Theorem coseq0negpitopi 16063

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 533	. . . . 5 |- ( ( ( A e. ( -u pi (,] pi ) /\ ( cos ` A ) = 0 ) /\ -u ( pi / 4 ) < A ) -> ( cos ` A ) = 0 )
2		pire 16013	. . . . . . . . . . . . 13 |- pi e. RR
3	2	renegcli 8595	. . . . . . . . . . . 12 |- -u pi e. RR
4	3	rexri 8389	. . . . . . . . . . 11 |- -u pi e. RR*
5		elioc2 10355	. . . . . . . . . . 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 430	. . . . . . . . . 10 |- ( A e. ( -u pi (,] pi ) <-> ( A e. RR /\ -u pi < A /\ A <_ pi ) )
7	6	simp1bi 1043	. . . . . . . . 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 16019	. . . . . . . . . 10 |- ( pi / 2 ) e. RR
11	10	renegcli 8595	. . . . . . . . 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 9390	. . . . . . . . . . 11 |- 4 e. RR
14		4ap0 9412	. . . . . . . . . . 11 |- 4 # 0
15	2, 13, 14	redivclapi 9117	. . . . . . . . . 10 |- ( pi / 4 ) e. RR
16	15	renegcli 8595	. . . . . . . . 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 9489	. . . . . . . . . . 11 |- 2 < 4
19		2re 9383	. . . . . . . . . . . . 13 |- 2 e. RR
20		2pos 9404	. . . . . . . . . . . . 13 |- 0 < 2
21	19, 20	pm3.2i 272	. . . . . . . . . . . 12 |- ( 2 e. RR /\ 0 < 2 )
22		4pos 9410	. . . . . . . . . . . . 13 |- 0 < 4
23	13, 22	pm3.2i 272	. . . . . . . . . . . 12 |- ( 4 e. RR /\ 0 < 4 )
24		pipos 16015	. . . . . . . . . . . . 13 |- 0 < pi
25	2, 24	pm3.2i 272	. . . . . . . . . . . 12 |- ( pi e. RR /\ 0 < pi )
26		ltdiv2 9225	. . . . . . . . . . . 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 1378	. . . . . . . . . . 11 |- ( 2 < 4 <-> ( pi / 4 ) < ( pi / 2 ) )
28	18, 27	mpbi 145	. . . . . . . . . 10 |- ( pi / 4 ) < ( pi / 2 )
29	15, 10	ltnegi 8828	. . . . . . . . . 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 8459	. . . . . . 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 9387	. . . . . . . . . 10 |- 3 e. RR
36	35, 10	remulcli 8346	. . . . . . . . 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 1045	. . . . . . . . 9 |- ( A e. ( -u pi (,] pi ) -> A <_ pi )
39	38	ad2antrr 492	. . . . . . . 8 |- ( ( ( A e. ( -u pi (,] pi ) /\ ( cos ` A ) = 0 ) /\ -u ( pi / 4 ) < A ) -> A <_ pi )
40		2lt3 9486	. . . . . . . . . . 11 |- 2 < 3
41		3pos 9407	. . . . . . . . . . . . 13 |- 0 < 3
42	35, 41	pm3.2i 272	. . . . . . . . . . . 12 |- ( 3 e. RR /\ 0 < 3 )
43		ltdiv2 9225	. . . . . . . . . . . 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 1378	. . . . . . . . . . 11 |- ( 2 < 3 <-> ( pi / 3 ) < ( pi / 2 ) )
45	40, 44	mpbi 145	. . . . . . . . . 10 |- ( pi / 3 ) < ( pi / 2 )
46		ltdivmul 9214	. . . . . . . . . . 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 1378	. . . . . . . . . 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 8458	. . . . . . 7 |- ( ( ( A e. ( -u pi (,] pi ) /\ ( cos ` A ) = 0 ) /\ -u ( pi / 4 ) < A ) -> A < ( 3 x. ( pi / 2 ) ) )
51	11	rexri 8389	. . . . . . . 8 |- -u ( pi / 2 ) e. RR*
52	36	rexri 8389	. . . . . . . 8 |- ( 3 x. ( pi / 2 ) ) e. RR*
53		elioo2 10340	. . . . . . . 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 430	. . . . . . 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 1212	. . . . . 6 |- ( ( ( A e. ( -u pi (,] pi ) /\ ( cos ` A ) = 0 ) /\ -u ( pi / 4 ) < A ) -> A e. ( -u ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) )
56		coseq0q4123 16061	. . . . . 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 3815	. . . . 5 |- ( ( pi / 2 ) e. RR -> ( pi / 2 ) e. { ( pi / 2 ) , -u ( pi / 2 ) } )
60		eleq1a 2310	. . . . 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 8360	. . . . . . 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 12519	. . . . . . . . 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 533	. . . . . . . 8 |- ( ( ( A e. ( -u pi (,] pi ) /\ ( cos ` A ) = 0 ) /\ A < 0 ) -> ( cos ` A ) = 0 )
68	66, 67	eqtrd 2271	. . . . . . 7 |- ( ( ( A e. ( -u pi (,] pi ) /\ ( cos ` A ) = 0 ) /\ A < 0 ) -> ( cos ` -u A ) = 0 )
69	8	renegcld 8714	. . . . . . . . . 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 8332	. . . . . . . . . . 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 8850	. . . . . . . . . . 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 8452	. . . . . . . . 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 1044	. . . . . . . . . . . . 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 8860	. . . . . . . . . . 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 8452	. . . . . . . . 9 |- ( ( ( A e. ( -u pi (,] pi ) /\ ( cos ` A ) = 0 ) /\ A < 0 ) -> -u A <_ pi )
85	71, 2	elicc2i 10358	. . . . . . . . 9 |- ( -u A e. ( 0 [,] pi ) <-> ( -u A e. RR /\ 0 <_ -u A /\ -u A <_ pi ) )
86	70, 77, 84, 85	syl3anbrc 1212	. . . . . . . 8 |- ( ( ( A e. ( -u pi (,] pi ) /\ ( cos ` A ) = 0 ) /\ A < 0 ) -> -u A e. ( 0 [,] pi ) )
87		coseq00topi 16062	. . . . . . . 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 8639	. . . . 5 |- ( ( ( A e. ( -u pi (,] pi ) /\ ( cos ` A ) = 0 ) /\ A < 0 ) -> -u ( pi / 2 ) = A )
91	90	eqcomd 2244	. . . 4 |- ( ( ( A e. ( -u pi (,] pi ) /\ ( cos ` A ) = 0 ) /\ A < 0 ) -> A = -u ( pi / 2 ) )
92		prid2g 3816	. . . . 5 |- ( -u ( pi / 2 ) e. RR -> -u ( pi / 2 ) e. { ( pi / 2 ) , -u ( pi / 2 ) } )
93		eleq1a 2310	. . . . 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 16016	. . . . . . 7 |- pi e. RR+
97	13, 22	elrpii 10074	. . . . . . 7 |- 4 e. RR+
98		rpdivcl 10097	. . . . . . 7 |- ( ( pi e. RR+ /\ 4 e. RR+ ) -> ( pi / 4 ) e. RR+ )
99	96, 97, 98	mp2an 430	. . . . . 6 |- ( pi / 4 ) e. RR+
100		rpgt0 10083	. . . . . 6 |- ( ( pi / 4 ) e. RR+ -> 0 < ( pi / 4 ) )
101	99, 100	ax-mp 5	. . . . 5 |- 0 < ( pi / 4 )
102		lt0neg2 8804	. . . . . 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 8399	. . . . 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 1382	. . . 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 810	. 2 |- ( ( A e. ( -u pi (,] pi ) /\ ( cos ` A ) = 0 ) -> A e. { ( pi / 2 ) , -u ( pi / 2 ) } )
109		elpri 3732	. . . 4 |- ( A e. { ( pi / 2 ) , -u ( pi / 2 ) } -> ( A = ( pi / 2 ) \/ A = -u ( pi / 2 ) ) )
110		fveq2 5698	. . . . . 6 |- ( A = ( pi / 2 ) -> ( cos ` A ) = ( cos ` ( pi / 2 ) ) )
111		coshalfpi 16024	. . . . . 6 |- ( cos ` ( pi / 2 ) ) = 0
112	110, 111	eqtrdi 2287	. . . . 5 |- ( A = ( pi / 2 ) -> ( cos ` A ) = 0 )
113		fveq2 5698	. . . . . 6 |- ( A = -u ( pi / 2 ) -> ( cos ` A ) = ( cos ` -u ( pi / 2 ) ) )
114		cosneghalfpi 16025	. . . . . 6 |- ( cos ` -u ( pi / 2 ) ) = 0
115	113, 114	eqtrdi 2287	. . . . 5 |- ( A = -u ( pi / 2 ) -> ( cos ` A ) = 0 )
116	112, 115	jaoi 728	. . . 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 604	1 |- ( A e. ( -u pi (,] pi ) -> ( ( cos ` A ) = 0 <-> A e. { ( pi / 2 ) , -u ( pi / 2 ) } ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 720   /\ w3a 1009   = wceq 1402   e. wcel 2209   {cpr 3710   class class class wbr 4131   ` cfv 5380  (class class class)co 6088   CCcc 8183   RRcr 8184   0cc0 8185   x. cmul 8190   RR*cxr 8365   < clt 8366   <_ cle 8367   -ucneg 8505   / cdiv 9010   2c2 9364   3c3 9365   4c4 9366   RR+crp 10071   (,)cioo 10307   (,]cioc 10308   [,]cicc 10310   cosccos 12437   picpi 12439

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-mulrcl 8284  ax-addcom 8285  ax-mulcom 8286  ax-addass 8287  ax-mulass 8288  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-1rid 8292  ax-0id 8293  ax-rnegex 8294  ax-precex 8295  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-apti 8300  ax-pre-ltadd 8301  ax-pre-mulgt0 8302  ax-pre-mulext 8303  ax-arch 8304  ax-caucvg 8305  ax-pre-suploc 8306  ax-addf 8307  ax-mulf 8308

This proof depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-disj 4107  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-po 4443  df-iso 4444  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-isom 5389  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-of 6305  df-1st 6378  df-2nd 6379  df-recs 6580  df-irdg 6645  df-frec 6666  df-1o 6691  df-oadd 6695  df-er 6811  df-map 6928  df-pm 6929  df-en 7027  df-dom 7028  df-fin 7029  df-sup 7330  df-inf 7331  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-reap 8911  df-ap 8918  df-div 9011  df-inn 9314  df-2 9372  df-3 9373  df-4 9374  df-5 9375  df-6 9376  df-7 9377  df-8 9378  df-9 9379  df-n0 9575  df-z 9656  df-uz 9938  df-q 10036  df-rp 10072  df-xneg 10191  df-xadd 10192  df-ioo 10311  df-ioc 10312  df-ico 10313  df-icc 10314  df-fz 10429  df-fzo 10567  df-seqfrec 10906  df-exp 10997  df-fac 11186  df-bc 11208  df-ihash 11237  df-shft 11602  df-cj 11629  df-re 11630  df-im 11631  df-rsqrt 11786  df-abs 11787  df-clim 12070  df-sumdc 12145  df-ef 12440  df-sin 12442  df-cos 12443  df-pi 12445  df-rest 13654  df-topgen 13673  df-psmet 14989  df-xmet 14990  df-met 14991  df-bl 14992  df-mopn 14993  df-top 15224  df-topon 15237  df-bases 15269  df-ntr 15322  df-cn 15414  df-cnp 15415  df-tx 15479  df-cncf 15797  df-limced 15882  df-dvap 15883

This theorem is used by: (None)