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Theorem absrdbnd 12100
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 10025 . . . . . . . . 9  |-  2  e.  RR
2 2ne0 10039 . . . . . . . . 9  |-  2  =/=  0
31, 2rereccli 9735 . . . . . . . 8  |-  ( 1  /  2 )  e.  RR
4 readdcl 9029 . . . . . . . 8  |-  ( ( A  e.  RR  /\  ( 1  /  2
)  e.  RR )  ->  ( A  +  ( 1  /  2
) )  e.  RR )
53, 4mpan2 653 . . . . . . 7  |-  ( A  e.  RR  ->  ( A  +  ( 1  /  2 ) )  e.  RR )
6 reflcl 11160 . . . . . . 7  |-  ( ( A  +  ( 1  /  2 ) )  e.  RR  ->  ( |_ `  ( A  +  ( 1  /  2
) ) )  e.  RR )
75, 6syl 16 . . . . . 6  |-  ( A  e.  RR  ->  ( |_ `  ( A  +  ( 1  /  2
) ) )  e.  RR )
87recnd 9070 . . . . 5  |-  ( A  e.  RR  ->  ( |_ `  ( A  +  ( 1  /  2
) ) )  e.  CC )
9 abscl 12038 . . . . 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 9036 . . . . 5  |-  ( A  e.  RR  ->  A  e.  CC )
12 abscl 12038 . . . . 5  |-  ( A  e.  CC  ->  ( abs `  A )  e.  RR )
1311, 12syl 16 . . . 4  |-  ( A  e.  RR  ->  ( abs `  A )  e.  RR )
14 1re 9046 . . . . 5  |-  1  e.  RR
1514a1i 11 . . . 4  |-  ( A  e.  RR  ->  1  e.  RR )
1610, 13resubcld 9421 . . . . 5  |-  ( A  e.  RR  ->  (
( abs `  ( |_ `  ( A  +  ( 1  /  2
) ) ) )  -  ( abs `  A
) )  e.  RR )
17 resubcl 9321 . . . . . . . 8  |-  ( ( ( |_ `  ( A  +  ( 1  /  2 ) ) )  e.  RR  /\  A  e.  RR )  ->  ( ( |_ `  ( A  +  (
1  /  2 ) ) )  -  A
)  e.  RR )
187, 17mpancom 651 . . . . . . 7  |-  ( A  e.  RR  ->  (
( |_ `  ( A  +  ( 1  /  2 ) ) )  -  A )  e.  RR )
1918recnd 9070 . . . . . 6  |-  ( A  e.  RR  ->  (
( |_ `  ( A  +  ( 1  /  2 ) ) )  -  A )  e.  CC )
20 abscl 12038 . . . . . 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 12091 . . . . . 6  |-  ( ( ( |_ `  ( A  +  ( 1  /  2 ) ) )  e.  CC  /\  A  e.  CC )  ->  ( ( abs `  ( |_ `  ( A  +  ( 1  /  2
) ) ) )  -  ( abs `  A
) )  <_  ( abs `  ( ( |_
`  ( A  +  ( 1  /  2
) ) )  -  A ) ) )
238, 11, 22syl2anc 643 . . . . 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 12099 . . . . . 6  |-  ( A  e.  RR  ->  ( abs `  ( ( |_
`  ( A  +  ( 1  /  2
) ) )  -  A ) )  <_ 
( 1  /  2
) )
26 halflt1 10145 . . . . . . . 8  |-  ( 1  /  2 )  <  1
273, 14, 26ltleii 9152 . . . . . . 7  |-  ( 1  /  2 )  <_ 
1
2827a1i 11 . . . . . 6  |-  ( A  e.  RR  ->  (
1  /  2 )  <_  1 )
2921, 24, 15, 25, 28letrd 9183 . . . . 5  |-  ( A  e.  RR  ->  ( abs `  ( ( |_
`  ( A  +  ( 1  /  2
) ) )  -  A ) )  <_ 
1 )
3016, 21, 15, 23, 29letrd 9183 . . . 4  |-  ( A  e.  RR  ->  (
( abs `  ( |_ `  ( A  +  ( 1  /  2
) ) ) )  -  ( abs `  A
) )  <_  1
)
3110, 13, 15, 30subled 9585 . . 3  |-  ( A  e.  RR  ->  (
( abs `  ( |_ `  ( A  +  ( 1  /  2
) ) ) )  -  1 )  <_ 
( abs `  A
) )
325flcld 11162 . . . . . . 7  |-  ( A  e.  RR  ->  ( |_ `  ( A  +  ( 1  /  2
) ) )  e.  ZZ )
33 nn0abscl 12072 . . . . . . 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 10329 . . . . 5  |-  ( A  e.  RR  ->  ( abs `  ( |_ `  ( A  +  (
1  /  2 ) ) ) )  e.  ZZ )
36 peano2zm 10276 . . . . 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 11169 . . . 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 643 . . 3  |-  ( A  e.  RR  ->  (
( ( abs `  ( |_ `  ( A  +  ( 1  /  2
) ) ) )  -  1 )  <_ 
( abs `  A
)  <->  ( ( abs `  ( |_ `  ( A  +  ( 1  /  2 ) ) ) )  -  1 )  <_  ( |_ `  ( abs `  A
) ) ) )
4031, 39mpbid 202 . 2  |-  ( A  e.  RR  ->  (
( abs `  ( |_ `  ( A  +  ( 1  /  2
) ) ) )  -  1 )  <_ 
( |_ `  ( abs `  A ) ) )
41 reflcl 11160 . . . 4  |-  ( ( abs `  A )  e.  RR  ->  ( |_ `  ( abs `  A
) )  e.  RR )
4213, 41syl 16 . . 3  |-  ( A  e.  RR  ->  ( |_ `  ( abs `  A
) )  e.  RR )
4310, 15, 42lesubaddd 9579 . 2  |-  ( A  e.  RR  ->  (
( ( abs `  ( |_ `  ( A  +  ( 1  /  2
) ) ) )  -  1 )  <_ 
( |_ `  ( abs `  A ) )  <-> 
( abs `  ( |_ `  ( A  +  ( 1  /  2
) ) ) )  <_  ( ( |_
`  ( abs `  A
) )  +  1 ) ) )
4440, 43mpbid 202 1  |-  ( A  e.  RR  ->  ( abs `  ( |_ `  ( A  +  (
1  /  2 ) ) ) )  <_ 
( ( |_ `  ( abs `  A ) )  +  1 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    e. wcel 1721   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   CCcc 8944   RRcr 8945   1c1 8947    + caddc 8949    <_ cle 9077    - cmin 9247    / cdiv 9633   2c2 10005   NN0cn0 10177   ZZcz 10238   |_cfl 11156   abscabs 11994
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-fl 11157  df-seq 11279  df-exp 11338  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996
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