . . . . . 10 |- ( ( 1 / 2 ) ^ 2 ) = ( 1 / ( 2 ^ 2 ) )
117	90	oveq2i 5793	. . . . . . . . . 10 |- ( 1 / ( 2 ^ 2 ) ) = ( 1 / 4 )
118	113, 116, 117	3eqtri 2165	. . . . . . . . 9 |- ( ( sin ` ( pi / 6 ) ) ^ 2 ) = ( 1 / 4 )
119	118	oveq1i 5792	. . . . . . . 8 |- ( ( ( sin ` ( pi / 6 ) ) ^ 2 ) + ( ( cos ` ( pi / 6 ) ) ^ 2 ) ) = ( ( 1 / 4 ) + ( ( cos ` ( pi / 6 ) ) ^ 2 ) )
120		sincossq 11491	. . . . . . . . 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 2163	. . . . . . 7 |- ( ( 1 / 4 ) + ( ( cos ` ( pi / 6 ) ) ^ 2 ) ) = 1
123	28, 111, 112, 122	subaddrii 8075	. . . . . 6 |- ( 1 - ( 1 / 4 ) ) = ( ( cos ` ( pi / 6 ) ) ^ 2 )
124	104, 110, 123	3eqtr3ri 2170	. . . . 5 |- ( ( cos ` ( pi / 6 ) ) ^ 2 ) = ( 3 / 4 )
125	124	fveq2i 5432	. . . 4 |- ( sqr ` ( ( cos ` ( pi / 6 ) ) ^ 2 ) ) = ( sqr ` ( 3 / 4 ) )
126	63, 15, 73	ltleii 7890	. . . . 5 |- 0 <_ ( cos ` ( pi / 6 ) )
127	15	sqrtsqi 10927	. . . . 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 2163	. . 3 |- ( sqr ` ( 3 / 4 ) ) = ( cos ` ( pi / 6 ) )
130	93, 100, 129	3eqtr3ri 2170	. 2 |- ( cos ` ( pi / 6 ) ) = ( ( sqr ` 3 ) / 2 )
131	81, 130	pm3.2i 270	1 |- ( ( sin ` ( pi / 6 ) ) = ( 1 / 2 ) /\ ( cos ` ( pi / 6 ) ) = ( ( sqr ` 3 ) / 2 ) )

Syntax hints:   /\ wa 103   <-> wb 104   /\ w3a 963   = wceq 1332   T. wtru 1333   e. wcel 1481   class class class wbr 3937   ` cfv 5131  (class class class)co 5782   CCcc 7642   RRcr 7643   0cc0 7644   1c1 7645   + caddc 7647   x. cmul 7649   RR*cxr 7823   < clt 7824   <_ cle 7825   - cmin 7957   # cap 8367   / cdiv 8456   2c2 8795   3c3 8796   4c4 8797   6c6 8799   ZZcz 9078   (,)cioo 9701   ^cexp 10323   sqrcsqrt 10800   sincsin 11387   cosccos 11388   picpi 11390

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-iinf 4510  ax-cnex 7735  ax-resscn 7736  ax-1cn 7737  ax-1re 7738  ax-icn 7739  ax-addcl 7740  ax-addrcl 7741  ax-mulcl 7742  ax-mulrcl 7743  ax-addcom 7744  ax-mulcom 7745  ax-addass 7746  ax-mulass 7747  ax-distr 7748  ax-i2m1 7749  ax-0lt1 7750  ax-1rid 7751  ax-0id 7752  ax-rnegex 7753  ax-precex 7754  ax-cnre 7755  ax-pre-ltirr 7756  ax-pre-ltwlin 7757  ax-pre-lttrn 7758  ax-pre-apti 7759  ax-pre-ltadd 7760  ax-pre-mulgt0 7761  ax-pre-mulext 7762  ax-arch 7763  ax-caucvg 7764  ax-pre-suploc 7765  ax-addf 7766  ax-mulf 7767

This theorem depends on definitions:  df-bi 116  df-stab 817  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rmo 2425  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-if 3480  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-disj 3915  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-id 4223  df-po 4226  df-iso 4227  df-iord 4296  df-on 4298  df-ilim 4299  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-isom 5140  df-riota 5738  df-ov 5785  df-oprab 5786  df-mpo 5787  df-of 5990  df-1st 6046  df-2nd 6047  df-recs 6210  df-irdg 6275  df-frec 6296  df-1o 6321  df-oadd 6325  df-er 6437  df-map 6552  df-pm 6553  df-en 6643  df-dom 6644  df-fin 6645  df-sup 6879  df-inf 6880  df-pnf 7826  df-mnf 7827  df-xr 7828  df-ltxr 7829  df-le 7830  df-sub 7959  df-neg 7960  df-reap 8361  df-ap 8368  df-div 8457  df-inn 8745  df-2 8803  df-3 8804  df-4 8805  df-5 8806  df-6 8807  df-7 8808  df-8 8809  df-9 8810  df-n0 9002  df-z 9079  df-uz 9351  df-q 9439  df-rp 9471  df-xneg 9589  df-xadd 9590  df-ioo 9705  df-ioc 9706  df-ico 9707  df-icc 9708  df-fz 9822  df-fzo 9951  df-seqfrec 10250  df-exp 10324  df-fac 10504  df-bc 10526  df-ihash 10554  df-shft 10619  df-cj 10646  df-re 10647  df-im 10648  df-rsqrt 10802  df-abs 10803  df-clim 11080  df-sumdc 11155  df-ef 11391  df-sin 11393  df-cos 11394  df-pi 11396  df-rest 12161  df-topgen 12180  df-psmet 12195  df-xmet 12196  df-met 12197  df-bl 12198  df-mopn 12199  df-top 12204  df-topon 12217  df-bases 12249  df-ntr 12304  df-cn 12396  df-cnp 12397  df-tx 12461  df-cncf 12766  df-limced 12833  df-dvap 12834

This theorem is referenced by:  sincos3rdpi  12972  pigt3  12973