Theorem coseq0negpitopi 15627

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 529	. . . . 5 |- ( ( ( A e. ( -u pi (,] pi ) /\ ( cos ` A ) = 0 ) /\ -u ( pi / 4 ) < A ) -> ( cos ` A ) = 0 )
2		pire 15577	. . . . . . . . . . . . 13 |- pi e. RR
3	2	renegcli 8484	. . . . . . . . . . . 12 |- -u pi e. RR
4	3	rexri 8280	. . . . . . . . . . 11 |- -u pi e. RR*
5		elioc2 10214	. . . . . . . . . . 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 1039	. . . . . . . . 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 15583	. . . . . . . . . 10 |- ( pi / 2 ) e. RR
11	10	renegcli 8484	. . . . . . . . 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 9263	. . . . . . . . . . 11 |- 4 e. RR
14		4ap0 9285	. . . . . . . . . . 11 |- 4 # 0
15	2, 13, 14	redivclapi 9002	. . . . . . . . . 10 |- ( pi / 4 ) e. RR
16	15	renegcli 8484	. . . . . . . . 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 9360	. . . . . . . . . . 11 |- 2 < 4
19		2re 9256	. . . . . . . . . . . . 13 |- 2 e. RR
20		2pos 9277	. . . . . . . . . . . . 13 |- 0 < 2
21	19, 20	pm3.2i 272	. . . . . . . . . . . 12 |- ( 2 e. RR /\ 0 < 2 )
22		4pos 9283	. . . . . . . . . . . . 13 |- 0 < 4
23	13, 22	pm3.2i 272	. . . . . . . . . . . 12 |- ( 4 e. RR /\ 0 < 4 )
24		pipos 15579	. . . . . . . . . . . . 13 |- 0 < pi
25	2, 24	pm3.2i 272	. . . . . . . . . . . 12 |- ( pi e. RR /\ 0 < pi )
26		ltdiv2 9110	. . . . . . . . . . . 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 1374	. . . . . . . . . . 11 |- ( 2 < 4 <-> ( pi / 4 ) < ( pi / 2 ) )
28	18, 27	mpbi 145	. . . . . . . . . 10 |- ( pi / 4 ) < ( pi / 2 )
29	15, 10	ltnegi 8716	. . . . . . . . . 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 8348	. . . . . . 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 9260	. . . . . . . . . 10 |- 3 e. RR
36	35, 10	remulcli 8236	. . . . . . . . 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 1041	. . . . . . . . 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 9357	. . . . . . . . . . 11 |- 2 < 3
41		3pos 9280	. . . . . . . . . . . . 13 |- 0 < 3
42	35, 41	pm3.2i 272	. . . . . . . . . . . 12 |- ( 3 e. RR /\ 0 < 3 )
43		ltdiv2 9110	. . . . . . . . . . . 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 1374	. . . . . . . . . . 11 |- ( 2 < 3 <-> ( pi / 3 ) < ( pi / 2 ) )
45	40, 44	mpbi 145	. . . . . . . . . 10 |- ( pi / 3 ) < ( pi / 2 )
46		ltdivmul 9099	. . . . . . . . . . 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 1374	. . . . . . . . . 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 8347	. . . . . . 7 |- ( ( ( A e. ( -u pi (,] pi ) /\ ( cos ` A ) = 0 ) /\ -u ( pi / 4 ) < A ) -> A < ( 3 x. ( pi / 2 ) ) )
51	11	rexri 8280	. . . . . . . 8 |- -u ( pi / 2 ) e. RR*
52	36	rexri 8280	. . . . . . . 8 |- ( 3 x. ( pi / 2 ) ) e. RR*
53		elioo2 10199	. . . . . . . 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 1208	. . . . . 6 |- ( ( ( A e. ( -u pi (,] pi ) /\ ( cos ` A ) = 0 ) /\ -u ( pi / 4 ) < A ) -> A e. ( -u ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) )
56		coseq0q4123 15625	. . . . . 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 3779	. . . . 5 |- ( ( pi / 2 ) e. RR -> ( pi / 2 ) e. { ( pi / 2 ) , -u ( pi / 2 ) } )
60		eleq1a 2303	. . . . 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 8251	. . . . . . 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 12349	. . . . . . . . 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 529	. . . . . . . 8 |- ( ( ( A e. ( -u pi (,] pi ) /\ ( cos ` A ) = 0 ) /\ A < 0 ) -> ( cos ` A ) = 0 )
68	66, 67	eqtrd 2264	. . . . . . 7 |- ( ( ( A e. ( -u pi (,] pi ) /\ ( cos ` A ) = 0 ) /\ A < 0 ) -> ( cos ` -u A ) = 0 )
69	8	renegcld 8602	. . . . . . . . . 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 8222	. . . . . . . . . . 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 8738	. . . . . . . . . . 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 8341	. . . . . . . . 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 1040	. . . . . . . . . . . . 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 8748	. . . . . . . . . . 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 8341	. . . . . . . . 9 |- ( ( ( A e. ( -u pi (,] pi ) /\ ( cos ` A ) = 0 ) /\ A < 0 ) -> -u A <_ pi )
85	71, 2	elicc2i 10217	. . . . . . . . 9 |- ( -u A e. ( 0 [,] pi ) <-> ( -u A e. RR /\ 0 <_ -u A /\ -u A <_ pi ) )
86	70, 77, 84, 85	syl3anbrc 1208	. . . . . . . 8 |- ( ( ( A e. ( -u pi (,] pi ) /\ ( cos ` A ) = 0 ) /\ A < 0 ) -> -u A e. ( 0 [,] pi ) )
87		coseq00topi 15626	. . . . . . . 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 8528	. . . . 5 |- ( ( ( A e. ( -u pi (,] pi ) /\ ( cos ` A ) = 0 ) /\ A < 0 ) -> -u ( pi / 2 ) = A )
91	90	eqcomd 2237	. . . 4 |- ( ( ( A e. ( -u pi (,] pi ) /\ ( cos ` A ) = 0 ) /\ A < 0 ) -> A = -u ( pi / 2 ) )
92		prid2g 3780	. . . . 5 |- ( -u ( pi / 2 ) e. RR -> -u ( pi / 2 ) e. { ( pi / 2 ) , -u ( pi / 2 ) } )
93		eleq1a 2303	. . . . 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 15580	. . . . . . 7 |- pi e. RR+
97	13, 22	elrpii 9934	. . . . . . 7 |- 4 e. RR+
98		rpdivcl 9957	. . . . . . 7 |- ( ( pi e. RR+ /\ 4 e. RR+ ) -> ( pi / 4 ) e. RR+ )
99	96, 97, 98	mp2an 426	. . . . . 6 |- ( pi / 4 ) e. RR+
100		rpgt0 9943	. . . . . 6 |- ( ( pi / 4 ) e. RR+ -> 0 < ( pi / 4 ) )
101	99, 100	ax-mp 5	. . . . 5 |- 0 < ( pi / 4 )
102		lt0neg2 8692	. . . . . 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 8290	. . . . 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 1378	. . . 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 806	. 2 |- ( ( A e. ( -u pi (,] pi ) /\ ( cos ` A ) = 0 ) -> A e. { ( pi / 2 ) , -u ( pi / 2 ) } )
109		elpri 3696	. . . 4 |- ( A e. { ( pi / 2 ) , -u ( pi / 2 ) } -> ( A = ( pi / 2 ) \/ A = -u ( pi / 2 ) ) )
110		fveq2 5648	. . . . . 6 |- ( A = ( pi / 2 ) -> ( cos ` A ) = ( cos ` ( pi / 2 ) ) )
111		coshalfpi 15588	. . . . . 6 |- ( cos ` ( pi / 2 ) ) = 0
112	110, 111	eqtrdi 2280	. . . . 5 |- ( A = ( pi / 2 ) -> ( cos ` A ) = 0 )
113		fveq2 5648	. . . . . 6 |- ( A = -u ( pi / 2 ) -> ( cos ` A ) = ( cos ` -u ( pi / 2 ) ) )
114		cosneghalfpi 15589	. . . . . 6 |- ( cos ` -u ( pi / 2 ) ) = 0
115	113, 114	eqtrdi 2280	. . . . 5 |- ( A = -u ( pi / 2 ) -> ( cos ` A ) = 0 )
116	112, 115	jaoi 724	. . . 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 600	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 716   /\ w3a 1005   = wceq 1398   e. wcel 2202   {cpr 3674   class class class wbr 4093   ` cfv 5333  (class class class)co 6028   CCcc 8073   RRcr 8074   0cc0 8075   x. cmul 8080   RR*cxr 8256   < clt 8257   <_ cle 8258   -ucneg 8394   / cdiv 8895   2c2 9237   3c3 9238   4c4 9239   RR+crp 9931   (,)cioo 10166   (,]cioc 10167   [,]cicc 10169   cosccos 12267   picpi 12269

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-mulrcl 8174  ax-addcom 8175  ax-mulcom 8176  ax-addass 8177  ax-mulass 8178  ax-distr 8179  ax-i2m1 8180  ax-0lt1 8181  ax-1rid 8182  ax-0id 8183  ax-rnegex 8184  ax-precex 8185  ax-cnre 8186  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189  ax-pre-apti 8190  ax-pre-ltadd 8191  ax-pre-mulgt0 8192  ax-pre-mulext 8193  ax-arch 8194  ax-caucvg 8195  ax-pre-suploc 8196  ax-addf 8197  ax-mulf 8198

This theorem depends on definitions:  df-bi 117  df-stab 839  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-disj 4070  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-isom 5342  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-of 6244  df-1st 6312  df-2nd 6313  df-recs 6514  df-irdg 6579  df-frec 6600  df-1o 6625  df-oadd 6629  df-er 6745  df-map 6862  df-pm 6863  df-en 6953  df-dom 6954  df-fin 6955  df-sup 7226  df-inf 7227  df-pnf 8259  df-mnf 8260  df-xr 8261  df-ltxr 8262  df-le 8263  df-sub 8395  df-neg 8396  df-reap 8798  df-ap 8805  df-div 8896  df-inn 9187  df-2 9245  df-3 9246  df-4 9247  df-5 9248  df-6 9249  df-7 9250  df-8 9251  df-9 9252  df-n0 9446  df-z 9523  df-uz 9799  df-q 9897  df-rp 9932  df-xneg 10050  df-xadd 10051  df-ioo 10170  df-ioc 10171  df-ico 10172  df-icc 10173  df-fz 10287  df-fzo 10421  df-seqfrec 10754  df-exp 10845  df-fac 11032  df-bc 11054  df-ihash 11082  df-shft 11436  df-cj 11463  df-re 11464  df-im 11465  df-rsqrt 11619  df-abs 11620  df-clim 11900  df-sumdc 11975  df-ef 12270  df-sin 12272  df-cos 12273  df-pi 12275  df-rest 13385  df-topgen 13404  df-psmet 14619  df-xmet 14620  df-met 14621  df-bl 14622  df-mopn 14623  df-top 14789  df-topon 14802  df-bases 14834  df-ntr 14887  df-cn 14979  df-cnp 14980  df-tx 15044  df-cncf 15362  df-limced 15447  df-dvap 15448

This theorem is referenced by: (None)