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Theorem absrdbnd 12147
Description: Bound on the absolute value of a real number rounded to the nearest integer. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 14-Sep-2015.)
Assertion
Ref Expression
absrdbnd  |-  ( A  e.  RR  ->  ( abs `  ( |_ `  ( A  +  (
1  /  2 ) ) ) )  <_ 
( ( |_ `  ( abs `  A ) )  +  1 ) )

Proof of Theorem absrdbnd
StepHypRef Expression
1 2re 10071 . . . . . . . . 9  |-  2  e.  RR
2 2ne0 10085 . . . . . . . . 9  |-  2  =/=  0
31, 2rereccli 9781 . . . . . . . 8  |-  ( 1  /  2 )  e.  RR
4 readdcl 9075 . . . . . . . 8  |-  ( ( A  e.  RR  /\  ( 1  /  2
)  e.  RR )  ->  ( A  +  ( 1  /  2
) )  e.  RR )
53, 4mpan2 654 . . . . . . 7  |-  ( A  e.  RR  ->  ( A  +  ( 1  /  2 ) )  e.  RR )
6 reflcl 11207 . . . . . . 7  |-  ( ( A  +  ( 1  /  2 ) )  e.  RR  ->  ( |_ `  ( A  +  ( 1  /  2
) ) )  e.  RR )
75, 6syl 16 . . . . . 6  |-  ( A  e.  RR  ->  ( |_ `  ( A  +  ( 1  /  2
) ) )  e.  RR )
87recnd 9116 . . . . 5  |-  ( A  e.  RR  ->  ( |_ `  ( A  +  ( 1  /  2
) ) )  e.  CC )
9 abscl 12085 . . . . 5  |-  ( ( |_ `  ( A  +  ( 1  / 
2 ) ) )  e.  CC  ->  ( abs `  ( |_ `  ( A  +  (
1  /  2 ) ) ) )  e.  RR )
108, 9syl 16 . . . 4  |-  ( A  e.  RR  ->  ( abs `  ( |_ `  ( A  +  (
1  /  2 ) ) ) )  e.  RR )
11 recn 9082 . . . . 5  |-  ( A  e.  RR  ->  A  e.  CC )
12 abscl 12085 . . . . 5  |-  ( A  e.  CC  ->  ( abs `  A )  e.  RR )
1311, 12syl 16 . . . 4  |-  ( A  e.  RR  ->  ( abs `  A )  e.  RR )
14 1re 9092 . . . . 5  |-  1  e.  RR
1514a1i 11 . . . 4  |-  ( A  e.  RR  ->  1  e.  RR )
1610, 13resubcld 9467 . . . . 5  |-  ( A  e.  RR  ->  (
( abs `  ( |_ `  ( A  +  ( 1  /  2
) ) ) )  -  ( abs `  A
) )  e.  RR )
17 resubcl 9367 . . . . . . . 8  |-  ( ( ( |_ `  ( A  +  ( 1  /  2 ) ) )  e.  RR  /\  A  e.  RR )  ->  ( ( |_ `  ( A  +  (
1  /  2 ) ) )  -  A
)  e.  RR )
187, 17mpancom 652 . . . . . . 7  |-  ( A  e.  RR  ->  (
( |_ `  ( A  +  ( 1  /  2 ) ) )  -  A )  e.  RR )
1918recnd 9116 . . . . . 6  |-  ( A  e.  RR  ->  (
( |_ `  ( A  +  ( 1  /  2 ) ) )  -  A )  e.  CC )
20 abscl 12085 . . . . . 6  |-  ( ( ( |_ `  ( A  +  ( 1  /  2 ) ) )  -  A )  e.  CC  ->  ( abs `  ( ( |_
`  ( A  +  ( 1  /  2
) ) )  -  A ) )  e.  RR )
2119, 20syl 16 . . . . 5  |-  ( A  e.  RR  ->  ( abs `  ( ( |_
`  ( A  +  ( 1  /  2
) ) )  -  A ) )  e.  RR )
22 abs2dif 12138 . . . . . 6  |-  ( ( ( |_ `  ( A  +  ( 1  /  2 ) ) )  e.  CC  /\  A  e.  CC )  ->  ( ( abs `  ( |_ `  ( A  +  ( 1  /  2
) ) ) )  -  ( abs `  A
) )  <_  ( abs `  ( ( |_
`  ( A  +  ( 1  /  2
) ) )  -  A ) ) )
238, 11, 22syl2anc 644 . . . . 5  |-  ( A  e.  RR  ->  (
( abs `  ( |_ `  ( A  +  ( 1  /  2
) ) ) )  -  ( abs `  A
) )  <_  ( abs `  ( ( |_
`  ( A  +  ( 1  /  2
) ) )  -  A ) ) )
243a1i 11 . . . . . 6  |-  ( A  e.  RR  ->  (
1  /  2 )  e.  RR )
25 rddif 12146 . . . . . 6  |-  ( A  e.  RR  ->  ( abs `  ( ( |_
`  ( A  +  ( 1  /  2
) ) )  -  A ) )  <_ 
( 1  /  2
) )
26 halflt1 10191 . . . . . . . 8  |-  ( 1  /  2 )  <  1
273, 14, 26ltleii 9198 . . . . . . 7  |-  ( 1  /  2 )  <_ 
1
2827a1i 11 . . . . . 6  |-  ( A  e.  RR  ->  (
1  /  2 )  <_  1 )
2921, 24, 15, 25, 28letrd 9229 . . . . 5  |-  ( A  e.  RR  ->  ( abs `  ( ( |_
`  ( A  +  ( 1  /  2
) ) )  -  A ) )  <_ 
1 )
3016, 21, 15, 23, 29letrd 9229 . . . 4  |-  ( A  e.  RR  ->  (
( abs `  ( |_ `  ( A  +  ( 1  /  2
) ) ) )  -  ( abs `  A
) )  <_  1
)
3110, 13, 15, 30subled 9631 . . 3  |-  ( A  e.  RR  ->  (
( abs `  ( |_ `  ( A  +  ( 1  /  2
) ) ) )  -  1 )  <_ 
( abs `  A
) )
325flcld 11209 . . . . . . 7  |-  ( A  e.  RR  ->  ( |_ `  ( A  +  ( 1  /  2
) ) )  e.  ZZ )
33 nn0abscl 12119 . . . . . . 7  |-  ( ( |_ `  ( A  +  ( 1  / 
2 ) ) )  e.  ZZ  ->  ( abs `  ( |_ `  ( A  +  (
1  /  2 ) ) ) )  e. 
NN0 )
3432, 33syl 16 . . . . . 6  |-  ( A  e.  RR  ->  ( abs `  ( |_ `  ( A  +  (
1  /  2 ) ) ) )  e. 
NN0 )
3534nn0zd 10375 . . . . 5  |-  ( A  e.  RR  ->  ( abs `  ( |_ `  ( A  +  (
1  /  2 ) ) ) )  e.  ZZ )
36 peano2zm 10322 . . . . 5  |-  ( ( abs `  ( |_
`  ( A  +  ( 1  /  2
) ) ) )  e.  ZZ  ->  (
( abs `  ( |_ `  ( A  +  ( 1  /  2
) ) ) )  -  1 )  e.  ZZ )
3735, 36syl 16 . . . 4  |-  ( A  e.  RR  ->  (
( abs `  ( |_ `  ( A  +  ( 1  /  2
) ) ) )  -  1 )  e.  ZZ )
38 flge 11216 . . . 4  |-  ( ( ( abs `  A
)  e.  RR  /\  ( ( abs `  ( |_ `  ( A  +  ( 1  /  2
) ) ) )  -  1 )  e.  ZZ )  ->  (
( ( abs `  ( |_ `  ( A  +  ( 1  /  2
) ) ) )  -  1 )  <_ 
( abs `  A
)  <->  ( ( abs `  ( |_ `  ( A  +  ( 1  /  2 ) ) ) )  -  1 )  <_  ( |_ `  ( abs `  A
) ) ) )
3913, 37, 38syl2anc 644 . . 3  |-  ( A  e.  RR  ->  (
( ( abs `  ( |_ `  ( A  +  ( 1  /  2
) ) ) )  -  1 )  <_ 
( abs `  A
)  <->  ( ( abs `  ( |_ `  ( A  +  ( 1  /  2 ) ) ) )  -  1 )  <_  ( |_ `  ( abs `  A
) ) ) )
4031, 39mpbid 203 . 2  |-  ( A  e.  RR  ->  (
( abs `  ( |_ `  ( A  +  ( 1  /  2
) ) ) )  -  1 )  <_ 
( |_ `  ( abs `  A ) ) )
41 reflcl 11207 . . . 4  |-  ( ( abs `  A )  e.  RR  ->  ( |_ `  ( abs `  A
) )  e.  RR )
4213, 41syl 16 . . 3  |-  ( A  e.  RR  ->  ( |_ `  ( abs `  A
) )  e.  RR )
4310, 15, 42lesubaddd 9625 . 2  |-  ( A  e.  RR  ->  (
( ( abs `  ( |_ `  ( A  +  ( 1  /  2
) ) ) )  -  1 )  <_ 
( |_ `  ( abs `  A ) )  <-> 
( abs `  ( |_ `  ( A  +  ( 1  /  2
) ) ) )  <_  ( ( |_
`  ( abs `  A
) )  +  1 ) ) )
4440, 43mpbid 203 1  |-  ( A  e.  RR  ->  ( abs `  ( |_ `  ( A  +  (
1  /  2 ) ) ) )  <_ 
( ( |_ `  ( abs `  A ) )  +  1 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    e. wcel 1726   class class class wbr 4214   ` cfv 5456  (class class class)co 6083   CCcc 8990   RRcr 8991   1c1 8993    + caddc 8995    <_ cle 9123    - cmin 9293    / cdiv 9679   2c2 10051   NN0cn0 10223   ZZcz 10284   |_cfl 11203   abscabs 12041
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-sup 7448  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-3 10061  df-n0 10224  df-z 10285  df-uz 10491  df-rp 10615  df-fl 11204  df-seq 11326  df-exp 11385  df-cj 11906  df-re 11907  df-im 11908  df-sqr 12042  df-abs 12043
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