Theorem sincos6thpi 14715

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 9019	. . . . 5 |- 2 e. CC
2	1	a1i 9	. . . 4 |- ( T. -> 2 e. CC )
3		pire 14659	. . . . . . . 8 |- pi e. RR
4		6re 9029	. . . . . . . 8 |- 6 e. RR
5		6pos 9049	. . . . . . . . 9 |- 0 < 6
6	4, 5	gt0ap0ii 8614	. . . . . . . 8 |- 6 # 0
7	3, 4, 6	redivclapi 8765	. . . . . . 7 |- ( pi / 6 ) e. RR
8	7	recni 7998	. . . . . 6 |- ( pi / 6 ) e. CC
9		sincl 11745	. . . . . 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 9041	. . . . 5 |- 2 # 0
13	12	a1i 9	. . . 4 |- ( T. -> 2 # 0 )
14		recoscl 11760	. . . . . . . . . . . 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 7998	. . . . . . . . . 10 |- ( cos ` ( pi / 6 ) ) e. CC
17	1, 10, 16	mulassi 7995	. . . . . . . . 9 |- ( ( 2 x. ( sin ` ( pi / 6 ) ) ) x. ( cos ` ( pi / 6 ) ) ) = ( 2 x. ( ( sin ` ( pi / 6 ) ) x. ( cos ` ( pi / 6 ) ) ) )
18		sin2t 11788	. . . . . . . . . 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 2213	. . . . . . . 8 |- ( ( 2 x. ( sin ` ( pi / 6 ) ) ) x. ( cos ` ( pi / 6 ) ) ) = ( sin ` ( 2 x. ( pi / 6 ) ) )
21		3cn 9023	. . . . . . . . . . . 12 |- 3 e. CC
22		3ap0 9044	. . . . . . . . . . . 12 |- 3 # 0
23	1, 21, 22	divclapi 8740	. . . . . . . . . . 11 |- ( 2 / 3 ) e. CC
24	21, 22	recclapi 8728	. . . . . . . . . . 11 |- ( 1 / 3 ) e. CC
25		df-3 9008	. . . . . . . . . . . . 13 |- 3 = ( 2 + 1 )
26	25	oveq1i 5905	. . . . . . . . . . . 12 |- ( 3 / 3 ) = ( ( 2 + 1 ) / 3 )
27	21, 22	dividapi 8731	. . . . . . . . . . . 12 |- ( 3 / 3 ) = 1
28		ax-1cn 7933	. . . . . . . . . . . . 13 |- 1 e. CC
29	1, 28, 21, 22	divdirapi 8755	. . . . . . . . . . . 12 |- ( ( 2 + 1 ) / 3 ) = ( ( 2 / 3 ) + ( 1 / 3 ) )
30	26, 27, 29	3eqtr3ri 2219	. . . . . . . . . . 11 |- ( ( 2 / 3 ) + ( 1 / 3 ) ) = 1
31		sincosq1eq 14712	. . . . . . . . . . 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 14660	. . . . . . . . . . . . 13 |- pi e. CC
34	1, 21, 33, 1, 22, 12	divmuldivapi 8758	. . . . . . . . . . . 12 |- ( ( 2 / 3 ) x. ( pi / 2 ) ) = ( ( 2 x. pi ) / ( 3 x. 2 ) )
35		3t2e6 9104	. . . . . . . . . . . . 13 |- ( 3 x. 2 ) = 6
36	35	oveq2i 5906	. . . . . . . . . . . 12 |- ( ( 2 x. pi ) / ( 3 x. 2 ) ) = ( ( 2 x. pi ) / 6 )
37		6cn 9030	. . . . . . . . . . . . 13 |- 6 e. CC
38	1, 33, 37, 6	divassapi 8754	. . . . . . . . . . . 12 |- ( ( 2 x. pi ) / 6 ) = ( 2 x. ( pi / 6 ) )
39	34, 36, 38	3eqtri 2214	. . . . . . . . . . 11 |- ( ( 2 / 3 ) x. ( pi / 2 ) ) = ( 2 x. ( pi / 6 ) )
40	39	fveq2i 5537	. . . . . . . . . 10 |- ( sin ` ( ( 2 / 3 ) x. ( pi / 2 ) ) ) = ( sin ` ( 2 x. ( pi / 6 ) ) )
41	32, 40	eqtr3i 2212	. . . . . . . . 9 |- ( cos ` ( ( 1 / 3 ) x. ( pi / 2 ) ) ) = ( sin ` ( 2 x. ( pi / 6 ) ) )
42	28, 21, 33, 1, 22, 12	divmuldivapi 8758	. . . . . . . . . . 11 |- ( ( 1 / 3 ) x. ( pi / 2 ) ) = ( ( 1 x. pi ) / ( 3 x. 2 ) )
43	33	mullidi 7989	. . . . . . . . . . . 12 |- ( 1 x. pi ) = pi
44	43, 35	oveq12i 5907	. . . . . . . . . . 11 |- ( ( 1 x. pi ) / ( 3 x. 2 ) ) = ( pi / 6 )
45	42, 44	eqtri 2210	. . . . . . . . . 10 |- ( ( 1 / 3 ) x. ( pi / 2 ) ) = ( pi / 6 )
46	45	fveq2i 5537	. . . . . . . . 9 |- ( cos ` ( ( 1 / 3 ) x. ( pi / 2 ) ) ) = ( cos ` ( pi / 6 ) )
47	41, 46	eqtr3i 2212	. . . . . . . 8 |- ( sin ` ( 2 x. ( pi / 6 ) ) ) = ( cos ` ( pi / 6 ) )
48	20, 47	eqtri 2210	. . . . . . 7 |- ( ( 2 x. ( sin ` ( pi / 6 ) ) ) x. ( cos ` ( pi / 6 ) ) ) = ( cos ` ( pi / 6 ) )
49	16	mullidi 7989	. . . . . . 7 |- ( 1 x. ( cos ` ( pi / 6 ) ) ) = ( cos ` ( pi / 6 ) )
50	48, 49	eqtr4i 2213	. . . . . 6 |- ( ( 2 x. ( sin ` ( pi / 6 ) ) ) x. ( cos ` ( pi / 6 ) ) ) = ( 1 x. ( cos ` ( pi / 6 ) ) )
51	1, 10	mulcli 7991	. . . . . . 7 |- ( 2 x. ( sin ` ( pi / 6 ) ) ) e. CC
52		pipos 14661	. . . . . . . . . . . . 13 |- 0 < pi
53	3, 4, 52, 5	divgt0ii 8905	. . . . . . . . . . . 12 |- 0 < ( pi / 6 )
54		2lt6 9130	. . . . . . . . . . . . 13 |- 2 < 6
55		2re 9018	. . . . . . . . . . . . . . 15 |- 2 e. RR
56		2pos 9039	. . . . . . . . . . . . . . 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 8873	. . . . . . . . . . . . . 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 7986	. . . . . . . . . . . . 13 |- 0 e. RR
64		halfpire 14665	. . . . . . . . . . . . 13 |- ( pi / 2 ) e. RR
65		rexr 8032	. . . . . . . . . . . . . 14 |- ( 0 e. RR -> 0 e. RR* )
66		rexr 8032	. . . . . . . . . . . . . 14 |- ( ( pi / 2 ) e. RR -> ( pi / 2 ) e. RR* )
67		elioo2 9950	. . . . . . . . . . . . . 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 14699	. . . . . . . . . . 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 8614	. . . . . . . 8 |- ( cos ` ( pi / 6 ) ) # 0
75	16, 74	pm3.2i 272	. . . . . . 7 |- ( ( cos ` ( pi / 6 ) ) e. CC /\ ( cos ` ( pi / 6 ) ) # 0 )
76		mulcanap2 8652	. . . . . . 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 8828	. . 3 |- ( T. -> ( sin ` ( pi / 6 ) ) = ( 1 / 2 ) )
81	80	mptru 1373	. 2 |- ( sin ` ( pi / 6 ) ) = ( 1 / 2 )
82		3re 9022	. . . . . . . 8 |- 3 e. RR
83		3pos 9042	. . . . . . . 8 |- 0 < 3
84	82, 83	sqrtpclii 11170	. . . . . . 7 |- ( sqr ` 3 ) e. RR
85	84	recni 7998	. . . . . 6 |- ( sqr ` 3 ) e. CC
86	85, 1, 12	sqdivapi 10634	. . . . 5 |- ( ( ( sqr ` 3 ) / 2 ) ^ 2 ) = ( ( ( sqr ` 3 ) ^ 2 ) / ( 2 ^ 2 ) )
87	63, 82, 83	ltleii 8089	. . . . . . 7 |- 0 <_ 3
88	82	sqsqrti 11164	. . . . . . 7 |- ( 0 <_ 3 -> ( ( sqr ` 3 ) ^ 2 ) = 3 )
89	87, 88	ax-mp 5	. . . . . 6 |- ( ( sqr ` 3 ) ^ 2 ) = 3
90		sq2 10646	. . . . . 6 |- ( 2 ^ 2 ) = 4
91	89, 90	oveq12i 5907	. . . . 5 |- ( ( ( sqr ` 3 ) ^ 2 ) / ( 2 ^ 2 ) ) = ( 3 / 4 )
92	86, 91	eqtri 2210	. . . 4 |- ( ( ( sqr ` 3 ) / 2 ) ^ 2 ) = ( 3 / 4 )
93	92	fveq2i 5537	. . 3 |- ( sqr ` ( ( ( sqr ` 3 ) / 2 ) ^ 2 ) ) = ( sqr ` ( 3 / 4 ) )
94	82	sqrtge0i 11165	. . . . . 6 |- ( 0 <_ 3 -> 0 <_ ( sqr ` 3 ) )
95	87, 94	ax-mp 5	. . . . 5 |- 0 <_ ( sqr ` 3 )
96	84, 55	divge0i 8897	. . . . 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 8765	. . . . 5 |- ( ( sqr ` 3 ) / 2 ) e. RR
99	98	sqrtsqi 11163	. . . 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 9026	. . . . . . . 8 |- 4 e. CC
102		4ap0 9047	. . . . . . . 8 |- 4 # 0
103	101, 102	dividapi 8731	. . . . . . 7 |- ( 4 / 4 ) = 1
104	103	oveq1i 5905	. . . . . 6 |- ( ( 4 / 4 ) - ( 1 / 4 ) ) = ( 1 - ( 1 / 4 ) )
105	101, 102	pm3.2i 272	. . . . . . . 8 |- ( 4 e. CC /\ 4 # 0 )
106		divsubdirap 8694	. . . . . . . 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 9069	. . . . . . . 8 |- ( 4 - 1 ) = 3
109	108	oveq1i 5905	. . . . . . 7 |- ( ( 4 - 1 ) / 4 ) = ( 3 / 4 )
110	107, 109	eqtr3i 2212	. . . . . 6 |- ( ( 4 / 4 ) - ( 1 / 4 ) ) = ( 3 / 4 )
111	101, 102	recclapi 8728	. . . . . . 7 |- ( 1 / 4 ) e. CC
112	16	sqcli 10631	. . . . . . 7 |- ( ( cos ` ( pi / 6 ) ) ^ 2 ) e. CC
113	81	oveq1i 5905	. . . . . . . . . 10 |- ( ( sin ` ( pi / 6 ) ) ^ 2 ) = ( ( 1 / 2 ) ^ 2 )
114		2z 9310	. . . . . . . . . . 11 |- 2 e. ZZ
115		exprecap 10591	. . . . . . . . . . 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 5906	. . . . . . . . . 10 |- ( 1 / ( 2 ^ 2 ) ) = ( 1 / 4 )
118	113, 116, 117	3eqtri 2214	. . . . . . . . 9 |- ( ( sin ` ( pi / 6 ) ) ^ 2 ) = ( 1 / 4 )
119	118	oveq1i 5905	. . . . . . . 8 |- ( ( ( sin ` ( pi / 6 ) ) ^ 2 ) + ( ( cos ` ( pi / 6 ) ) ^ 2 ) ) = ( ( 1 / 4 ) + ( ( cos ` ( pi / 6 ) ) ^ 2 ) )
120		sincossq 11787	. . . . . . . . 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 2212	. . . . . . 7 |- ( ( 1 / 4 ) + ( ( cos ` ( pi / 6 ) ) ^ 2 ) ) = 1
123	28, 111, 112, 122	subaddrii 8275	. . . . . 6 |- ( 1 - ( 1 / 4 ) ) = ( ( cos ` ( pi / 6 ) ) ^ 2 )
124	104, 110, 123	3eqtr3ri 2219	. . . . 5 |- ( ( cos ` ( pi / 6 ) ) ^ 2 ) = ( 3 / 4 )
125	124	fveq2i 5537	. . . 4 |- ( sqr ` ( ( cos ` ( pi / 6 ) ) ^ 2 ) ) = ( sqr ` ( 3 / 4 ) )
126	63, 15, 73	ltleii 8089	. . . . 5 |- 0 <_ ( cos ` ( pi / 6 ) )
127	15	sqrtsqi 11163	. . . . 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 2212	. . 3 |- ( sqr ` ( 3 / 4 ) ) = ( cos ` ( pi / 6 ) )
130	93, 100, 129	3eqtr3ri 2219	. 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 2160   class class class wbr 4018   ` cfv 5235  (class class class)co 5895   CCcc 7838   RRcr 7839   0cc0 7840   1c1 7841   + caddc 7843   x. cmul 7845   RR*cxr 8020   < clt 8021   <_ cle 8022   - cmin 8157   # cap 8567   / cdiv 8658   2c2 8999   3c3 9000   4c4 9001   6c6 9003   ZZcz 9282   (,)cioo 9917   ^cexp 10549   sqrcsqrt 11036   sincsin 11683   cosccos 11684   picpi 11686

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605  ax-cnex 7931  ax-resscn 7932  ax-1cn 7933  ax-1re 7934  ax-icn 7935  ax-addcl 7936  ax-addrcl 7937  ax-mulcl 7938  ax-mulrcl 7939  ax-addcom 7940  ax-mulcom 7941  ax-addass 7942  ax-mulass 7943  ax-distr 7944  ax-i2m1 7945  ax-0lt1 7946  ax-1rid 7947  ax-0id 7948  ax-rnegex 7949  ax-precex 7950  ax-cnre 7951  ax-pre-ltirr 7952  ax-pre-ltwlin 7953  ax-pre-lttrn 7954  ax-pre-apti 7955  ax-pre-ltadd 7956  ax-pre-mulgt0 7957  ax-pre-mulext 7958  ax-arch 7959  ax-caucvg 7960  ax-pre-suploc 7961  ax-addf 7962  ax-mulf 7963

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 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-disj 3996  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-po 4314  df-iso 4315  df-iord 4384  df-on 4386  df-ilim 4387  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-isom 5244  df-riota 5851  df-ov 5898  df-oprab 5899  df-mpo 5900  df-of 6105  df-1st 6164  df-2nd 6165  df-recs 6329  df-irdg 6394  df-frec 6415  df-1o 6440  df-oadd 6444  df-er 6558  df-map 6675  df-pm 6676  df-en 6766  df-dom 6767  df-fin 6768  df-sup 7012  df-inf 7013  df-pnf 8023  df-mnf 8024  df-xr 8025  df-ltxr 8026  df-le 8027  df-sub 8159  df-neg 8160  df-reap 8561  df-ap 8568  df-div 8659  df-inn 8949  df-2 9007  df-3 9008  df-4 9009  df-5 9010  df-6 9011  df-7 9012  df-8 9013  df-9 9014  df-n0 9206  df-z 9283  df-uz 9558  df-q 9649  df-rp 9683  df-xneg 9801  df-xadd 9802  df-ioo 9921  df-ioc 9922  df-ico 9923  df-icc 9924  df-fz 10038  df-fzo 10172  df-seqfrec 10476  df-exp 10550  df-fac 10737  df-bc 10759  df-ihash 10787  df-shft 10855  df-cj 10882  df-re 10883  df-im 10884  df-rsqrt 11038  df-abs 11039  df-clim 11318  df-sumdc 11393  df-ef 11687  df-sin 11689  df-cos 11690  df-pi 11692  df-rest 12743  df-topgen 12762  df-psmet 13853  df-xmet 13854  df-met 13855  df-bl 13856  df-mopn 13857  df-top 13950  df-topon 13963  df-bases 13995  df-ntr 14048  df-cn 14140  df-cnp 14141  df-tx 14205  df-cncf 14510  df-limced 14577  df-dvap 14578

This theorem is referenced by:  sincos3rdpi  14716  pigt3  14717