Theorem coseq0negpitopi 15937

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 15887	. . . . . . . . . . . . 13 |- pi e. RR
3	2	renegcli 8588	. . . . . . . . . . . 12 |- -u pi e. RR
4	3	rexri 8383	. . . . . . . . . . 11 |- -u pi e. RR*
5		elioc2 10338	. . . . . . . . . . 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 15893	. . . . . . . . . 10 |- ( pi / 2 ) e. RR
11	10	renegcli 8588	. . . . . . . . 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 9381	. . . . . . . . . . 11 |- 4 e. RR
14		4ap0 9403	. . . . . . . . . . 11 |- 4 # 0
15	2, 13, 14	redivclapi 9109	. . . . . . . . . 10 |- ( pi / 4 ) e. RR
16	15	renegcli 8588	. . . . . . . . 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 9478	. . . . . . . . . . 11 |- 2 < 4
19		2re 9374	. . . . . . . . . . . . 13 |- 2 e. RR
20		2pos 9395	. . . . . . . . . . . . 13 |- 0 < 2
21	19, 20	pm3.2i 272	. . . . . . . . . . . 12 |- ( 2 e. RR /\ 0 < 2 )
22		4pos 9401	. . . . . . . . . . . . 13 |- 0 < 4
23	13, 22	pm3.2i 272	. . . . . . . . . . . 12 |- ( 4 e. RR /\ 0 < 4 )
24		pipos 15889	. . . . . . . . . . . . 13 |- 0 < pi
25	2, 24	pm3.2i 272	. . . . . . . . . . . 12 |- ( pi e. RR /\ 0 < pi )
26		ltdiv2 9217	. . . . . . . . . . . 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 8821	. . . . . . . . . 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 8452	. . . . . . 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 9378	. . . . . . . . . 10 |- 3 e. RR
36	35, 10	remulcli 8340	. . . . . . . . 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 9475	. . . . . . . . . . 11 |- 2 < 3
41		3pos 9398	. . . . . . . . . . . . 13 |- 0 < 3
42	35, 41	pm3.2i 272	. . . . . . . . . . . 12 |- ( 3 e. RR /\ 0 < 3 )
43		ltdiv2 9217	. . . . . . . . . . . 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 9206	. . . . . . . . . . 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 8451	. . . . . . 7 |- ( ( ( A e. ( -u pi (,] pi ) /\ ( cos ` A ) = 0 ) /\ -u ( pi / 4 ) < A ) -> A < ( 3 x. ( pi / 2 ) ) )
51	11	rexri 8383	. . . . . . . 8 |- -u ( pi / 2 ) e. RR*
52	36	rexri 8383	. . . . . . . 8 |- ( 3 x. ( pi / 2 ) ) e. RR*
53		elioo2 10323	. . . . . . . 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 15935	. . . . . 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 8354	. . . . . . 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 12494	. . . . . . . . 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 8707	. . . . . . . . . 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 8326	. . . . . . . . . . 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 8843	. . . . . . . . . . 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 8445	. . . . . . . . 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 8853	. . . . . . . . . . 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 8445	. . . . . . . . 9 |- ( ( ( A e. ( -u pi (,] pi ) /\ ( cos ` A ) = 0 ) /\ A < 0 ) -> -u A <_ pi )
85	71, 2	elicc2i 10341	. . . . . . . . 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 15936	. . . . . . . 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 8632	. . . . 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 15890	. . . . . . 7 |- pi e. RR+
97	13, 22	elrpii 10057	. . . . . . 7 |- 4 e. RR+
98		rpdivcl 10080	. . . . . . 7 |- ( ( pi e. RR+ /\ 4 e. RR+ ) -> ( pi / 4 ) e. RR+ )
99	96, 97, 98	mp2an 430	. . . . . 6 |- ( pi / 4 ) e. RR+
100		rpgt0 10066	. . . . . 6 |- ( ( pi / 4 ) e. RR+ -> 0 < ( pi / 4 ) )
101	99, 100	ax-mp 5	. . . . 5 |- 0 < ( pi / 4 )
102		lt0neg2 8797	. . . . . 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 8393	. . . . 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 5695	. . . . . 6 |- ( A = ( pi / 2 ) -> ( cos ` A ) = ( cos ` ( pi / 2 ) ) )
111		coshalfpi 15898	. . . . . 6 |- ( cos ` ( pi / 2 ) ) = 0
112	110, 111	eqtrdi 2287	. . . . 5 |- ( A = ( pi / 2 ) -> ( cos ` A ) = 0 )
113		fveq2 5695	. . . . . 6 |- ( A = -u ( pi / 2 ) -> ( cos ` A ) = ( cos ` -u ( pi / 2 ) ) )
114		cosneghalfpi 15899	. . . . . 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 4130   ` cfv 5377  (class class class)co 6085   CCcc 8177   RRcr 8178   0cc0 8179   x. cmul 8184   RR*cxr 8359   < clt 8360   <_ cle 8361   -ucneg 8498   / cdiv 9002   2c2 9355   3c3 9356   4c4 9357   RR+crp 10054   (,)cioo 10290   (,]cioc 10291   [,]cicc 10293   cosccos 12412   picpi 12414

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 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298  ax-caucvg 8299  ax-pre-suploc 8300  ax-addf 8301  ax-mulf 8302

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 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-isom 5386  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-of 6302  df-1st 6374  df-2nd 6375  df-recs 6576  df-irdg 6641  df-frec 6662  df-1o 6687  df-oadd 6691  df-er 6807  df-map 6924  df-pm 6925  df-en 7023  df-dom 7024  df-fin 7025  df-sup 7324  df-inf 7325  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-div 9003  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-5 9366  df-6 9367  df-7 9368  df-8 9369  df-9 9370  df-n0 9564  df-z 9645  df-uz 9922  df-q 10020  df-rp 10055  df-xneg 10174  df-xadd 10175  df-ioo 10294  df-ioc 10295  df-ico 10296  df-icc 10297  df-fz 10412  df-fzo 10550  df-seqfrec 10885  df-exp 10976  df-fac 11164  df-bc 11186  df-ihash 11215  df-shft 11580  df-cj 11607  df-re 11608  df-im 11609  df-rsqrt 11764  df-abs 11765  df-clim 12045  df-sumdc 12120  df-ef 12415  df-sin 12417  df-cos 12418  df-pi 12420  df-rest 13595  df-topgen 13614  df-psmet 14880  df-xmet 14881  df-met 14882  df-bl 14883  df-mopn 14884  df-top 15099  df-topon 15112  df-bases 15144  df-ntr 15197  df-cn 15289  df-cnp 15290  df-tx 15354  df-cncf 15672  df-limced 15757  df-dvap 15758

This theorem is used by: (None)