Theorem absrdbnd 15799

Description: Bound on the absolute value of a real number rounded to the nearest integer.

Assertion

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

Proof of Theorem absrdbnd

Step	Hyp	Ref	Expression
1		2re 7163	. . . . . . . . 9 |- 2 e. RR
2		2ne0 7174	. . . . . . . . 9 |- 2 =/= 0
3	1, 2	rereccli 6979	. . . . . . . 8 |- (1 / 2) e. RR
4		readdcl 6455	. . . . . . . 8 |- ((A e. RR /\ (1 / 2) e. RR) -> (A + (1 / 2)) e. RR)
5	3, 4	mpan2 760	. . . . . . 7 |- (A e. RR -> (A + (1 / 2)) e. RR)
6		flcl 7465	. . . . . . 7 |- ((A + (1 / 2)) e. RR -> (|_` (A + (1 / 2))) e. ZZ)
7		zre 7348	. . . . . . 7 |- ((|_` (A + (1 / 2))) e. ZZ -> (|_` (A + (1 / 2))) e. RR)
8	5, 6, 7	3syl 24	. . . . . 6 |- (A e. RR -> (|_` (A + (1 / 2))) e. RR)
9		renegcl 6600	. . . . . 6 |- ((|_` (A + (1 / 2))) e. RR -> -u(|_` (A + (1 / 2))) e. RR)
10	8, 9	syl 12	. . . . 5 |- (A e. RR -> -u(|_` (A + (1 / 2))) e. RR)
11	10	adantr 425	. . . 4 |- ((A e. RR /\ A < 0) -> -u(|_` (A + (1 / 2))) e. RR)
12		reflcl 7466	. . . . . 6 |- (A e. RR -> (|_` A) e. RR)
13		renegcl 6600	. . . . . 6 |- ((|_` A) e. RR -> -u(|_` A) e. RR)
14	12, 13	syl 12	. . . . 5 |- (A e. RR -> -u(|_` A) e. RR)
15	14	adantr 425	. . . 4 |- ((A e. RR /\ A < 0) -> -u(|_` A) e. RR)
16		renegcl 6600	. . . . . . 7 |- (A e. RR -> -uA e. RR)
17		flcl 7465	. . . . . . 7 |- (-uA e. RR -> (|_` -uA) e. ZZ)
18		zre 7348	. . . . . . 7 |- ((|_` -uA) e. ZZ -> (|_` -uA) e. RR)
19	16, 17, 18	3syl 24	. . . . . 6 |- (A e. RR -> (|_` -uA) e. RR)
20		peano2re 6599	. . . . . 6 |- ((|_` -uA) e. RR -> ((|_` -uA) + 1) e. RR)
21	19, 20	syl 12	. . . . 5 |- (A e. RR -> ((|_` -uA) + 1) e. RR)
22	21	adantr 425	. . . 4 |- ((A e. RR /\ A < 0) -> ((|_` -uA) + 1) e. RR)
23		recn 6466	. . . . . . . . 9 |- (A e. RR -> A e. CC)
24		addid1 6463	. . . . . . . . 9 |- (A e. CC -> (A + 0) = A)
25	23, 24	syl 12	. . . . . . . 8 |- (A e. RR -> (A + 0) = A)
26		0re 6603	. . . . . . . . . . 11 |- 0 e. RR
27		1re 6598	. . . . . . . . . . 11 |- 1 e. RR
28		lt01 6871	. . . . . . . . . . 11 |- 0 < 1
29	26, 27, 28	ltleii 6756	. . . . . . . . . 10 |- 0 <_ 1
30		2pos 7173	. . . . . . . . . 10 |- 0 < 2
31	27, 1	divge0i 7040	. . . . . . . . . 10 |- ((0 <_ 1 /\ 0 < 2) -> 0 <_ (1 / 2))
32	29, 30, 31	mp2an 761	. . . . . . . . 9 |- 0 <_ (1 / 2)
33		leadd2 6809	. . . . . . . . . 10 |- ((0 e. RR /\ (1 / 2) e. RR /\ A e. RR) -> (0 <_ (1 / 2) <-> (A + 0) <_ (A + (1 / 2))))
34	26, 3, 33	mp3an12 1181	. . . . . . . . 9 |- (A e. RR -> (0 <_ (1 / 2) <-> (A + 0) <_ (A + (1 / 2))))
35	32, 34	mpbii 210	. . . . . . . 8 |- (A e. RR -> (A + 0) <_ (A + (1 / 2)))
36	25, 35	eqbrtrrd 3359	. . . . . . 7 |- (A e. RR -> A <_ (A + (1 / 2)))
37		flwordi 7477	. . . . . . 7 |- ((A e. RR /\ (A + (1 / 2)) e. RR /\ A <_ (A + (1 / 2))) -> (|_` A) <_ (|_` (A + (1 / 2))))
38	5, 36, 37	mpd3an23 1193	. . . . . 6 |- (A e. RR -> (|_` A) <_ (|_` (A + (1 / 2))))
39		leneg 6846	. . . . . . 7 |- (((|_` A) e. RR /\ (|_` (A + (1 / 2))) e. RR) -> ((|_` A) <_ (|_` (A + (1 / 2))) <-> -u(|_` (A + (1 / 2))) <_ -u(|_` A)))
40	12, 8, 39	syl11anc 524	. . . . . 6 |- (A e. RR -> ((|_` A) <_ (|_` (A + (1 / 2))) <-> -u(|_` (A + (1 / 2))) <_ -u(|_` A)))
41	38, 40	mpbid 212	. . . . 5 |- (A e. RR -> -u(|_` (A + (1 / 2))) <_ -u(|_` A))
42	41	adantr 425	. . . 4 |- ((A e. RR /\ A < 0) -> -u(|_` (A + (1 / 2))) <_ -u(|_` A))
43		flcl 7465	. . . . . . . . . 10 |- (A e. RR -> (|_` A) e. ZZ)
44	43	peano2zdi 7376	. . . . . . . . 9 |- (A e. RR -> ((|_` A) + 1) e. ZZ)
45		flltp1 7469	. . . . . . . . . 10 |- (A e. RR -> A < ((|_` A) + 1))
46		peano2re 6599	. . . . . . . . . . . 12 |- ((|_` A) e. RR -> ((|_` A) + 1) e. RR)
47	12, 46	syl 12	. . . . . . . . . . 11 |- (A e. RR -> ((|_` A) + 1) e. RR)
48		ltle 6690	. . . . . . . . . . 11 |- ((A e. RR /\ ((|_` A) + 1) e. RR) -> (A < ((|_` A) + 1) -> A <_ ((|_` A) + 1)))
49	47, 48	mpdan 768	. . . . . . . . . 10 |- (A e. RR -> (A < ((|_` A) + 1) -> A <_ ((|_` A) + 1)))
50	45, 49	mpd 29	. . . . . . . . 9 |- (A e. RR -> A <_ ((|_` A) + 1))
51		ceile 7491	. . . . . . . . 9 |- ((A e. RR /\ ((|_` A) + 1) e. ZZ /\ A <_ ((|_` A) + 1)) -> -u(|_` -uA) <_ ((|_` A) + 1))
52	44, 50, 51	mpd3an23 1193	. . . . . . . 8 |- (A e. RR -> -u(|_` -uA) <_ ((|_` A) + 1))
53		renegcl 6600	. . . . . . . . . 10 |- ((|_` -uA) e. RR -> -u(|_` -uA) e. RR)
54	19, 53	syl 12	. . . . . . . . 9 |- (A e. RR -> -u(|_` -uA) e. RR)
55		leneg 6846	. . . . . . . . 9 |- ((-u(|_` -uA) e. RR /\ ((|_` A) + 1) e. RR) -> (-u(|_` -uA) <_ ((|_` A) + 1) <-> -u((|_` A) + 1) <_ -u-u(|_` -uA)))
56	54, 47, 55	syl11anc 524	. . . . . . . 8 |- (A e. RR -> (-u(|_` -uA) <_ ((|_` A) + 1) <-> -u((|_` A) + 1) <_ -u-u(|_` -uA)))
57	52, 56	mpbid 212	. . . . . . 7 |- (A e. RR -> -u((|_` A) + 1) <_ -u-u(|_` -uA))
58		negdi2 6621	. . . . . . . 8 |- (((|_` A) e. CC /\ 1 e. CC) -> -u((|_` A) + 1) = (-u(|_` A) - 1))
59	12	recnd 6468	. . . . . . . 8 |- (A e. RR -> (|_` A) e. CC)
60		ax1cn 6422	. . . . . . . 8 |- 1 e. CC
61	58, 59, 60	sylancl 525	. . . . . . 7 |- (A e. RR -> -u((|_` A) + 1) = (-u(|_` A) - 1))
62		zcn 7349	. . . . . . . . 9 |- ((|_` -uA) e. ZZ -> (|_` -uA) e. CC)
63	16, 17, 62	3syl 24	. . . . . . . 8 |- (A e. RR -> (|_` -uA) e. CC)
64		negneg 6553	. . . . . . . 8 |- ((|_` -uA) e. CC -> -u-u(|_` -uA) = (|_` -uA))
65	63, 64	syl 12	. . . . . . 7 |- (A e. RR -> -u-u(|_` -uA) = (|_` -uA))
66	57, 61, 65	3brtr3d 3366	. . . . . 6 |- (A e. RR -> (-u(|_` A) - 1) <_ (|_` -uA))
67	27	a1i 8	. . . . . . 7 |- (A e. RR -> 1 e. RR)
68		lesubadd 6812	. . . . . . 7 |- ((-u(|_` A) e. RR /\ 1 e. RR /\ (|_` -uA) e. RR) -> ((-u(|_` A) - 1) <_ (|_` -uA) <-> -u(|_` A) <_ ((|_` -uA) + 1)))
69	14, 67, 19, 68	syl111anc 1100	. . . . . 6 |- (A e. RR -> ((-u(|_` A) - 1) <_ (|_` -uA) <-> -u(|_` A) <_ ((|_` -uA) + 1)))
70	66, 69	mpbid 212	. . . . 5 |- (A e. RR -> -u(|_` A) <_ ((|_` -uA) + 1))
71	70	adantr 425	. . . 4 |- ((A e. RR /\ A < 0) -> -u(|_` A) <_ ((|_` -uA) + 1))
72	11, 15, 22, 42, 71	letrd 6696	. . 3 |- ((A e. RR /\ A < 0) -> -u(|_` (A + (1 / 2))) <_ ((|_` -uA) + 1))
73	5	adantr 425	. . . . 5 |- ((A e. RR /\ A < 0) -> (A + (1 / 2)) e. RR)
74	73, 6, 7	3syl 24	. . . 4 |- ((A e. RR /\ A < 0) -> (|_` (A + (1 / 2))) e. RR)
75	3	a1i 8	. . . . . 6 |- ((A e. RR /\ A < 0) -> (1 / 2) e. RR)
76		ltle 6690	. . . . . . . . . 10 |- ((A e. RR /\ 0 e. RR) -> (A < 0 -> A <_ 0))
77	26, 76	mpan2 760	. . . . . . . . 9 |- (A e. RR -> (A < 0 -> A <_ 0))
78	77	imp 377	. . . . . . . 8 |- ((A e. RR /\ A < 0) -> A <_ 0)
79		leadd1 6808	. . . . . . . . . 10 |- ((A e. RR /\ 0 e. RR /\ (1 / 2) e. RR) -> (A <_ 0 <-> (A + (1 / 2)) <_ (0 + (1 / 2))))
80	26, 3, 79	mp3an23 1183	. . . . . . . . 9 |- (A e. RR -> (A <_ 0 <-> (A + (1 / 2)) <_ (0 + (1 / 2))))
81	80	adantr 425	. . . . . . . 8 |- ((A e. RR /\ A < 0) -> (A <_ 0 <-> (A + (1 / 2)) <_ (0 + (1 / 2))))
82	78, 81	mpbid 212	. . . . . . 7 |- ((A e. RR /\ A < 0) -> (A + (1 / 2)) <_ (0 + (1 / 2)))
83	3	recni 6467	. . . . . . . 8 |- (1 / 2) e. CC
84	83	addid2i 6484	. . . . . . 7 |- (0 + (1 / 2)) = (1 / 2)
85	82, 84	syl6breq 3376	. . . . . 6 |- ((A e. RR /\ A < 0) -> (A + (1 / 2)) <_ (1 / 2))
86		flwordi 7477	. . . . . 6 |- (((A + (1 / 2)) e. RR /\ (1 / 2) e. RR /\ (A + (1 / 2)) <_ (1 / 2)) -> (|_` (A + (1 / 2))) <_ (|_` (1 / 2)))
87	73, 75, 85, 86	syl111anc 1100	. . . . 5 |- ((A e. RR /\ A < 0) -> (|_` (A + (1 / 2))) <_ (|_` (1 / 2)))
88		0z 7355	. . . . . . 7 |- 0 e. ZZ
89		flbi 7480	. . . . . . 7 |- (((1 / 2) e. RR /\ 0 e. ZZ) -> ((|_` (1 / 2)) = 0 <-> (0 <_ (1 / 2) /\ (1 / 2) < (0 + 1))))
90	3, 88, 89	mp2an 761	. . . . . 6 |- ((|_` (1 / 2)) = 0 <-> (0 <_ (1 / 2) /\ (1 / 2) < (0 + 1)))
91		halflt1 7216	. . . . . . 7 |- (1 / 2) < 1
92	60	addid2i 6484	. . . . . . 7 |- (0 + 1) = 1
93	91, 92	breqtrri 3362	. . . . . 6 |- (1 / 2) < (0 + 1)
94	90, 32, 93	mpbir2an 800	. . . . 5 |- (|_` (1 / 2)) = 0
95	87, 94	syl6breq 3376	. . . 4 |- ((A e. RR /\ A < 0) -> (|_` (A + (1 / 2))) <_ 0)
96		absnid 8114	. . . 4 |- (((|_` (A + (1 / 2))) e. RR /\ (|_` (A + (1 / 2))) <_ 0) -> (abs` (|_` (A + (1 / 2)))) = -u(|_` (A + (1 / 2))))
97	74, 95, 96	syl11anc 524	. . 3 |- ((A e. RR /\ A < 0) -> (abs` (|_` (A + (1 / 2)))) = -u(|_` (A + (1 / 2))))
98		absnid 8114	. . . . . 6 |- ((A e. RR /\ A <_ 0) -> (abs` A) = -uA)
99	78, 98	syldan 516	. . . . 5 |- ((A e. RR /\ A < 0) -> (abs` A) = -uA)
100	99	fveq2d 4685	. . . 4 |- ((A e. RR /\ A < 0) -> (|_` (abs` A)) = (|_` -uA))
101	100	opreq1d 4897	. . 3 |- ((A e. RR /\ A < 0) -> ((|_` (abs` A)) + 1) = ((|_` -uA) + 1))
102	72, 97, 101	3brtr4d 3367	. 2 |- ((A e. RR /\ A < 0) -> (abs` (|_` (A + (1 / 2)))) <_ ((|_` (abs` A)) + 1))
103		peano2re 6599	. . . . . 6 |- (A e. RR -> (A + 1) e. RR)
104	3, 27, 91	ltleii 6756	. . . . . . 7 |- (1 / 2) <_ 1
105		leadd2 6809	. . . . . . . 8 |- (((1 / 2) e. RR /\ 1 e. RR /\ A e. RR) -> ((1 / 2) <_ 1 <-> (A + (1 / 2)) <_ (A + 1)))
106	3, 27, 105	mp3an12 1181	. . . . . . 7 |- (A e. RR -> ((1 / 2) <_ 1 <-> (A + (1 / 2)) <_ (A + 1)))
107	104, 106	mpbii 210	. . . . . 6 |- (A e. RR -> (A + (1 / 2)) <_ (A + 1))
108		flwordi 7477	. . . . . 6 |- (((A + (1 / 2)) e. RR /\ (A + 1) e. RR /\ (A + (1 / 2)) <_ (A + 1)) -> (|_` (A + (1 / 2))) <_ (|_` (A + 1)))
109	5, 103, 107, 108	syl111anc 1100	. . . . 5 |- (A e. RR -> (|_` (A + (1 / 2))) <_ (|_` (A + 1)))
110		1z 7368	. . . . . 6 |- 1 e. ZZ
111		fladdz 7484	. . . . . 6 |- ((A e. RR /\ 1 e. ZZ) -> (|_` (A + 1)) = ((|_` A) + 1))
112	110, 111	mpan2 760	. . . . 5 |- (A e. RR -> (|_` (A + 1)) = ((|_` A) + 1))
113	109, 112	breqtrd 3361	. . . 4 |- (A e. RR -> (|_` (A + (1 / 2))) <_ ((|_` A) + 1))
114	113	adantr 425	. . 3 |- ((A e. RR /\ 0 <_ A) -> (|_` (A + (1 / 2))) <_ ((|_` A) + 1))
115	5	adantr 425	. . . . 5 |- ((A e. RR /\ 0 <_ A) -> (A + (1 / 2)) e. RR)
116	115, 6, 7	3syl 24	. . . 4 |- ((A e. RR /\ 0 <_ A) -> (|_` (A + (1 / 2))) e. RR)
117		addge0 6837	. . . . . . 7 |- (((A e. RR /\ (1 / 2) e. RR) /\ (0 <_ A /\ 0 <_ (1 / 2))) -> 0 <_ (A + (1 / 2)))
118	117	an4s 566	. . . . . 6 |- (((A e. RR /\ 0 <_ A) /\ ((1 / 2) e. RR /\ 0 <_ (1 / 2))) -> 0 <_ (A + (1 / 2)))
119	3, 32, 118	mpanr12 778	. . . . 5 |- ((A e. RR /\ 0 <_ A) -> 0 <_ (A + (1 / 2)))
120		flge 7472	. . . . . 6 |- (((A + (1 / 2)) e. RR /\ 0 e. ZZ) -> (0 <_ (A + (1 / 2)) <-> 0 <_ (|_` (A + (1 / 2)))))
121	120, 115, 88	sylancl 525	. . . . 5 |- ((A e. RR /\ 0 <_ A) -> (0 <_ (A + (1 / 2)) <-> 0 <_ (|_` (A + (1 / 2)))))
122	119, 121	mpbid 212	. . . 4 |- ((A e. RR /\ 0 <_ A) -> 0 <_ (|_` (A + (1 / 2))))
123		absid 8113	. . . 4 |- (((|_` (A + (1 / 2))) e. RR /\ 0 <_ (|_` (A + (1 / 2)))) -> (abs` (|_` (A + (1 / 2)))) = (|_` (A + (1 / 2))))
124	116, 122, 123	syl11anc 524	. . 3 |- ((A e. RR /\ 0 <_ A) -> (abs` (|_` (A + (1 / 2)))) = (|_` (A + (1 / 2))))
125		absid 8113	. . . . 5 |- ((A e. RR /\ 0 <_ A) -> (abs` A) = A)
126	125	fveq2d 4685	. . . 4 |- ((A e. RR /\ 0 <_ A) -> (|_` (abs` A)) = (|_` A))
127	126	opreq1d 4897	. . 3 |- ((A e. RR /\ 0 <_ A) -> ((|_` (abs` A)) + 1) = ((|_` A) + 1))
128	114, 124, 127	3brtr4d 3367	. 2 |- ((A e. RR /\ 0 <_ A) -> (abs` (|_` (A + (1 / 2)))) <_ ((|_` (abs` A)) + 1))
129		id 73	. 2 |- (A e. RR -> A e. RR)
130	26	a1i 8	. 2 |- (A e. RR -> 0 e. RR)
131	102, 128, 129, 130	pm2.61ltlei 6705	1 |- (A e. RR -> (abs` (|_` (A + (1 / 2)))) <_ ((|_` (abs` A)) + 1))

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   - cmin 6445  -ucneg 6446   / cdiv 6447   <_ cle 6448  ZZcz 6451   < clt 6653  2c2 7145  |_cfl 7462  abscabs 8000

This theorem is referenced by:  rrntotbndlem1 16020

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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