Theorem coseq0negpitopi 15550

Description: Location of the zeroes of cosine in ( -u pi (,] pi ). (Contributed by David Moews, 28-Feb-2017.)

Assertion

Ref	Expression
coseq0negpitopi	|- ( A e. ( -u pi (,] pi ) -> ( ( cos ` A ) = 0 <-> A e. { ( pi / 2 ) , -u ( pi / 2 ) } ) )

Proof of Theorem coseq0negpitopi

Step	Hyp	Ref	Expression
1		simplr 528	. . . . 5 |- ( ( ( A e. ( -u pi (,] pi ) /\ ( cos ` A ) = 0 ) /\ -u ( pi / 4 ) < A ) -> ( cos ` A ) = 0 )
2		pire 15500	. . . . . . . . . . . . 13 |- pi e. RR
3	2	renegcli 8431	. . . . . . . . . . . 12 |- -u pi e. RR
4	3	rexri 8227	. . . . . . . . . . 11 |- -u pi e. RR*
5		elioc2 10161	. . . . . . . . . . 11 |- ( ( -u pi e. RR* /\ pi e. RR ) -> ( A e. ( -u pi (,] pi ) <-> ( A e. RR /\ -u pi < A /\ A <_ pi ) ) )
6	4, 2, 5	mp2an 426	. . . . . . . . . 10 |- ( A e. ( -u pi (,] pi ) <-> ( A e. RR /\ -u pi < A /\ A <_ pi ) )
7	6	simp1bi 1036	. . . . . . . . 9 |- ( A e. ( -u pi (,] pi ) -> A e. RR )
8	7	adantr 276	. . . . . . . 8 |- ( ( A e. ( -u pi (,] pi ) /\ ( cos ` A ) = 0 ) -> A e. RR )
9	8	adantr 276	. . . . . . 7 |- ( ( ( A e. ( -u pi (,] pi ) /\ ( cos ` A ) = 0 ) /\ -u ( pi / 4 ) < A ) -> A e. RR )
10		halfpire 15506	. . . . . . . . . 10 |- ( pi / 2 ) e. RR
11	10	renegcli 8431	. . . . . . . . 9 |- -u ( pi / 2 ) e. RR
12	11	a1i 9	. . . . . . . 8 |- ( ( ( A e. ( -u pi (,] pi ) /\ ( cos ` A ) = 0 ) /\ -u ( pi / 4 ) < A ) -> -u ( pi / 2 ) e. RR )
13		4re 9210	. . . . . . . . . . 11 |- 4 e. RR
14		4ap0 9232	. . . . . . . . . . 11 |- 4 # 0
15	2, 13, 14	redivclapi 8949	. . . . . . . . . 10 |- ( pi / 4 ) e. RR
16	15	renegcli 8431	. . . . . . . . 9 |- -u ( pi / 4 ) e. RR
17	16	a1i 9	. . . . . . . 8 |- ( ( ( A e. ( -u pi (,] pi ) /\ ( cos ` A ) = 0 ) /\ -u ( pi / 4 ) < A ) -> -u ( pi / 4 ) e. RR )
18		2lt4 9307	. . . . . . . . . . 11 |- 2 < 4
19		2re 9203	. . . . . . . . . . . . 13 |- 2 e. RR
20		2pos 9224	. . . . . . . . . . . . 13 |- 0 < 2
21	19, 20	pm3.2i 272	. . . . . . . . . . . 12 |- ( 2 e. RR /\ 0 < 2 )
22		4pos 9230	. . . . . . . . . . . . 13 |- 0 < 4
23	13, 22	pm3.2i 272	. . . . . . . . . . . 12 |- ( 4 e. RR /\ 0 < 4 )
24		pipos 15502	. . . . . . . . . . . . 13 |- 0 < pi
25	2, 24	pm3.2i 272	. . . . . . . . . . . 12 |- ( pi e. RR /\ 0 < pi )
26		ltdiv2 9057	. . . . . . . . . . . 12 |- ( ( ( 2 e. RR /\ 0 < 2 ) /\ ( 4 e. RR /\ 0 < 4 ) /\ ( pi e. RR /\ 0 < pi ) ) -> ( 2 < 4 <-> ( pi / 4 ) < ( pi / 2 ) ) )
27	21, 23, 25, 26	mp3an 1371	. . . . . . . . . . 11 |- ( 2 < 4 <-> ( pi / 4 ) < ( pi / 2 ) )
28	18, 27	mpbi 145	. . . . . . . . . 10 |- ( pi / 4 ) < ( pi / 2 )
29	15, 10	ltnegi 8663	. . . . . . . . . 10 |- ( ( pi / 4 ) < ( pi / 2 ) <-> -u ( pi / 2 ) < -u ( pi / 4 ) )
30	28, 29	mpbi 145	. . . . . . . . 9 |- -u ( pi / 2 ) < -u ( pi / 4 )
31	30	a1i 9	. . . . . . . 8 |- ( ( ( A e. ( -u pi (,] pi ) /\ ( cos ` A ) = 0 ) /\ -u ( pi / 4 ) < A ) -> -u ( pi / 2 ) < -u ( pi / 4 ) )
32		simpr 110	. . . . . . . 8 |- ( ( ( A e. ( -u pi (,] pi ) /\ ( cos ` A ) = 0 ) /\ -u ( pi / 4 ) < A ) -> -u ( pi / 4 ) < A )
33	12, 17, 9, 31, 32	lttrd 8295	. . . . . . 7 |- ( ( ( A e. ( -u pi (,] pi ) /\ ( cos ` A ) = 0 ) /\ -u ( pi / 4 ) < A ) -> -u ( pi / 2 ) < A )
34	2	a1i 9	. . . . . . . 8 |- ( ( ( A e. ( -u pi (,] pi ) /\ ( cos ` A ) = 0 ) /\ -u ( pi / 4 ) < A ) -> pi e. RR )
35		3re 9207	. . . . . . . . . 10 |- 3 e. RR
36	35, 10	remulcli 8183	. . . . . . . . 9 |- ( 3 x. ( pi / 2 ) ) e. RR
37	36	a1i 9	. . . . . . . 8 |- ( ( ( A e. ( -u pi (,] pi ) /\ ( cos ` A ) = 0 ) /\ -u ( pi / 4 ) < A ) -> ( 3 x. ( pi / 2 ) ) e. RR )
38	6	simp3bi 1038	. . . . . . . . 9 |- ( A e. ( -u pi (,] pi ) -> A <_ pi )
39	38	ad2antrr 488	. . . . . . . 8 |- ( ( ( A e. ( -u pi (,] pi ) /\ ( cos ` A ) = 0 ) /\ -u ( pi / 4 ) < A ) -> A <_ pi )
40		2lt3 9304	. . . . . . . . . . 11 |- 2 < 3
41		3pos 9227	. . . . . . . . . . . . 13 |- 0 < 3
42	35, 41	pm3.2i 272	. . . . . . . . . . . 12 |- ( 3 e. RR /\ 0 < 3 )
43		ltdiv2 9057	. . . . . . . . . . . 12 |- ( ( ( 2 e. RR /\ 0 < 2 ) /\ ( 3 e. RR /\ 0 < 3 ) /\ ( pi e. RR /\ 0 < pi ) ) -> ( 2 < 3 <-> ( pi / 3 ) < ( pi / 2 ) ) )
44	21, 42, 25, 43	mp3an 1371	. . . . . . . . . . 11 |- ( 2 < 3 <-> ( pi / 3 ) < ( pi / 2 ) )
45	40, 44	mpbi 145	. . . . . . . . . 10 |- ( pi / 3 ) < ( pi / 2 )
46		ltdivmul 9046	. . . . . . . . . . 11 |- ( ( pi e. RR /\ ( pi / 2 ) e. RR /\ ( 3 e. RR /\ 0 < 3 ) ) -> ( ( pi / 3 ) < ( pi / 2 ) <-> pi < ( 3 x. ( pi / 2 ) ) ) )
47	2, 10, 42, 46	mp3an 1371	. . . . . . . . . 10 |- ( ( pi / 3 ) < ( pi / 2 ) <-> pi < ( 3 x. ( pi / 2 ) ) )
48	45, 47	mpbi 145	. . . . . . . . 9 |- pi < ( 3 x. ( pi / 2 ) )
49	48	a1i 9	. . . . . . . 8 |- ( ( ( A e. ( -u pi (,] pi ) /\ ( cos ` A ) = 0 ) /\ -u ( pi / 4 ) < A ) -> pi < ( 3 x. ( pi / 2 ) ) )
50	9, 34, 37, 39, 49	lelttrd 8294	. . . . . . 7 |- ( ( ( A e. ( -u pi (,] pi ) /\ ( cos ` A ) = 0 ) /\ -u ( pi / 4 ) < A ) -> A < ( 3 x. ( pi / 2 ) ) )
51	11	rexri 8227	. . . . . . . 8 |- -u ( pi / 2 ) e. RR*
52	36	rexri 8227	. . . . . . . 8 |- ( 3 x. ( pi / 2 ) ) e. RR*
53		elioo2 10146	. . . . . . . 8 |- ( ( -u ( pi / 2 ) e. RR* /\ ( 3 x. ( pi / 2 ) ) e. RR* ) -> ( A e. ( -u ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) <-> ( A e. RR /\ -u ( pi / 2 ) < A /\ A < ( 3 x. ( pi / 2 ) ) ) ) )
54	51, 52, 53	mp2an 426	. . . . . . 7 |- ( A e. ( -u ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) <-> ( A e. RR /\ -u ( pi / 2 ) < A /\ A < ( 3 x. ( pi / 2 ) ) ) )
55	9, 33, 50, 54	syl3anbrc 1205	. . . . . 6 |- ( ( ( A e. ( -u pi (,] pi ) /\ ( cos ` A ) = 0 ) /\ -u ( pi / 4 ) < A ) -> A e. ( -u ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) )
56		coseq0q4123 15548	. . . . . 6 |- ( A e. ( -u ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) -> ( ( cos ` A ) = 0 <-> A = ( pi / 2 ) ) )
57	55, 56	syl 14	. . . . 5 |- ( ( ( A e. ( -u pi (,] pi ) /\ ( cos ` A ) = 0 ) /\ -u ( pi / 4 ) < A ) -> ( ( cos ` A ) = 0 <-> A = ( pi / 2 ) ) )
58	1, 57	mpbid 147	. . . 4 |- ( ( ( A e. ( -u pi (,] pi ) /\ ( cos ` A ) = 0 ) /\ -u ( pi / 4 ) < A ) -> A = ( pi / 2 ) )
59		prid1g 3773	. . . . 5 |- ( ( pi / 2 ) e. RR -> ( pi / 2 ) e. { ( pi / 2 ) , -u ( pi / 2 ) } )
60		eleq1a 2301	. . . . 5 |- ( ( pi / 2 ) e. { ( pi / 2 ) , -u ( pi / 2 ) } -> ( A = ( pi / 2 ) -> A e. { ( pi / 2 ) , -u ( pi / 2 ) } ) )
61	10, 59, 60	mp2b 8	. . . 4 |- ( A = ( pi / 2 ) -> A e. { ( pi / 2 ) , -u ( pi / 2 ) } )
62	58, 61	syl 14	. . 3 |- ( ( ( A e. ( -u pi (,] pi ) /\ ( cos ` A ) = 0 ) /\ -u ( pi / 4 ) < A ) -> A e. { ( pi / 2 ) , -u ( pi / 2 ) } )
63	8	recnd 8198	. . . . . . 7 |- ( ( A e. ( -u pi (,] pi ) /\ ( cos ` A ) = 0 ) -> A e. CC )
64	63	adantr 276	. . . . . 6 |- ( ( ( A e. ( -u pi (,] pi ) /\ ( cos ` A ) = 0 ) /\ A < 0 ) -> A e. CC )
65		cosneg 12278	. . . . . . . . 9 |- ( A e. CC -> ( cos ` -u A ) = ( cos ` A ) )
66	64, 65	syl 14	. . . . . . . 8 |- ( ( ( A e. ( -u pi (,] pi ) /\ ( cos ` A ) = 0 ) /\ A < 0 ) -> ( cos ` -u A ) = ( cos ` A ) )
67		simplr 528	. . . . . . . 8 |- ( ( ( A e. ( -u pi (,] pi ) /\ ( cos ` A ) = 0 ) /\ A < 0 ) -> ( cos ` A ) = 0 )
68	66, 67	eqtrd 2262	. . . . . . 7 |- ( ( ( A e. ( -u pi (,] pi ) /\ ( cos ` A ) = 0 ) /\ A < 0 ) -> ( cos ` -u A ) = 0 )
69	8	renegcld 8549	. . . . . . . . . 10 |- ( ( A e. ( -u pi (,] pi ) /\ ( cos ` A ) = 0 ) -> -u A e. RR )
70	69	adantr 276	. . . . . . . . 9 |- ( ( ( A e. ( -u pi (,] pi ) /\ ( cos ` A ) = 0 ) /\ A < 0 ) -> -u A e. RR )
71		0re 8169	. . . . . . . . . . 11 |- 0 e. RR
72	71	a1i 9	. . . . . . . . . 10 |- ( ( ( A e. ( -u pi (,] pi ) /\ ( cos ` A ) = 0 ) /\ A < 0 ) -> 0 e. RR )
73		simpr 110	. . . . . . . . . . 11 |- ( ( ( A e. ( -u pi (,] pi ) /\ ( cos ` A ) = 0 ) /\ A < 0 ) -> A < 0 )
74	8	adantr 276	. . . . . . . . . . . 12 |- ( ( ( A e. ( -u pi (,] pi ) /\ ( cos ` A ) = 0 ) /\ A < 0 ) -> A e. RR )
75	74	lt0neg1d 8685	. . . . . . . . . . 11 |- ( ( ( A e. ( -u pi (,] pi ) /\ ( cos ` A ) = 0 ) /\ A < 0 ) -> ( A < 0 <-> 0 < -u A ) )
76	73, 75	mpbid 147	. . . . . . . . . 10 |- ( ( ( A e. ( -u pi (,] pi ) /\ ( cos ` A ) = 0 ) /\ A < 0 ) -> 0 < -u A )
77	72, 70, 76	ltled 8288	. . . . . . . . 9 |- ( ( ( A e. ( -u pi (,] pi ) /\ ( cos ` A ) = 0 ) /\ A < 0 ) -> 0 <_ -u A )
78	2	a1i 9	. . . . . . . . . 10 |- ( ( ( A e. ( -u pi (,] pi ) /\ ( cos ` A ) = 0 ) /\ A < 0 ) -> pi e. RR )
79	2	a1i 9	. . . . . . . . . . . 12 |- ( ( A e. ( -u pi (,] pi ) /\ ( cos ` A ) = 0 ) -> pi e. RR )
80	6	simp2bi 1037	. . . . . . . . . . . . 13 |- ( A e. ( -u pi (,] pi ) -> -u pi < A )
81	80	adantr 276	. . . . . . . . . . . 12 |- ( ( A e. ( -u pi (,] pi ) /\ ( cos ` A ) = 0 ) -> -u pi < A )
82	79, 8, 81	ltnegcon1d 8695	. . . . . . . . . . 11 |- ( ( A e. ( -u pi (,] pi ) /\ ( cos ` A ) = 0 ) -> -u A < pi )
83	82	adantr 276	. . . . . . . . . 10 |- ( ( ( A e. ( -u pi (,] pi ) /\ ( cos ` A ) = 0 ) /\ A < 0 ) -> -u A < pi )
84	70, 78, 83	ltled 8288	. . . . . . . . 9 |- ( ( ( A e. ( -u pi (,] pi ) /\ ( cos ` A ) = 0 ) /\ A < 0 ) -> -u A <_ pi )
85	71, 2	elicc2i 10164	. . . . . . . . 9 |- ( -u A e. ( 0 [,] pi ) <-> ( -u A e. RR /\ 0 <_ -u A /\ -u A <_ pi ) )
86	70, 77, 84, 85	syl3anbrc 1205	. . . . . . . 8 |- ( ( ( A e. ( -u pi (,] pi ) /\ ( cos ` A ) = 0 ) /\ A < 0 ) -> -u A e. ( 0 [,] pi ) )
87		coseq00topi 15549	. . . . . . . 8 |- ( -u A e. ( 0 [,] pi ) -> ( ( cos ` -u A ) = 0 <-> -u A = ( pi / 2 ) ) )
88	86, 87	syl 14	. . . . . . 7 |- ( ( ( A e. ( -u pi (,] pi ) /\ ( cos ` A ) = 0 ) /\ A < 0 ) -> ( ( cos ` -u A ) = 0 <-> -u A = ( pi / 2 ) ) )
89	68, 88	mpbid 147	. . . . . 6 |- ( ( ( A e. ( -u pi (,] pi ) /\ ( cos ` A ) = 0 ) /\ A < 0 ) -> -u A = ( pi / 2 ) )
90	64, 89	negcon1ad 8475	. . . . 5 |- ( ( ( A e. ( -u pi (,] pi ) /\ ( cos ` A ) = 0 ) /\ A < 0 ) -> -u ( pi / 2 ) = A )
91	90	eqcomd 2235	. . . 4 |- ( ( ( A e. ( -u pi (,] pi ) /\ ( cos ` A ) = 0 ) /\ A < 0 ) -> A = -u ( pi / 2 ) )
92		prid2g 3774	. . . . 5 |- ( -u ( pi / 2 ) e. RR -> -u ( pi / 2 ) e. { ( pi / 2 ) , -u ( pi / 2 ) } )
93		eleq1a 2301	. . . . 5 |- ( -u ( pi / 2 ) e. { ( pi / 2 ) , -u ( pi / 2 ) } -> ( A = -u ( pi / 2 ) -> A e. { ( pi / 2 ) , -u ( pi / 2 ) } ) )
94	11, 92, 93	mp2b 8	. . . 4 |- ( A = -u ( pi / 2 ) -> A e. { ( pi / 2 ) , -u ( pi / 2 ) } )
95	91, 94	syl 14	. . 3 |- ( ( ( A e. ( -u pi (,] pi ) /\ ( cos ` A ) = 0 ) /\ A < 0 ) -> A e. { ( pi / 2 ) , -u ( pi / 2 ) } )
96		pirp 15503	. . . . . . 7 |- pi e. RR+
97	13, 22	elrpii 9881	. . . . . . 7 |- 4 e. RR+
98		rpdivcl 9904	. . . . . . 7 |- ( ( pi e. RR+ /\ 4 e. RR+ ) -> ( pi / 4 ) e. RR+ )
99	96, 97, 98	mp2an 426	. . . . . 6 |- ( pi / 4 ) e. RR+
100		rpgt0 9890	. . . . . 6 |- ( ( pi / 4 ) e. RR+ -> 0 < ( pi / 4 ) )
101	99, 100	ax-mp 5	. . . . 5 |- 0 < ( pi / 4 )
102		lt0neg2 8639	. . . . . 6 |- ( ( pi / 4 ) e. RR -> ( 0 < ( pi / 4 ) <-> -u ( pi / 4 ) < 0 ) )
103	15, 102	ax-mp 5	. . . . 5 |- ( 0 < ( pi / 4 ) <-> -u ( pi / 4 ) < 0 )
104	101, 103	mpbi 145	. . . 4 |- -u ( pi / 4 ) < 0
105		axltwlin 8237	. . . . 5 |- ( ( -u ( pi / 4 ) e. RR /\ 0 e. RR /\ A e. RR ) -> ( -u ( pi / 4 ) < 0 -> ( -u ( pi / 4 ) < A \/ A < 0 ) ) )
106	16, 71, 8, 105	mp3an12i 1375	. . . 4 |- ( ( A e. ( -u pi (,] pi ) /\ ( cos ` A ) = 0 ) -> ( -u ( pi / 4 ) < 0 -> ( -u ( pi / 4 ) < A \/ A < 0 ) ) )
107	104, 106	mpi 15	. . 3 |- ( ( A e. ( -u pi (,] pi ) /\ ( cos ` A ) = 0 ) -> ( -u ( pi / 4 ) < A \/ A < 0 ) )
108	62, 95, 107	mpjaodan 803	. 2 |- ( ( A e. ( -u pi (,] pi ) /\ ( cos ` A ) = 0 ) -> A e. { ( pi / 2 ) , -u ( pi / 2 ) } )
109		elpri 3690	. . . 4 |- ( A e. { ( pi / 2 ) , -u ( pi / 2 ) } -> ( A = ( pi / 2 ) \/ A = -u ( pi / 2 ) ) )
110		fveq2 5635	. . . . . 6 |- ( A = ( pi / 2 ) -> ( cos ` A ) = ( cos ` ( pi / 2 ) ) )
111		coshalfpi 15511	. . . . . 6 |- ( cos ` ( pi / 2 ) ) = 0
112	110, 111	eqtrdi 2278	. . . . 5 |- ( A = ( pi / 2 ) -> ( cos ` A ) = 0 )
113		fveq2 5635	. . . . . 6 |- ( A = -u ( pi / 2 ) -> ( cos ` A ) = ( cos ` -u ( pi / 2 ) ) )
114		cosneghalfpi 15512	. . . . . 6 |- ( cos ` -u ( pi / 2 ) ) = 0
115	113, 114	eqtrdi 2278	. . . . 5 |- ( A = -u ( pi / 2 ) -> ( cos ` A ) = 0 )
116	112, 115	jaoi 721	. . . 4 |- ( ( A = ( pi / 2 ) \/ A = -u ( pi / 2 ) ) -> ( cos ` A ) = 0 )
117	109, 116	syl 14	. . 3 |- ( A e. { ( pi / 2 ) , -u ( pi / 2 ) } -> ( cos ` A ) = 0 )
118	117	adantl 277	. 2 |- ( ( A e. ( -u pi (,] pi ) /\ A e. { ( pi / 2 ) , -u ( pi / 2 ) } ) -> ( cos ` A ) = 0 )
119	108, 118	impbida 598	1 |- ( A e. ( -u pi (,] pi ) -> ( ( cos ` A ) = 0 <-> A e. { ( pi / 2 ) , -u ( pi / 2 ) } ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 713   /\ w3a 1002   = wceq 1395   e. wcel 2200   {cpr 3668   class class class wbr 4086   ` cfv 5324  (class class class)co 6013   CCcc 8020   RRcr 8021   0cc0 8022   x. cmul 8027   RR*cxr 8203   < clt 8204   <_ cle 8205   -ucneg 8341   / cdiv 8842   2c2 9184   3c3 9185   4c4 9186   RR+crp 9878   (,)cioo 10113   (,]cioc 10114   [,]cicc 10116   cosccos 12196   picpi 12198

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 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-mulrcl 8121  ax-addcom 8122  ax-mulcom 8123  ax-addass 8124  ax-mulass 8125  ax-distr 8126  ax-i2m1 8127  ax-0lt1 8128  ax-1rid 8129  ax-0id 8130  ax-rnegex 8131  ax-precex 8132  ax-cnre 8133  ax-pre-ltirr 8134  ax-pre-ltwlin 8135  ax-pre-lttrn 8136  ax-pre-apti 8137  ax-pre-ltadd 8138  ax-pre-mulgt0 8139  ax-pre-mulext 8140  ax-arch 8141  ax-caucvg 8142  ax-pre-suploc 8143  ax-addf 8144  ax-mulf 8145

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 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-disj 4063  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-po 4391  df-iso 4392  df-iord 4461  df-on 4463  df-ilim 4464  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-isom 5333  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-of 6230  df-1st 6298  df-2nd 6299  df-recs 6466  df-irdg 6531  df-frec 6552  df-1o 6577  df-oadd 6581  df-er 6697  df-map 6814  df-pm 6815  df-en 6905  df-dom 6906  df-fin 6907  df-sup 7174  df-inf 7175  df-pnf 8206  df-mnf 8207  df-xr 8208  df-ltxr 8209  df-le 8210  df-sub 8342  df-neg 8343  df-reap 8745  df-ap 8752  df-div 8843  df-inn 9134  df-2 9192  df-3 9193  df-4 9194  df-5 9195  df-6 9196  df-7 9197  df-8 9198  df-9 9199  df-n0 9393  df-z 9470  df-uz 9746  df-q 9844  df-rp 9879  df-xneg 9997  df-xadd 9998  df-ioo 10117  df-ioc 10118  df-ico 10119  df-icc 10120  df-fz 10234  df-fzo 10368  df-seqfrec 10700  df-exp 10791  df-fac 10978  df-bc 11000  df-ihash 11028  df-shft 11366  df-cj 11393  df-re 11394  df-im 11395  df-rsqrt 11549  df-abs 11550  df-clim 11830  df-sumdc 11905  df-ef 12199  df-sin 12201  df-cos 12202  df-pi 12204  df-rest 13314  df-topgen 13333  df-psmet 14547  df-xmet 14548  df-met 14549  df-bl 14550  df-mopn 14551  df-top 14712  df-topon 14725  df-bases 14757  df-ntr 14810  df-cn 14902  df-cnp 14903  df-tx 14967  df-cncf 15285  df-limced 15370  df-dvap 15371

This theorem is referenced by: (None)