Theorem sincos6thpi 15516

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 9181	. . . . 5 |- 2 e. CC
2	1	a1i 9	. . . 4 |- ( T. -> 2 e. CC )
3		pire 15460	. . . . . . . 8 |- pi e. RR
4		6re 9191	. . . . . . . 8 |- 6 e. RR
5		6pos 9211	. . . . . . . . 9 |- 0 < 6
6	4, 5	gt0ap0ii 8775	. . . . . . . 8 |- 6 # 0
7	3, 4, 6	redivclapi 8926	. . . . . . 7 |- ( pi / 6 ) e. RR
8	7	recni 8158	. . . . . 6 |- ( pi / 6 ) e. CC
9		sincl 12217	. . . . . 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 9203	. . . . 5 |- 2 # 0
13	12	a1i 9	. . . 4 |- ( T. -> 2 # 0 )
14		recoscl 12232	. . . . . . . . . . . 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 8158	. . . . . . . . . 10 |- ( cos ` ( pi / 6 ) ) e. CC
17	1, 10, 16	mulassi 8155	. . . . . . . . 9 |- ( ( 2 x. ( sin ` ( pi / 6 ) ) ) x. ( cos ` ( pi / 6 ) ) ) = ( 2 x. ( ( sin ` ( pi / 6 ) ) x. ( cos ` ( pi / 6 ) ) ) )
18		sin2t 12260	. . . . . . . . . 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 2253	. . . . . . . 8 |- ( ( 2 x. ( sin ` ( pi / 6 ) ) ) x. ( cos ` ( pi / 6 ) ) ) = ( sin ` ( 2 x. ( pi / 6 ) ) )
21		3cn 9185	. . . . . . . . . . . 12 |- 3 e. CC
22		3ap0 9206	. . . . . . . . . . . 12 |- 3 # 0
23	1, 21, 22	divclapi 8901	. . . . . . . . . . 11 |- ( 2 / 3 ) e. CC
24	21, 22	recclapi 8889	. . . . . . . . . . 11 |- ( 1 / 3 ) e. CC
25		df-3 9170	. . . . . . . . . . . . 13 |- 3 = ( 2 + 1 )
26	25	oveq1i 6011	. . . . . . . . . . . 12 |- ( 3 / 3 ) = ( ( 2 + 1 ) / 3 )
27	21, 22	dividapi 8892	. . . . . . . . . . . 12 |- ( 3 / 3 ) = 1
28		ax-1cn 8092	. . . . . . . . . . . . 13 |- 1 e. CC
29	1, 28, 21, 22	divdirapi 8916	. . . . . . . . . . . 12 |- ( ( 2 + 1 ) / 3 ) = ( ( 2 / 3 ) + ( 1 / 3 ) )
30	26, 27, 29	3eqtr3ri 2259	. . . . . . . . . . 11 |- ( ( 2 / 3 ) + ( 1 / 3 ) ) = 1
31		sincosq1eq 15513	. . . . . . . . . . 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 1371	. . . . . . . . . 10 |- ( sin ` ( ( 2 / 3 ) x. ( pi / 2 ) ) ) = ( cos ` ( ( 1 / 3 ) x. ( pi / 2 ) ) )
33		picn 15461	. . . . . . . . . . . . 13 |- pi e. CC
34	1, 21, 33, 1, 22, 12	divmuldivapi 8919	. . . . . . . . . . . 12 |- ( ( 2 / 3 ) x. ( pi / 2 ) ) = ( ( 2 x. pi ) / ( 3 x. 2 ) )
35		3t2e6 9267	. . . . . . . . . . . . 13 |- ( 3 x. 2 ) = 6
36	35	oveq2i 6012	. . . . . . . . . . . 12 |- ( ( 2 x. pi ) / ( 3 x. 2 ) ) = ( ( 2 x. pi ) / 6 )
37		6cn 9192	. . . . . . . . . . . . 13 |- 6 e. CC
38	1, 33, 37, 6	divassapi 8915	. . . . . . . . . . . 12 |- ( ( 2 x. pi ) / 6 ) = ( 2 x. ( pi / 6 ) )
39	34, 36, 38	3eqtri 2254	. . . . . . . . . . 11 |- ( ( 2 / 3 ) x. ( pi / 2 ) ) = ( 2 x. ( pi / 6 ) )
40	39	fveq2i 5630	. . . . . . . . . 10 |- ( sin ` ( ( 2 / 3 ) x. ( pi / 2 ) ) ) = ( sin ` ( 2 x. ( pi / 6 ) ) )
41	32, 40	eqtr3i 2252	. . . . . . . . 9 |- ( cos ` ( ( 1 / 3 ) x. ( pi / 2 ) ) ) = ( sin ` ( 2 x. ( pi / 6 ) ) )
42	28, 21, 33, 1, 22, 12	divmuldivapi 8919	. . . . . . . . . . 11 |- ( ( 1 / 3 ) x. ( pi / 2 ) ) = ( ( 1 x. pi ) / ( 3 x. 2 ) )
43	33	mullidi 8149	. . . . . . . . . . . 12 |- ( 1 x. pi ) = pi
44	43, 35	oveq12i 6013	. . . . . . . . . . 11 |- ( ( 1 x. pi ) / ( 3 x. 2 ) ) = ( pi / 6 )
45	42, 44	eqtri 2250	. . . . . . . . . 10 |- ( ( 1 / 3 ) x. ( pi / 2 ) ) = ( pi / 6 )
46	45	fveq2i 5630	. . . . . . . . 9 |- ( cos ` ( ( 1 / 3 ) x. ( pi / 2 ) ) ) = ( cos ` ( pi / 6 ) )
47	41, 46	eqtr3i 2252	. . . . . . . 8 |- ( sin ` ( 2 x. ( pi / 6 ) ) ) = ( cos ` ( pi / 6 ) )
48	20, 47	eqtri 2250	. . . . . . 7 |- ( ( 2 x. ( sin ` ( pi / 6 ) ) ) x. ( cos ` ( pi / 6 ) ) ) = ( cos ` ( pi / 6 ) )
49	16	mullidi 8149	. . . . . . 7 |- ( 1 x. ( cos ` ( pi / 6 ) ) ) = ( cos ` ( pi / 6 ) )
50	48, 49	eqtr4i 2253	. . . . . 6 |- ( ( 2 x. ( sin ` ( pi / 6 ) ) ) x. ( cos ` ( pi / 6 ) ) ) = ( 1 x. ( cos ` ( pi / 6 ) ) )
51	1, 10	mulcli 8151	. . . . . . 7 |- ( 2 x. ( sin ` ( pi / 6 ) ) ) e. CC
52		pipos 15462	. . . . . . . . . . . . 13 |- 0 < pi
53	3, 4, 52, 5	divgt0ii 9066	. . . . . . . . . . . 12 |- 0 < ( pi / 6 )
54		2lt6 9293	. . . . . . . . . . . . 13 |- 2 < 6
55		2re 9180	. . . . . . . . . . . . . . 15 |- 2 e. RR
56		2pos 9201	. . . . . . . . . . . . . . 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 9034	. . . . . . . . . . . . . 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 1371	. . . . . . . . . . . . 13 |- ( 2 < 6 <-> ( pi / 6 ) < ( pi / 2 ) )
62	54, 61	mpbi 145	. . . . . . . . . . . 12 |- ( pi / 6 ) < ( pi / 2 )
63		0re 8146	. . . . . . . . . . . . 13 |- 0 e. RR
64		halfpire 15466	. . . . . . . . . . . . 13 |- ( pi / 2 ) e. RR
65		rexr 8192	. . . . . . . . . . . . . 14 |- ( 0 e. RR -> 0 e. RR* )
66		rexr 8192	. . . . . . . . . . . . . 14 |- ( ( pi / 2 ) e. RR -> ( pi / 2 ) e. RR* )
67		elioo2 10117	. . . . . . . . . . . . . 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 1203	. . . . . . . . . . 11 |- ( pi / 6 ) e. ( 0 (,) ( pi / 2 ) )
71		sincosq1sgn 15500	. . . . . . . . . . 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 8775	. . . . . . . 8 |- ( cos ` ( pi / 6 ) ) # 0
75	16, 74	pm3.2i 272	. . . . . . 7 |- ( ( cos ` ( pi / 6 ) ) e. CC /\ ( cos ` ( pi / 6 ) ) # 0 )
76		mulcanap2 8813	. . . . . . 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 1371	. . . . . 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 8989	. . 3 |- ( T. -> ( sin ` ( pi / 6 ) ) = ( 1 / 2 ) )
81	80	mptru 1404	. 2 |- ( sin ` ( pi / 6 ) ) = ( 1 / 2 )
82		3re 9184	. . . . . . . 8 |- 3 e. RR
83		3pos 9204	. . . . . . . 8 |- 0 < 3
84	82, 83	sqrtpclii 11641	. . . . . . 7 |- ( sqr ` 3 ) e. RR
85	84	recni 8158	. . . . . 6 |- ( sqr ` 3 ) e. CC
86	85, 1, 12	sqdivapi 10845	. . . . 5 |- ( ( ( sqr ` 3 ) / 2 ) ^ 2 ) = ( ( ( sqr ` 3 ) ^ 2 ) / ( 2 ^ 2 ) )
87	63, 82, 83	ltleii 8249	. . . . . . 7 |- 0 <_ 3
88	82	sqsqrti 11635	. . . . . . 7 |- ( 0 <_ 3 -> ( ( sqr ` 3 ) ^ 2 ) = 3 )
89	87, 88	ax-mp 5	. . . . . 6 |- ( ( sqr ` 3 ) ^ 2 ) = 3
90		sq2 10857	. . . . . 6 |- ( 2 ^ 2 ) = 4
91	89, 90	oveq12i 6013	. . . . 5 |- ( ( ( sqr ` 3 ) ^ 2 ) / ( 2 ^ 2 ) ) = ( 3 / 4 )
92	86, 91	eqtri 2250	. . . 4 |- ( ( ( sqr ` 3 ) / 2 ) ^ 2 ) = ( 3 / 4 )
93	92	fveq2i 5630	. . 3 |- ( sqr ` ( ( ( sqr ` 3 ) / 2 ) ^ 2 ) ) = ( sqr ` ( 3 / 4 ) )
94	82	sqrtge0i 11636	. . . . . 6 |- ( 0 <_ 3 -> 0 <_ ( sqr ` 3 ) )
95	87, 94	ax-mp 5	. . . . 5 |- 0 <_ ( sqr ` 3 )
96	84, 55	divge0i 9058	. . . . 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 8926	. . . . 5 |- ( ( sqr ` 3 ) / 2 ) e. RR
99	98	sqrtsqi 11634	. . . 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 9188	. . . . . . . 8 |- 4 e. CC
102		4ap0 9209	. . . . . . . 8 |- 4 # 0
103	101, 102	dividapi 8892	. . . . . . 7 |- ( 4 / 4 ) = 1
104	103	oveq1i 6011	. . . . . 6 |- ( ( 4 / 4 ) - ( 1 / 4 ) ) = ( 1 - ( 1 / 4 ) )
105	101, 102	pm3.2i 272	. . . . . . . 8 |- ( 4 e. CC /\ 4 # 0 )
106		divsubdirap 8855	. . . . . . . 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 1371	. . . . . . 7 |- ( ( 4 - 1 ) / 4 ) = ( ( 4 / 4 ) - ( 1 / 4 ) )
108		4m1e3 9231	. . . . . . . 8 |- ( 4 - 1 ) = 3
109	108	oveq1i 6011	. . . . . . 7 |- ( ( 4 - 1 ) / 4 ) = ( 3 / 4 )
110	107, 109	eqtr3i 2252	. . . . . 6 |- ( ( 4 / 4 ) - ( 1 / 4 ) ) = ( 3 / 4 )
111	101, 102	recclapi 8889	. . . . . . 7 |- ( 1 / 4 ) e. CC
112	16	sqcli 10842	. . . . . . 7 |- ( ( cos ` ( pi / 6 ) ) ^ 2 ) e. CC
113	81	oveq1i 6011	. . . . . . . . . 10 |- ( ( sin ` ( pi / 6 ) ) ^ 2 ) = ( ( 1 / 2 ) ^ 2 )
114		2z 9474	. . . . . . . . . . 11 |- 2 e. ZZ
115		exprecap 10802	. . . . . . . . . . 11 |- ( ( 2 e. CC /\ 2 # 0 /\ 2 e. ZZ ) -> ( ( 1 / 2 ) ^ 2 ) = ( 1 / ( 2 ^ 2 ) ) )
116	1, 12, 114, 115	mp3an 1371	. . . . . . . . . 10 |- ( ( 1 / 2 ) ^ 2 ) = ( 1 / ( 2 ^ 2 ) )
117	90	oveq2i 6012	. . . . . . . . . 10 |- ( 1 / ( 2 ^ 2 ) ) = ( 1 / 4 )
118	113, 116, 117	3eqtri 2254	. . . . . . . . 9 |- ( ( sin ` ( pi / 6 ) ) ^ 2 ) = ( 1 / 4 )
119	118	oveq1i 6011	. . . . . . . 8 |- ( ( ( sin ` ( pi / 6 ) ) ^ 2 ) + ( ( cos ` ( pi / 6 ) ) ^ 2 ) ) = ( ( 1 / 4 ) + ( ( cos ` ( pi / 6 ) ) ^ 2 ) )
120		sincossq 12259	. . . . . . . . 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 2252	. . . . . . 7 |- ( ( 1 / 4 ) + ( ( cos ` ( pi / 6 ) ) ^ 2 ) ) = 1
123	28, 111, 112, 122	subaddrii 8435	. . . . . 6 |- ( 1 - ( 1 / 4 ) ) = ( ( cos ` ( pi / 6 ) ) ^ 2 )
124	104, 110, 123	3eqtr3ri 2259	. . . . 5 |- ( ( cos ` ( pi / 6 ) ) ^ 2 ) = ( 3 / 4 )
125	124	fveq2i 5630	. . . 4 |- ( sqr ` ( ( cos ` ( pi / 6 ) ) ^ 2 ) ) = ( sqr ` ( 3 / 4 ) )
126	63, 15, 73	ltleii 8249	. . . . 5 |- 0 <_ ( cos ` ( pi / 6 ) )
127	15	sqrtsqi 11634	. . . . 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 2252	. . 3 |- ( sqr ` ( 3 / 4 ) ) = ( cos ` ( pi / 6 ) )
130	93, 100, 129	3eqtr3ri 2259	. 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 1002   = wceq 1395   T. wtru 1396   e. wcel 2200   class class class wbr 4083   ` cfv 5318  (class class class)co 6001   CCcc 7997   RRcr 7998   0cc0 7999   1c1 8000   + caddc 8002   x. cmul 8004   RR*cxr 8180   < clt 8181   <_ cle 8182   - cmin 8317   # cap 8728   / cdiv 8819   2c2 9161   3c3 9162   4c4 9163   6c6 9165   ZZcz 9446   (,)cioo 10084   ^cexp 10760   sqrcsqrt 11507   sincsin 12155   cosccos 12156   picpi 12158

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-mulrcl 8098  ax-addcom 8099  ax-mulcom 8100  ax-addass 8101  ax-mulass 8102  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-1rid 8106  ax-0id 8107  ax-rnegex 8108  ax-precex 8109  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-pre-apti 8114  ax-pre-ltadd 8115  ax-pre-mulgt0 8116  ax-pre-mulext 8117  ax-arch 8118  ax-caucvg 8119  ax-pre-suploc 8120  ax-addf 8121  ax-mulf 8122

This theorem depends on definitions:  df-bi 117  df-stab 836  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-disj 4060  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-isom 5327  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-of 6218  df-1st 6286  df-2nd 6287  df-recs 6451  df-irdg 6516  df-frec 6537  df-1o 6562  df-oadd 6566  df-er 6680  df-map 6797  df-pm 6798  df-en 6888  df-dom 6889  df-fin 6890  df-sup 7151  df-inf 7152  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-sub 8319  df-neg 8320  df-reap 8722  df-ap 8729  df-div 8820  df-inn 9111  df-2 9169  df-3 9170  df-4 9171  df-5 9172  df-6 9173  df-7 9174  df-8 9175  df-9 9176  df-n0 9370  df-z 9447  df-uz 9723  df-q 9815  df-rp 9850  df-xneg 9968  df-xadd 9969  df-ioo 10088  df-ioc 10089  df-ico 10090  df-icc 10091  df-fz 10205  df-fzo 10339  df-seqfrec 10670  df-exp 10761  df-fac 10948  df-bc 10970  df-ihash 10998  df-shft 11326  df-cj 11353  df-re 11354  df-im 11355  df-rsqrt 11509  df-abs 11510  df-clim 11790  df-sumdc 11865  df-ef 12159  df-sin 12161  df-cos 12162  df-pi 12164  df-rest 13274  df-topgen 13293  df-psmet 14507  df-xmet 14508  df-met 14509  df-bl 14510  df-mopn 14511  df-top 14672  df-topon 14685  df-bases 14717  df-ntr 14770  df-cn 14862  df-cnp 14863  df-tx 14927  df-cncf 15245  df-limced 15330  df-dvap 15331

This theorem is referenced by:  sincos3rdpi  15517  pigt3  15518