MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  absrdbnd Unicode version

Theorem absrdbnd 11825
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 9815 . . . . . . . . 9  |-  2  e.  RR
2 2ne0 9829 . . . . . . . . 9  |-  2  =/=  0
31, 2rereccli 9525 . . . . . . . 8  |-  ( 1  /  2 )  e.  RR
4 readdcl 8820 . . . . . . . 8  |-  ( ( A  e.  RR  /\  ( 1  /  2
)  e.  RR )  ->  ( A  +  ( 1  /  2
) )  e.  RR )
53, 4mpan2 652 . . . . . . 7  |-  ( A  e.  RR  ->  ( A  +  ( 1  /  2 ) )  e.  RR )
6 reflcl 10928 . . . . . . 7  |-  ( ( A  +  ( 1  /  2 ) )  e.  RR  ->  ( |_ `  ( A  +  ( 1  /  2
) ) )  e.  RR )
75, 6syl 15 . . . . . 6  |-  ( A  e.  RR  ->  ( |_ `  ( A  +  ( 1  /  2
) ) )  e.  RR )
87recnd 8861 . . . . 5  |-  ( A  e.  RR  ->  ( |_ `  ( A  +  ( 1  /  2
) ) )  e.  CC )
9 abscl 11763 . . . . 5  |-  ( ( |_ `  ( A  +  ( 1  / 
2 ) ) )  e.  CC  ->  ( abs `  ( |_ `  ( A  +  (
1  /  2 ) ) ) )  e.  RR )
108, 9syl 15 . . . 4  |-  ( A  e.  RR  ->  ( abs `  ( |_ `  ( A  +  (
1  /  2 ) ) ) )  e.  RR )
11 recn 8827 . . . . 5  |-  ( A  e.  RR  ->  A  e.  CC )
12 abscl 11763 . . . . 5  |-  ( A  e.  CC  ->  ( abs `  A )  e.  RR )
1311, 12syl 15 . . . 4  |-  ( A  e.  RR  ->  ( abs `  A )  e.  RR )
14 1re 8837 . . . . 5  |-  1  e.  RR
1514a1i 10 . . . 4  |-  ( A  e.  RR  ->  1  e.  RR )
1610, 13resubcld 9211 . . . . 5  |-  ( A  e.  RR  ->  (
( abs `  ( |_ `  ( A  +  ( 1  /  2
) ) ) )  -  ( abs `  A
) )  e.  RR )
17 resubcl 9111 . . . . . . . 8  |-  ( ( ( |_ `  ( A  +  ( 1  /  2 ) ) )  e.  RR  /\  A  e.  RR )  ->  ( ( |_ `  ( A  +  (
1  /  2 ) ) )  -  A
)  e.  RR )
187, 17mpancom 650 . . . . . . 7  |-  ( A  e.  RR  ->  (
( |_ `  ( A  +  ( 1  /  2 ) ) )  -  A )  e.  RR )
1918recnd 8861 . . . . . 6  |-  ( A  e.  RR  ->  (
( |_ `  ( A  +  ( 1  /  2 ) ) )  -  A )  e.  CC )
20 abscl 11763 . . . . . 6  |-  ( ( ( |_ `  ( A  +  ( 1  /  2 ) ) )  -  A )  e.  CC  ->  ( abs `  ( ( |_
`  ( A  +  ( 1  /  2
) ) )  -  A ) )  e.  RR )
2119, 20syl 15 . . . . 5  |-  ( A  e.  RR  ->  ( abs `  ( ( |_
`  ( A  +  ( 1  /  2
) ) )  -  A ) )  e.  RR )
22 abs2dif 11816 . . . . . 6  |-  ( ( ( |_ `  ( A  +  ( 1  /  2 ) ) )  e.  CC  /\  A  e.  CC )  ->  ( ( abs `  ( |_ `  ( A  +  ( 1  /  2
) ) ) )  -  ( abs `  A
) )  <_  ( abs `  ( ( |_
`  ( A  +  ( 1  /  2
) ) )  -  A ) ) )
238, 11, 22syl2anc 642 . . . . 5  |-  ( A  e.  RR  ->  (
( abs `  ( |_ `  ( A  +  ( 1  /  2
) ) ) )  -  ( abs `  A
) )  <_  ( abs `  ( ( |_
`  ( A  +  ( 1  /  2
) ) )  -  A ) ) )
243a1i 10 . . . . . 6  |-  ( A  e.  RR  ->  (
1  /  2 )  e.  RR )
25 rddif 11824 . . . . . 6  |-  ( A  e.  RR  ->  ( abs `  ( ( |_
`  ( A  +  ( 1  /  2
) ) )  -  A ) )  <_ 
( 1  /  2
) )
26 halflt1 9933 . . . . . . . 8  |-  ( 1  /  2 )  <  1
273, 14, 26ltleii 8941 . . . . . . 7  |-  ( 1  /  2 )  <_ 
1
2827a1i 10 . . . . . 6  |-  ( A  e.  RR  ->  (
1  /  2 )  <_  1 )
2921, 24, 15, 25, 28letrd 8973 . . . . 5  |-  ( A  e.  RR  ->  ( abs `  ( ( |_
`  ( A  +  ( 1  /  2
) ) )  -  A ) )  <_ 
1 )
3016, 21, 15, 23, 29letrd 8973 . . . 4  |-  ( A  e.  RR  ->  (
( abs `  ( |_ `  ( A  +  ( 1  /  2
) ) ) )  -  ( abs `  A
) )  <_  1
)
3110, 13, 15, 30subled 9375 . . 3  |-  ( A  e.  RR  ->  (
( abs `  ( |_ `  ( A  +  ( 1  /  2
) ) ) )  -  1 )  <_ 
( abs `  A
) )
325flcld 10930 . . . . . . 7  |-  ( A  e.  RR  ->  ( |_ `  ( A  +  ( 1  /  2
) ) )  e.  ZZ )
33 nn0abscl 11797 . . . . . . 7  |-  ( ( |_ `  ( A  +  ( 1  / 
2 ) ) )  e.  ZZ  ->  ( abs `  ( |_ `  ( A  +  (
1  /  2 ) ) ) )  e. 
NN0 )
3432, 33syl 15 . . . . . 6  |-  ( A  e.  RR  ->  ( abs `  ( |_ `  ( A  +  (
1  /  2 ) ) ) )  e. 
NN0 )
3534nn0zd 10115 . . . . 5  |-  ( A  e.  RR  ->  ( abs `  ( |_ `  ( A  +  (
1  /  2 ) ) ) )  e.  ZZ )
36 peano2zm 10062 . . . . 5  |-  ( ( abs `  ( |_
`  ( A  +  ( 1  /  2
) ) ) )  e.  ZZ  ->  (
( abs `  ( |_ `  ( A  +  ( 1  /  2
) ) ) )  -  1 )  e.  ZZ )
3735, 36syl 15 . . . 4  |-  ( A  e.  RR  ->  (
( abs `  ( |_ `  ( A  +  ( 1  /  2
) ) ) )  -  1 )  e.  ZZ )
38 flge 10937 . . . 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 642 . . 3  |-  ( A  e.  RR  ->  (
( ( abs `  ( |_ `  ( A  +  ( 1  /  2
) ) ) )  -  1 )  <_ 
( abs `  A
)  <->  ( ( abs `  ( |_ `  ( A  +  ( 1  /  2 ) ) ) )  -  1 )  <_  ( |_ `  ( abs `  A
) ) ) )
4031, 39mpbid 201 . 2  |-  ( A  e.  RR  ->  (
( abs `  ( |_ `  ( A  +  ( 1  /  2
) ) ) )  -  1 )  <_ 
( |_ `  ( abs `  A ) ) )
41 reflcl 10928 . . . 4  |-  ( ( abs `  A )  e.  RR  ->  ( |_ `  ( abs `  A
) )  e.  RR )
4213, 41syl 15 . . 3  |-  ( A  e.  RR  ->  ( |_ `  ( abs `  A
) )  e.  RR )
4310, 15, 42lesubaddd 9369 . 2  |-  ( A  e.  RR  ->  (
( ( abs `  ( |_ `  ( A  +  ( 1  /  2
) ) ) )  -  1 )  <_ 
( |_ `  ( abs `  A ) )  <-> 
( abs `  ( |_ `  ( A  +  ( 1  /  2
) ) ) )  <_  ( ( |_
`  ( abs `  A
) )  +  1 ) ) )
4440, 43mpbid 201 1  |-  ( A  e.  RR  ->  ( abs `  ( |_ `  ( A  +  (
1  /  2 ) ) ) )  <_ 
( ( |_ `  ( abs `  A ) )  +  1 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    e. wcel 1684   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   1c1 8738    + caddc 8740    <_ cle 8868    - cmin 9037    / cdiv 9423   2c2 9795   NN0cn0 9965   ZZcz 10024   |_cfl 10924   abscabs 11719
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fl 10925  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721
  Copyright terms: Public domain W3C validator