Theorem rddif 15798

Description: The difference between a real number and its nearest integer is less than or equal to one half.

Assertion

Ref	Expression
rddif	|- (A e. RR -> (abs` ((|_` (A + (1 / 2))) - A)) <_ (1 / 2))

Proof of Theorem rddif

Step	Hyp	Ref	Expression
1		2re 7163	. . . . . . . . . . 11 |- 2 e. RR
2		2ne0 7174	. . . . . . . . . . 11 |- 2 =/= 0
3	1, 2	rereccli 6979	. . . . . . . . . 10 |- (1 / 2) e. RR
4		readdcl 6455	. . . . . . . . . 10 |- ((A e. RR /\ (1 / 2) e. RR) -> (A + (1 / 2)) e. RR)
5	3, 4	mpan2 760	. . . . . . . . 9 |- (A e. RR -> (A + (1 / 2)) e. RR)
6		flcl 7465	. . . . . . . . 9 |- (A e. RR -> (|_` A) e. ZZ)
7		flbi 7480	. . . . . . . . 9 |- (((A + (1 / 2)) e. RR /\ (|_` A) e. ZZ) -> ((|_` (A + (1 / 2))) = (|_` A) <-> ((|_` A) <_ (A + (1 / 2)) /\ (A + (1 / 2)) < ((|_` A) + 1))))
8	5, 6, 7	syl11anc 524	. . . . . . . 8 |- (A e. RR -> ((|_` (A + (1 / 2))) = (|_` A) <-> ((|_` A) <_ (A + (1 / 2)) /\ (A + (1 / 2)) < ((|_` A) + 1))))
9	8	adantr 425	. . . . . . 7 |- ((A e. RR /\ (A - (|_` A)) < (1 / 2)) -> ((|_` (A + (1 / 2))) = (|_` A) <-> ((|_` A) <_ (A + (1 / 2)) /\ (A + (1 / 2)) < ((|_` A) + 1))))
10		reflcl 7466	. . . . . . . . 9 |- (A e. RR -> (|_` A) e. RR)
11		id 73	. . . . . . . . 9 |- (A e. RR -> A e. RR)
12		flle 7468	. . . . . . . . 9 |- (A e. RR -> (|_` A) <_ A)
13		recn 6466	. . . . . . . . . . 11 |- (A e. RR -> A e. CC)
14		ax0id 6434	. . . . . . . . . . 11 |- (A e. CC -> (A + 0) = A)
15	13, 14	syl 12	. . . . . . . . . 10 |- (A e. RR -> (A + 0) = A)
16		0re 6603	. . . . . . . . . . . 12 |- 0 e. RR
17		halfgt0 7215	. . . . . . . . . . . 12 |- 0 < (1 / 2)
18	16, 3, 17	ltleii 6756	. . . . . . . . . . 11 |- 0 <_ (1 / 2)
19		leadd2 6809	. . . . . . . . . . . 12 |- ((0 e. RR /\ (1 / 2) e. RR /\ A e. RR) -> (0 <_ (1 / 2) <-> (A + 0) <_ (A + (1 / 2))))
20	16, 3, 19	mp3an12 1181	. . . . . . . . . . 11 |- (A e. RR -> (0 <_ (1 / 2) <-> (A + 0) <_ (A + (1 / 2))))
21	18, 20	mpbii 210	. . . . . . . . . 10 |- (A e. RR -> (A + 0) <_ (A + (1 / 2)))
22	15, 21	eqbrtrrd 3359	. . . . . . . . 9 |- (A e. RR -> A <_ (A + (1 / 2)))
23	10, 11, 5, 12, 22	letrd 6696	. . . . . . . 8 |- (A e. RR -> (|_` A) <_ (A + (1 / 2)))
24	23	adantr 425	. . . . . . 7 |- ((A e. RR /\ (A - (|_` A)) < (1 / 2)) -> (|_` A) <_ (A + (1 / 2)))
25	3	a1i 8	. . . . . . . . . . 11 |- (A e. RR -> (1 / 2) e. RR)
26		ltsubadd2 6811	. . . . . . . . . . 11 |- ((A e. RR /\ (|_` A) e. RR /\ (1 / 2) e. RR) -> ((A - (|_` A)) < (1 / 2) <-> A < ((|_` A) + (1 / 2))))
27	10, 25, 26	mpd3an23 1193	. . . . . . . . . 10 |- (A e. RR -> ((A - (|_` A)) < (1 / 2) <-> A < ((|_` A) + (1 / 2))))
28	27	biimpa 460	. . . . . . . . 9 |- ((A e. RR /\ (A - (|_` A)) < (1 / 2)) -> A < ((|_` A) + (1 / 2)))
29		readdcl 6455	. . . . . . . . . . . 12 |- (((|_` A) e. RR /\ (1 / 2) e. RR) -> ((|_` A) + (1 / 2)) e. RR)
30	29, 10, 3	sylancl 525	. . . . . . . . . . 11 |- (A e. RR -> ((|_` A) + (1 / 2)) e. RR)
31		ltadd1 6806	. . . . . . . . . . 11 |- ((A e. RR /\ ((|_` A) + (1 / 2)) e. RR /\ (1 / 2) e. RR) -> (A < ((|_` A) + (1 / 2)) <-> (A + (1 / 2)) < (((|_` A) + (1 / 2)) + (1 / 2))))
32	30, 25, 31	mpd3an23 1193	. . . . . . . . . 10 |- (A e. RR -> (A < ((|_` A) + (1 / 2)) <-> (A + (1 / 2)) < (((|_` A) + (1 / 2)) + (1 / 2))))
33	32	adantr 425	. . . . . . . . 9 |- ((A e. RR /\ (A - (|_` A)) < (1 / 2)) -> (A < ((|_` A) + (1 / 2)) <-> (A + (1 / 2)) < (((|_` A) + (1 / 2)) + (1 / 2))))
34	28, 33	mpbid 212	. . . . . . . 8 |- ((A e. RR /\ (A - (|_` A)) < (1 / 2)) -> (A + (1 / 2)) < (((|_` A) + (1 / 2)) + (1 / 2)))
35	10	recnd 6468	. . . . . . . . . . 11 |- (A e. RR -> (|_` A) e. CC)
36	3	recni 6467	. . . . . . . . . . . 12 |- (1 / 2) e. CC
37		addass 6460	. . . . . . . . . . . 12 |- (((|_` A) e. CC /\ (1 / 2) e. CC /\ (1 / 2) e. CC) -> (((|_` A) + (1 / 2)) + (1 / 2)) = ((|_` A) + ((1 / 2) + (1 / 2))))
38	36, 36, 37	mp3an23 1183	. . . . . . . . . . 11 |- ((|_` A) e. CC -> (((|_` A) + (1 / 2)) + (1 / 2)) = ((|_` A) + ((1 / 2) + (1 / 2))))
39	35, 38	syl 12	. . . . . . . . . 10 |- (A e. RR -> (((|_` A) + (1 / 2)) + (1 / 2)) = ((|_` A) + ((1 / 2) + (1 / 2))))
40	36	2timesi 7187	. . . . . . . . . . . . 13 |- (2 x. (1 / 2)) = ((1 / 2) + (1 / 2))
41		2cn 7164	. . . . . . . . . . . . . 14 |- 2 e. CC
42	41, 2	recidi 6916	. . . . . . . . . . . . 13 |- (2 x. (1 / 2)) = 1
43	40, 42	eqtr3i 1910	. . . . . . . . . . . 12 |- ((1 / 2) + (1 / 2)) = 1
44	43	a1i 8	. . . . . . . . . . 11 |- (A e. RR -> ((1 / 2) + (1 / 2)) = 1)
45	44	opreq2d 4898	. . . . . . . . . 10 |- (A e. RR -> ((|_` A) + ((1 / 2) + (1 / 2))) = ((|_` A) + 1))
46	39, 45	eqtrd 1925	. . . . . . . . 9 |- (A e. RR -> (((|_` A) + (1 / 2)) + (1 / 2)) = ((|_` A) + 1))
47	46	adantr 425	. . . . . . . 8 |- ((A e. RR /\ (A - (|_` A)) < (1 / 2)) -> (((|_` A) + (1 / 2)) + (1 / 2)) = ((|_` A) + 1))
48	34, 47	breqtrd 3361	. . . . . . 7 |- ((A e. RR /\ (A - (|_` A)) < (1 / 2)) -> (A + (1 / 2)) < ((|_` A) + 1))
49	9, 24, 48	mpbir2and 802	. . . . . 6 |- ((A e. RR /\ (A - (|_` A)) < (1 / 2)) -> (|_` (A + (1 / 2))) = (|_` A))
50	49	opreq1d 4897	. . . . 5 |- ((A e. RR /\ (A - (|_` A)) < (1 / 2)) -> ((|_` (A + (1 / 2))) - A) = ((|_` A) - A))
51	50	fveq2d 4685	. . . 4 |- ((A e. RR /\ (A - (|_` A)) < (1 / 2)) -> (abs` ((|_` (A + (1 / 2))) - A)) = (abs` ((|_` A) - A)))
52		abssuble0 8148	. . . . . 6 |- (((|_` A) e. RR /\ A e. RR /\ (|_` A) <_ A) -> (abs` ((|_` A) - A)) = (A - (|_` A)))
53	10, 11, 12, 52	syl111anc 1100	. . . . 5 |- (A e. RR -> (abs` ((|_` A) - A)) = (A - (|_` A)))
54	53	adantr 425	. . . 4 |- ((A e. RR /\ (A - (|_` A)) < (1 / 2)) -> (abs` ((|_` A) - A)) = (A - (|_` A)))
55	51, 54	eqtrd 1925	. . 3 |- ((A e. RR /\ (A - (|_` A)) < (1 / 2)) -> (abs` ((|_` (A + (1 / 2))) - A)) = (A - (|_` A)))
56		ltle 6690	. . . . 5 |- (((A - (|_` A)) e. RR /\ (1 / 2) e. RR) -> ((A - (|_` A)) < (1 / 2) -> (A - (|_` A)) <_ (1 / 2)))
57		resubcl 6601	. . . . . 6 |- ((A e. RR /\ (|_` A) e. RR) -> (A - (|_` A)) e. RR)
58	10, 57	mpdan 768	. . . . 5 |- (A e. RR -> (A - (|_` A)) e. RR)
59	56, 58, 3	sylancl 525	. . . 4 |- (A e. RR -> ((A - (|_` A)) < (1 / 2) -> (A - (|_` A)) <_ (1 / 2)))
60	59	imp 377	. . 3 |- ((A e. RR /\ (A - (|_` A)) < (1 / 2)) -> (A - (|_` A)) <_ (1 / 2))
61	55, 60	eqbrtrd 3357	. 2 |- ((A e. RR /\ (A - (|_` A)) < (1 / 2)) -> (abs` ((|_` (A + (1 / 2))) - A)) <_ (1 / 2))
62		zaddcl 7374	. . . . . . . . . . 11 |- ((1 e. ZZ /\ (|_` A) e. ZZ) -> (1 + (|_` A)) e. ZZ)
63		1z 7368	. . . . . . . . . . 11 |- 1 e. ZZ
64	62, 63, 6	sylancr 526	. . . . . . . . . 10 |- (A e. RR -> (1 + (|_` A)) e. ZZ)
65		flbi 7480	. . . . . . . . . 10 |- (((A + (1 / 2)) e. RR /\ (1 + (|_` A)) e. ZZ) -> ((|_` (A + (1 / 2))) = (1 + (|_` A)) <-> ((1 + (|_` A)) <_ (A + (1 / 2)) /\ (A + (1 / 2)) < ((1 + (|_` A)) + 1))))
66	5, 64, 65	syl11anc 524	. . . . . . . . 9 |- (A e. RR -> ((|_` (A + (1 / 2))) = (1 + (|_` A)) <-> ((1 + (|_` A)) <_ (A + (1 / 2)) /\ (A + (1 / 2)) < ((1 + (|_` A)) + 1))))
67	66	adantr 425	. . . . . . . 8 |- ((A e. RR /\ (1 / 2) <_ (A - (|_` A))) -> ((|_` (A + (1 / 2))) = (1 + (|_` A)) <-> ((1 + (|_` A)) <_ (A + (1 / 2)) /\ (A + (1 / 2)) < ((1 + (|_` A)) + 1))))
68	36	a1i 8	. . . . . . . . . . . 12 |- (A e. RR -> (1 / 2) e. CC)
69	35, 68, 68, 37	syl111anc 1100	. . . . . . . . . . 11 |- (A e. RR -> (((|_` A) + (1 / 2)) + (1 / 2)) = ((|_` A) + ((1 / 2) + (1 / 2))))
70		addcom 6458	. . . . . . . . . . . 12 |- (((|_` A) e. CC /\ 1 e. CC) -> ((|_` A) + 1) = (1 + (|_` A)))
71		ax1cn 6422	. . . . . . . . . . . 12 |- 1 e. CC
72	70, 35, 71	sylancl 525	. . . . . . . . . . 11 |- (A e. RR -> ((|_` A) + 1) = (1 + (|_` A)))
73	69, 45, 72	3eqtrd 1929	. . . . . . . . . 10 |- (A e. RR -> (((|_` A) + (1 / 2)) + (1 / 2)) = (1 + (|_` A)))
74	73	adantr 425	. . . . . . . . 9 |- ((A e. RR /\ (1 / 2) <_ (A - (|_` A))) -> (((|_` A) + (1 / 2)) + (1 / 2)) = (1 + (|_` A)))
75		leaddsub2 6817	. . . . . . . . . . . 12 |- (((|_` A) e. RR /\ (1 / 2) e. RR /\ A e. RR) -> (((|_` A) + (1 / 2)) <_ A <-> (1 / 2) <_ (A - (|_` A))))
76	10, 25, 11, 75	syl111anc 1100	. . . . . . . . . . 11 |- (A e. RR -> (((|_` A) + (1 / 2)) <_ A <-> (1 / 2) <_ (A - (|_` A))))
77	76	biimpar 461	. . . . . . . . . 10 |- ((A e. RR /\ (1 / 2) <_ (A - (|_` A))) -> ((|_` A) + (1 / 2)) <_ A)
78		leadd1 6808	. . . . . . . . . . . 12 |- ((((|_` A) + (1 / 2)) e. RR /\ A e. RR /\ (1 / 2) e. RR) -> (((|_` A) + (1 / 2)) <_ A <-> (((|_` A) + (1 / 2)) + (1 / 2)) <_ (A + (1 / 2))))
79	30, 11, 25, 78	syl111anc 1100	. . . . . . . . . . 11 |- (A e. RR -> (((|_` A) + (1 / 2)) <_ A <-> (((|_` A) + (1 / 2)) + (1 / 2)) <_ (A + (1 / 2))))
80	79	adantr 425	. . . . . . . . . 10 |- ((A e. RR /\ (1 / 2) <_ (A - (|_` A))) -> (((|_` A) + (1 / 2)) <_ A <-> (((|_` A) + (1 / 2)) + (1 / 2)) <_ (A + (1 / 2))))
81	77, 80	mpbid 212	. . . . . . . . 9 |- ((A e. RR /\ (1 / 2) <_ (A - (|_` A))) -> (((|_` A) + (1 / 2)) + (1 / 2)) <_ (A + (1 / 2)))
82	74, 81	eqbrtrrd 3359	. . . . . . . 8 |- ((A e. RR /\ (1 / 2) <_ (A - (|_` A))) -> (1 + (|_` A)) <_ (A + (1 / 2)))
83		fraclt1 7470	. . . . . . . . . . . 12 |- (A e. RR -> (A - (|_` A)) < 1)
84		1re 6598	. . . . . . . . . . . . . 14 |- 1 e. RR
85	84	a1i 8	. . . . . . . . . . . . 13 |- (A e. RR -> 1 e. RR)
86		ltsubadd 6810	. . . . . . . . . . . . 13 |- ((A e. RR /\ (|_` A) e. RR /\ 1 e. RR) -> ((A - (|_` A)) < 1 <-> A < (1 + (|_` A))))
87	10, 85, 86	mpd3an23 1193	. . . . . . . . . . . 12 |- (A e. RR -> ((A - (|_` A)) < 1 <-> A < (1 + (|_` A))))
88	83, 87	mpbid 212	. . . . . . . . . . 11 |- (A e. RR -> A < (1 + (|_` A)))
89		halflt1 7216	. . . . . . . . . . 11 |- (1 / 2) < 1
90	88, 89	jctir 317	. . . . . . . . . 10 |- (A e. RR -> (A < (1 + (|_` A)) /\ (1 / 2) < 1))
91		readdcl 6455	. . . . . . . . . . . 12 |- ((1 e. RR /\ (|_` A) e. RR) -> (1 + (|_` A)) e. RR)
92	91, 84, 10	sylancr 526	. . . . . . . . . . 11 |- (A e. RR -> (1 + (|_` A)) e. RR)
93		lt2add 6827	. . . . . . . . . . 11 |- (((A e. RR /\ (1 / 2) e. RR) /\ ((1 + (|_` A)) e. RR /\ 1 e. RR)) -> ((A < (1 + (|_` A)) /\ (1 / 2) < 1) -> (A + (1 / 2)) < ((1 + (|_` A)) + 1)))
94	11, 25, 92, 85, 93	syl22anc 1101	. . . . . . . . . 10 |- (A e. RR -> ((A < (1 + (|_` A)) /\ (1 / 2) < 1) -> (A + (1 / 2)) < ((1 + (|_` A)) + 1)))
95	90, 94	mpd 29	. . . . . . . . 9 |- (A e. RR -> (A + (1 / 2)) < ((1 + (|_` A)) + 1))
96	95	adantr 425	. . . . . . . 8 |- ((A e. RR /\ (1 / 2) <_ (A - (|_` A))) -> (A + (1 / 2)) < ((1 + (|_` A)) + 1))
97	67, 82, 96	mpbir2and 802	. . . . . . 7 |- ((A e. RR /\ (1 / 2) <_ (A - (|_` A))) -> (|_` (A + (1 / 2))) = (1 + (|_` A)))
98	97	opreq1d 4897	. . . . . 6 |- ((A e. RR /\ (1 / 2) <_ (A - (|_` A))) -> ((|_` (A + (1 / 2))) - A) = ((1 + (|_` A)) - A))
99	71	a1i 8	. . . . . . . 8 |- (A e. RR -> 1 e. CC)
100		subsub3 6628	. . . . . . . 8 |- ((1 e. CC /\ A e. CC /\ (|_` A) e. CC) -> (1 - (A - (|_` A))) = ((1 + (|_` A)) - A))
101	99, 13, 35, 100	syl111anc 1100	. . . . . . 7 |- (A e. RR -> (1 - (A - (|_` A))) = ((1 + (|_` A)) - A))
102	101	adantr 425	. . . . . 6 |- ((A e. RR /\ (1 / 2) <_ (A - (|_` A))) -> (1 - (A - (|_` A))) = ((1 + (|_` A)) - A))
103	98, 102	eqtr4d 1928	. . . . 5 |- ((A e. RR /\ (1 / 2) <_ (A - (|_` A))) -> ((|_` (A + (1 / 2))) - A) = (1 - (A - (|_` A))))
104	103	fveq2d 4685	. . . 4 |- ((A e. RR /\ (1 / 2) <_ (A - (|_` A))) -> (abs` ((|_` (A + (1 / 2))) - A)) = (abs` (1 - (A - (|_` A)))))
105		resubcl 6601	. . . . . . 7 |- ((1 e. RR /\ (A - (|_` A)) e. RR) -> (1 - (A - (|_` A))) e. RR)
106	105, 84, 58	sylancr 526	. . . . . 6 |- (A e. RR -> (1 - (A - (|_` A))) e. RR)
107		ltle 6690	. . . . . . . . 9 |- (((A - (|_` A)) e. RR /\ 1 e. RR) -> ((A - (|_` A)) < 1 -> (A - (|_` A)) <_ 1))
108	107, 58, 84	sylancl 525	. . . . . . . 8 |- (A e. RR -> ((A - (|_` A)) < 1 -> (A - (|_` A)) <_ 1))
109	83, 108	mpd 29	. . . . . . 7 |- (A e. RR -> (A - (|_` A)) <_ 1)
110		subge0 6863	. . . . . . . 8 |- ((1 e. RR /\ (A - (|_` A)) e. RR) -> (0 <_ (1 - (A - (|_` A))) <-> (A - (|_` A)) <_ 1))
111	110, 84, 58	sylancr 526	. . . . . . 7 |- (A e. RR -> (0 <_ (1 - (A - (|_` A))) <-> (A - (|_` A)) <_ 1))
112	109, 111	mpbird 213	. . . . . 6 |- (A e. RR -> 0 <_ (1 - (A - (|_` A))))
113		absid 8113	. . . . . 6 |- (((1 - (A - (|_` A))) e. RR /\ 0 <_ (1 - (A - (|_` A)))) -> (abs` (1 - (A - (|_` A)))) = (1 - (A - (|_` A))))
114	106, 112, 113	syl11anc 524	. . . . 5 |- (A e. RR -> (abs` (1 - (A - (|_` A)))) = (1 - (A - (|_` A))))
115	114	adantr 425	. . . 4 |- ((A e. RR /\ (1 / 2) <_ (A - (|_` A))) -> (abs` (1 - (A - (|_` A)))) = (1 - (A - (|_` A))))
116	104, 115	eqtrd 1925	. . 3 |- ((A e. RR /\ (1 / 2) <_ (A - (|_` A))) -> (abs` ((|_` (A + (1 / 2))) - A)) = (1 - (A - (|_` A))))
117		lesub2 6850	. . . . . 6 |- (((1 / 2) e. RR /\ (A - (|_` A)) e. RR /\ 1 e. RR) -> ((1 / 2) <_ (A - (|_` A)) <-> (1 - (A - (|_` A))) <_ (1 - (1 / 2))))
118	25, 58, 85, 117	syl111anc 1100	. . . . 5 |- (A e. RR -> ((1 / 2) <_ (A - (|_` A)) <-> (1 - (A - (|_` A))) <_ (1 - (1 / 2))))
119	118	biimpa 460	. . . 4 |- ((A e. RR /\ (1 / 2) <_ (A - (|_` A))) -> (1 - (A - (|_` A))) <_ (1 - (1 / 2)))
120	71, 36, 36, 43	subaddrii 6529	. . . 4 |- (1 - (1 / 2)) = (1 / 2)
121	119, 120	syl6breq 3376	. . 3 |- ((A e. RR /\ (1 / 2) <_ (A - (|_` A))) -> (1 - (A - (|_` A))) <_ (1 / 2))
122	116, 121	eqbrtrd 3357	. 2 |- ((A e. RR /\ (1 / 2) <_ (A - (|_` A))) -> (abs` ((|_` (A + (1 / 2))) - A)) <_ (1 / 2))
123	61, 122, 58, 25	pm2.61ltlei 6705	1 |- (A e. RR -> (abs` ((|_` (A + (1 / 2))) - A)) <_ (1 / 2))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300   class class class wbr 3338  ` cfv 3998  (class class class)co 4884  CCcc 6384  RRcr 6385  0cc0 6386  1c1 6387   + caddc 6389   x. cmul 6391   - cmin 6445   / cdiv 6447   <_ cle 6448  ZZcz 6451   < clt 6653  2c2 7145  |_cfl 7462  abscabs 8000

This theorem is referenced by:  rrntotbndlem2 16021

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-sup 5664  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-div 6892  df-n 7108  df-2 7154  df-n0 7309  df-z 7345  df-fl 7463  df-sqr 7920  df-re 8001  df-im 8002  df-cj 8003  df-abs 8004

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