Theorem coseq0negpitopi 15423

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 15373	. . . . . . . . . . . . 13 |- pi e. RR
3	2	renegcli 8369	. . . . . . . . . . . 12 |- -u pi e. RR
4	3	rexri 8165	. . . . . . . . . . 11 |- -u pi e. RR*
5		elioc2 10093	. . . . . . . . . . 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 1015	. . . . . . . . 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 15379	. . . . . . . . . 10 |- ( pi / 2 ) e. RR
11	10	renegcli 8369	. . . . . . . . 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 9148	. . . . . . . . . . 11 |- 4 e. RR
14		4ap0 9170	. . . . . . . . . . 11 |- 4 # 0
15	2, 13, 14	redivclapi 8887	. . . . . . . . . 10 |- ( pi / 4 ) e. RR
16	15	renegcli 8369	. . . . . . . . 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 9245	. . . . . . . . . . 11 |- 2 < 4
19		2re 9141	. . . . . . . . . . . . 13 |- 2 e. RR
20		2pos 9162	. . . . . . . . . . . . 13 |- 0 < 2
21	19, 20	pm3.2i 272	. . . . . . . . . . . 12 |- ( 2 e. RR /\ 0 < 2 )
22		4pos 9168	. . . . . . . . . . . . 13 |- 0 < 4
23	13, 22	pm3.2i 272	. . . . . . . . . . . 12 |- ( 4 e. RR /\ 0 < 4 )
24		pipos 15375	. . . . . . . . . . . . 13 |- 0 < pi
25	2, 24	pm3.2i 272	. . . . . . . . . . . 12 |- ( pi e. RR /\ 0 < pi )
26		ltdiv2 8995	. . . . . . . . . . . 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 1350	. . . . . . . . . . 11 |- ( 2 < 4 <-> ( pi / 4 ) < ( pi / 2 ) )
28	18, 27	mpbi 145	. . . . . . . . . 10 |- ( pi / 4 ) < ( pi / 2 )
29	15, 10	ltnegi 8601	. . . . . . . . . 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 8233	. . . . . . 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 9145	. . . . . . . . . 10 |- 3 e. RR
36	35, 10	remulcli 8121	. . . . . . . . 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 1017	. . . . . . . . 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 9242	. . . . . . . . . . 11 |- 2 < 3
41		3pos 9165	. . . . . . . . . . . . 13 |- 0 < 3
42	35, 41	pm3.2i 272	. . . . . . . . . . . 12 |- ( 3 e. RR /\ 0 < 3 )
43		ltdiv2 8995	. . . . . . . . . . . 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 1350	. . . . . . . . . . 11 |- ( 2 < 3 <-> ( pi / 3 ) < ( pi / 2 ) )
45	40, 44	mpbi 145	. . . . . . . . . 10 |- ( pi / 3 ) < ( pi / 2 )
46		ltdivmul 8984	. . . . . . . . . . 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 1350	. . . . . . . . . 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 8232	. . . . . . 7 |- ( ( ( A e. ( -u pi (,] pi ) /\ ( cos ` A ) = 0 ) /\ -u ( pi / 4 ) < A ) -> A < ( 3 x. ( pi / 2 ) ) )
51	11	rexri 8165	. . . . . . . 8 |- -u ( pi / 2 ) e. RR*
52	36	rexri 8165	. . . . . . . 8 |- ( 3 x. ( pi / 2 ) ) e. RR*
53		elioo2 10078	. . . . . . . 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 1184	. . . . . 6 |- ( ( ( A e. ( -u pi (,] pi ) /\ ( cos ` A ) = 0 ) /\ -u ( pi / 4 ) < A ) -> A e. ( -u ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) )
56		coseq0q4123 15421	. . . . . 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 3747	. . . . 5 |- ( ( pi / 2 ) e. RR -> ( pi / 2 ) e. { ( pi / 2 ) , -u ( pi / 2 ) } )
60		eleq1a 2279	. . . . 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 8136	. . . . . . 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 12153	. . . . . . . . 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 2240	. . . . . . 7 |- ( ( ( A e. ( -u pi (,] pi ) /\ ( cos ` A ) = 0 ) /\ A < 0 ) -> ( cos ` -u A ) = 0 )
69	8	renegcld 8487	. . . . . . . . . 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 8107	. . . . . . . . . . 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 8623	. . . . . . . . . . 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 8226	. . . . . . . . 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 1016	. . . . . . . . . . . . 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 8633	. . . . . . . . . . 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 8226	. . . . . . . . 9 |- ( ( ( A e. ( -u pi (,] pi ) /\ ( cos ` A ) = 0 ) /\ A < 0 ) -> -u A <_ pi )
85	71, 2	elicc2i 10096	. . . . . . . . 9 |- ( -u A e. ( 0 [,] pi ) <-> ( -u A e. RR /\ 0 <_ -u A /\ -u A <_ pi ) )
86	70, 77, 84, 85	syl3anbrc 1184	. . . . . . . 8 |- ( ( ( A e. ( -u pi (,] pi ) /\ ( cos ` A ) = 0 ) /\ A < 0 ) -> -u A e. ( 0 [,] pi ) )
87		coseq00topi 15422	. . . . . . . 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 8413	. . . . 5 |- ( ( ( A e. ( -u pi (,] pi ) /\ ( cos ` A ) = 0 ) /\ A < 0 ) -> -u ( pi / 2 ) = A )
91	90	eqcomd 2213	. . . 4 |- ( ( ( A e. ( -u pi (,] pi ) /\ ( cos ` A ) = 0 ) /\ A < 0 ) -> A = -u ( pi / 2 ) )
92		prid2g 3748	. . . . 5 |- ( -u ( pi / 2 ) e. RR -> -u ( pi / 2 ) e. { ( pi / 2 ) , -u ( pi / 2 ) } )
93		eleq1a 2279	. . . . 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 15376	. . . . . . 7 |- pi e. RR+
97	13, 22	elrpii 9813	. . . . . . 7 |- 4 e. RR+
98		rpdivcl 9836	. . . . . . 7 |- ( ( pi e. RR+ /\ 4 e. RR+ ) -> ( pi / 4 ) e. RR+ )
99	96, 97, 98	mp2an 426	. . . . . 6 |- ( pi / 4 ) e. RR+
100		rpgt0 9822	. . . . . 6 |- ( ( pi / 4 ) e. RR+ -> 0 < ( pi / 4 ) )
101	99, 100	ax-mp 5	. . . . 5 |- 0 < ( pi / 4 )
102		lt0neg2 8577	. . . . . 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 8175	. . . . 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 1354	. . . 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 800	. 2 |- ( ( A e. ( -u pi (,] pi ) /\ ( cos ` A ) = 0 ) -> A e. { ( pi / 2 ) , -u ( pi / 2 ) } )
109		elpri 3666	. . . 4 |- ( A e. { ( pi / 2 ) , -u ( pi / 2 ) } -> ( A = ( pi / 2 ) \/ A = -u ( pi / 2 ) ) )
110		fveq2 5599	. . . . . 6 |- ( A = ( pi / 2 ) -> ( cos ` A ) = ( cos ` ( pi / 2 ) ) )
111		coshalfpi 15384	. . . . . 6 |- ( cos ` ( pi / 2 ) ) = 0
112	110, 111	eqtrdi 2256	. . . . 5 |- ( A = ( pi / 2 ) -> ( cos ` A ) = 0 )
113		fveq2 5599	. . . . . 6 |- ( A = -u ( pi / 2 ) -> ( cos ` A ) = ( cos ` -u ( pi / 2 ) ) )
114		cosneghalfpi 15385	. . . . . 6 |- ( cos ` -u ( pi / 2 ) ) = 0
115	113, 114	eqtrdi 2256	. . . . 5 |- ( A = -u ( pi / 2 ) -> ( cos ` A ) = 0 )
116	112, 115	jaoi 718	. . . 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 596	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 710   /\ w3a 981   = wceq 1373   e. wcel 2178   {cpr 3644   class class class wbr 4059   ` cfv 5290  (class class class)co 5967   CCcc 7958   RRcr 7959   0cc0 7960   x. cmul 7965   RR*cxr 8141   < clt 8142   <_ cle 8143   -ucneg 8279   / cdiv 8780   2c2 9122   3c3 9123   4c4 9124   RR+crp 9810   (,)cioo 10045   (,]cioc 10046   [,]cicc 10048   cosccos 12071   picpi 12073

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-mulrcl 8059  ax-addcom 8060  ax-mulcom 8061  ax-addass 8062  ax-mulass 8063  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-1rid 8067  ax-0id 8068  ax-rnegex 8069  ax-precex 8070  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-apti 8075  ax-pre-ltadd 8076  ax-pre-mulgt0 8077  ax-pre-mulext 8078  ax-arch 8079  ax-caucvg 8080  ax-pre-suploc 8081  ax-addf 8082  ax-mulf 8083

This theorem depends on definitions:  df-bi 117  df-stab 833  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-if 3580  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-disj 4036  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-id 4358  df-po 4361  df-iso 4362  df-iord 4431  df-on 4433  df-ilim 4434  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-isom 5299  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-of 6181  df-1st 6249  df-2nd 6250  df-recs 6414  df-irdg 6479  df-frec 6500  df-1o 6525  df-oadd 6529  df-er 6643  df-map 6760  df-pm 6761  df-en 6851  df-dom 6852  df-fin 6853  df-sup 7112  df-inf 7113  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-sub 8280  df-neg 8281  df-reap 8683  df-ap 8690  df-div 8781  df-inn 9072  df-2 9130  df-3 9131  df-4 9132  df-5 9133  df-6 9134  df-7 9135  df-8 9136  df-9 9137  df-n0 9331  df-z 9408  df-uz 9684  df-q 9776  df-rp 9811  df-xneg 9929  df-xadd 9930  df-ioo 10049  df-ioc 10050  df-ico 10051  df-icc 10052  df-fz 10166  df-fzo 10300  df-seqfrec 10630  df-exp 10721  df-fac 10908  df-bc 10930  df-ihash 10958  df-shft 11241  df-cj 11268  df-re 11269  df-im 11270  df-rsqrt 11424  df-abs 11425  df-clim 11705  df-sumdc 11780  df-ef 12074  df-sin 12076  df-cos 12077  df-pi 12079  df-rest 13188  df-topgen 13207  df-psmet 14420  df-xmet 14421  df-met 14422  df-bl 14423  df-mopn 14424  df-top 14585  df-topon 14598  df-bases 14630  df-ntr 14683  df-cn 14775  df-cnp 14776  df-tx 14840  df-cncf 15158  df-limced 15243  df-dvap 15244

This theorem is referenced by: (None)