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Theorem absrdbnd 12821
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 halfre 10532 . . . . . . . 8  |-  ( 1  /  2 )  e.  RR
2 readdcl 9357 . . . . . . . 8  |-  ( ( A  e.  RR  /\  ( 1  /  2
)  e.  RR )  ->  ( A  +  ( 1  /  2
) )  e.  RR )
31, 2mpan2 671 . . . . . . 7  |-  ( A  e.  RR  ->  ( A  +  ( 1  /  2 ) )  e.  RR )
4 reflcl 11638 . . . . . . 7  |-  ( ( A  +  ( 1  /  2 ) )  e.  RR  ->  ( |_ `  ( A  +  ( 1  /  2
) ) )  e.  RR )
53, 4syl 16 . . . . . 6  |-  ( A  e.  RR  ->  ( |_ `  ( A  +  ( 1  /  2
) ) )  e.  RR )
65recnd 9404 . . . . 5  |-  ( A  e.  RR  ->  ( |_ `  ( A  +  ( 1  /  2
) ) )  e.  CC )
7 abscl 12759 . . . . 5  |-  ( ( |_ `  ( A  +  ( 1  / 
2 ) ) )  e.  CC  ->  ( abs `  ( |_ `  ( A  +  (
1  /  2 ) ) ) )  e.  RR )
86, 7syl 16 . . . 4  |-  ( A  e.  RR  ->  ( abs `  ( |_ `  ( A  +  (
1  /  2 ) ) ) )  e.  RR )
9 recn 9364 . . . . 5  |-  ( A  e.  RR  ->  A  e.  CC )
10 abscl 12759 . . . . 5  |-  ( A  e.  CC  ->  ( abs `  A )  e.  RR )
119, 10syl 16 . . . 4  |-  ( A  e.  RR  ->  ( abs `  A )  e.  RR )
12 1re 9377 . . . . 5  |-  1  e.  RR
1312a1i 11 . . . 4  |-  ( A  e.  RR  ->  1  e.  RR )
148, 11resubcld 9768 . . . . 5  |-  ( A  e.  RR  ->  (
( abs `  ( |_ `  ( A  +  ( 1  /  2
) ) ) )  -  ( abs `  A
) )  e.  RR )
15 resubcl 9665 . . . . . . . 8  |-  ( ( ( |_ `  ( A  +  ( 1  /  2 ) ) )  e.  RR  /\  A  e.  RR )  ->  ( ( |_ `  ( A  +  (
1  /  2 ) ) )  -  A
)  e.  RR )
165, 15mpancom 669 . . . . . . 7  |-  ( A  e.  RR  ->  (
( |_ `  ( A  +  ( 1  /  2 ) ) )  -  A )  e.  RR )
1716recnd 9404 . . . . . 6  |-  ( A  e.  RR  ->  (
( |_ `  ( A  +  ( 1  /  2 ) ) )  -  A )  e.  CC )
18 abscl 12759 . . . . . 6  |-  ( ( ( |_ `  ( A  +  ( 1  /  2 ) ) )  -  A )  e.  CC  ->  ( abs `  ( ( |_
`  ( A  +  ( 1  /  2
) ) )  -  A ) )  e.  RR )
1917, 18syl 16 . . . . 5  |-  ( A  e.  RR  ->  ( abs `  ( ( |_
`  ( A  +  ( 1  /  2
) ) )  -  A ) )  e.  RR )
20 abs2dif 12812 . . . . . 6  |-  ( ( ( |_ `  ( A  +  ( 1  /  2 ) ) )  e.  CC  /\  A  e.  CC )  ->  ( ( abs `  ( |_ `  ( A  +  ( 1  /  2
) ) ) )  -  ( abs `  A
) )  <_  ( abs `  ( ( |_
`  ( A  +  ( 1  /  2
) ) )  -  A ) ) )
216, 9, 20syl2anc 661 . . . . 5  |-  ( A  e.  RR  ->  (
( abs `  ( |_ `  ( A  +  ( 1  /  2
) ) ) )  -  ( abs `  A
) )  <_  ( abs `  ( ( |_
`  ( A  +  ( 1  /  2
) ) )  -  A ) ) )
221a1i 11 . . . . . 6  |-  ( A  e.  RR  ->  (
1  /  2 )  e.  RR )
23 rddif 12820 . . . . . 6  |-  ( A  e.  RR  ->  ( abs `  ( ( |_
`  ( A  +  ( 1  /  2
) ) )  -  A ) )  <_ 
( 1  /  2
) )
24 halflt1 10535 . . . . . . . 8  |-  ( 1  /  2 )  <  1
251, 12, 24ltleii 9489 . . . . . . 7  |-  ( 1  /  2 )  <_ 
1
2625a1i 11 . . . . . 6  |-  ( A  e.  RR  ->  (
1  /  2 )  <_  1 )
2719, 22, 13, 23, 26letrd 9520 . . . . 5  |-  ( A  e.  RR  ->  ( abs `  ( ( |_
`  ( A  +  ( 1  /  2
) ) )  -  A ) )  <_ 
1 )
2814, 19, 13, 21, 27letrd 9520 . . . 4  |-  ( A  e.  RR  ->  (
( abs `  ( |_ `  ( A  +  ( 1  /  2
) ) ) )  -  ( abs `  A
) )  <_  1
)
298, 11, 13, 28subled 9934 . . 3  |-  ( A  e.  RR  ->  (
( abs `  ( |_ `  ( A  +  ( 1  /  2
) ) ) )  -  1 )  <_ 
( abs `  A
) )
303flcld 11640 . . . . . . 7  |-  ( A  e.  RR  ->  ( |_ `  ( A  +  ( 1  /  2
) ) )  e.  ZZ )
31 nn0abscl 12793 . . . . . . 7  |-  ( ( |_ `  ( A  +  ( 1  / 
2 ) ) )  e.  ZZ  ->  ( abs `  ( |_ `  ( A  +  (
1  /  2 ) ) ) )  e. 
NN0 )
3230, 31syl 16 . . . . . 6  |-  ( A  e.  RR  ->  ( abs `  ( |_ `  ( A  +  (
1  /  2 ) ) ) )  e. 
NN0 )
3332nn0zd 10737 . . . . 5  |-  ( A  e.  RR  ->  ( abs `  ( |_ `  ( A  +  (
1  /  2 ) ) ) )  e.  ZZ )
34 peano2zm 10680 . . . . 5  |-  ( ( abs `  ( |_
`  ( A  +  ( 1  /  2
) ) ) )  e.  ZZ  ->  (
( abs `  ( |_ `  ( A  +  ( 1  /  2
) ) ) )  -  1 )  e.  ZZ )
3533, 34syl 16 . . . 4  |-  ( A  e.  RR  ->  (
( abs `  ( |_ `  ( A  +  ( 1  /  2
) ) ) )  -  1 )  e.  ZZ )
36 flge 11647 . . . 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
) ) ) )
3711, 35, 36syl2anc 661 . . 3  |-  ( A  e.  RR  ->  (
( ( abs `  ( |_ `  ( A  +  ( 1  /  2
) ) ) )  -  1 )  <_ 
( abs `  A
)  <->  ( ( abs `  ( |_ `  ( A  +  ( 1  /  2 ) ) ) )  -  1 )  <_  ( |_ `  ( abs `  A
) ) ) )
3829, 37mpbid 210 . 2  |-  ( A  e.  RR  ->  (
( abs `  ( |_ `  ( A  +  ( 1  /  2
) ) ) )  -  1 )  <_ 
( |_ `  ( abs `  A ) ) )
39 reflcl 11638 . . . 4  |-  ( ( abs `  A )  e.  RR  ->  ( |_ `  ( abs `  A
) )  e.  RR )
4011, 39syl 16 . . 3  |-  ( A  e.  RR  ->  ( |_ `  ( abs `  A
) )  e.  RR )
418, 13, 40lesubaddd 9928 . 2  |-  ( A  e.  RR  ->  (
( ( abs `  ( |_ `  ( A  +  ( 1  /  2
) ) ) )  -  1 )  <_ 
( |_ `  ( abs `  A ) )  <-> 
( abs `  ( |_ `  ( A  +  ( 1  /  2
) ) ) )  <_  ( ( |_
`  ( abs `  A
) )  +  1 ) ) )
4238, 41mpbid 210 1  |-  ( A  e.  RR  ->  ( abs `  ( |_ `  ( A  +  (
1  /  2 ) ) ) )  <_ 
( ( |_ `  ( abs `  A ) )  +  1 ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    e. wcel 1756   class class class wbr 4287   ` cfv 5413  (class class class)co 6086   CCcc 9272   RRcr 9273   1c1 9275    + caddc 9277    <_ cle 9411    - cmin 9587    / cdiv 9985   2c2 10363   NN0cn0 10571   ZZcz 10638   |_cfl 11632   abscabs 12715
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-2nd 6573  df-recs 6824  df-rdg 6858  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-sup 7683  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-n0 10572  df-z 10639  df-uz 10854  df-rp 10984  df-fl 11634  df-seq 11799  df-exp 11858  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717
This theorem is referenced by: (None)
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