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Theorem rddif 12144
Description: The difference between a real number and its nearest integer is less than or equal to one half. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 14-Sep-2015.)
Assertion
Ref Expression
rddif  |-  ( A  e.  RR  ->  ( abs `  ( ( |_
`  ( A  +  ( 1  /  2
) ) )  -  A ) )  <_ 
( 1  /  2
) )

Proof of Theorem rddif
StepHypRef Expression
1 2re 10069 . . . . . . . . . 10  |-  2  e.  RR
2 2ne0 10083 . . . . . . . . . 10  |-  2  =/=  0
31, 2rereccli 9779 . . . . . . . . 9  |-  ( 1  /  2 )  e.  RR
43recni 9102 . . . . . . . 8  |-  ( 1  /  2 )  e.  CC
542timesi 10101 . . . . . . 7  |-  ( 2  x.  ( 1  / 
2 ) )  =  ( ( 1  / 
2 )  +  ( 1  /  2 ) )
6 2cn 10070 . . . . . . . 8  |-  2  e.  CC
76, 2recidi 9745 . . . . . . 7  |-  ( 2  x.  ( 1  / 
2 ) )  =  1
85, 7eqtr3i 2458 . . . . . 6  |-  ( ( 1  /  2 )  +  ( 1  / 
2 ) )  =  1
98oveq2i 6092 . . . . 5  |-  ( ( A  -  ( 1  /  2 ) )  +  ( ( 1  /  2 )  +  ( 1  /  2
) ) )  =  ( ( A  -  ( 1  /  2
) )  +  1 )
10 recn 9080 . . . . . 6  |-  ( A  e.  RR  ->  A  e.  CC )
114a1i 11 . . . . . 6  |-  ( A  e.  RR  ->  (
1  /  2 )  e.  CC )
1210, 11, 11nppcan3d 9438 . . . . 5  |-  ( A  e.  RR  ->  (
( A  -  (
1  /  2 ) )  +  ( ( 1  /  2 )  +  ( 1  / 
2 ) ) )  =  ( A  +  ( 1  /  2
) ) )
139, 12syl5eqr 2482 . . . 4  |-  ( A  e.  RR  ->  (
( A  -  (
1  /  2 ) )  +  1 )  =  ( A  +  ( 1  /  2
) ) )
14 readdcl 9073 . . . . . 6  |-  ( ( A  e.  RR  /\  ( 1  /  2
)  e.  RR )  ->  ( A  +  ( 1  /  2
) )  e.  RR )
153, 14mpan2 653 . . . . 5  |-  ( A  e.  RR  ->  ( A  +  ( 1  /  2 ) )  e.  RR )
16 fllep1 11210 . . . . 5  |-  ( ( A  +  ( 1  /  2 ) )  e.  RR  ->  ( A  +  ( 1  /  2 ) )  <_  ( ( |_
`  ( A  +  ( 1  /  2
) ) )  +  1 ) )
1715, 16syl 16 . . . 4  |-  ( A  e.  RR  ->  ( A  +  ( 1  /  2 ) )  <_  ( ( |_
`  ( A  +  ( 1  /  2
) ) )  +  1 ) )
1813, 17eqbrtrd 4232 . . 3  |-  ( A  e.  RR  ->  (
( A  -  (
1  /  2 ) )  +  1 )  <_  ( ( |_
`  ( A  +  ( 1  /  2
) ) )  +  1 ) )
19 resubcl 9365 . . . . 5  |-  ( ( A  e.  RR  /\  ( 1  /  2
)  e.  RR )  ->  ( A  -  ( 1  /  2
) )  e.  RR )
203, 19mpan2 653 . . . 4  |-  ( A  e.  RR  ->  ( A  -  ( 1  /  2 ) )  e.  RR )
21 reflcl 11205 . . . . 5  |-  ( ( A  +  ( 1  /  2 ) )  e.  RR  ->  ( |_ `  ( A  +  ( 1  /  2
) ) )  e.  RR )
2215, 21syl 16 . . . 4  |-  ( A  e.  RR  ->  ( |_ `  ( A  +  ( 1  /  2
) ) )  e.  RR )
23 1re 9090 . . . . 5  |-  1  e.  RR
2423a1i 11 . . . 4  |-  ( A  e.  RR  ->  1  e.  RR )
2520, 22, 24leadd1d 9620 . . 3  |-  ( A  e.  RR  ->  (
( A  -  (
1  /  2 ) )  <_  ( |_ `  ( A  +  ( 1  /  2 ) ) )  <->  ( ( A  -  ( 1  /  2 ) )  +  1 )  <_ 
( ( |_ `  ( A  +  (
1  /  2 ) ) )  +  1 ) ) )
2618, 25mpbird 224 . 2  |-  ( A  e.  RR  ->  ( A  -  ( 1  /  2 ) )  <_  ( |_ `  ( A  +  (
1  /  2 ) ) ) )
27 flle 11208 . . 3  |-  ( ( A  +  ( 1  /  2 ) )  e.  RR  ->  ( |_ `  ( A  +  ( 1  /  2
) ) )  <_ 
( A  +  ( 1  /  2 ) ) )
2815, 27syl 16 . 2  |-  ( A  e.  RR  ->  ( |_ `  ( A  +  ( 1  /  2
) ) )  <_ 
( A  +  ( 1  /  2 ) ) )
29 id 20 . . 3  |-  ( A  e.  RR  ->  A  e.  RR )
303a1i 11 . . 3  |-  ( A  e.  RR  ->  (
1  /  2 )  e.  RR )
31 absdifle 12122 . . 3  |-  ( ( ( |_ `  ( A  +  ( 1  /  2 ) ) )  e.  RR  /\  A  e.  RR  /\  (
1  /  2 )  e.  RR )  -> 
( ( abs `  (
( |_ `  ( A  +  ( 1  /  2 ) ) )  -  A ) )  <_  ( 1  /  2 )  <->  ( ( A  -  ( 1  /  2 ) )  <_  ( |_ `  ( A  +  (
1  /  2 ) ) )  /\  ( |_ `  ( A  +  ( 1  /  2
) ) )  <_ 
( A  +  ( 1  /  2 ) ) ) ) )
3222, 29, 30, 31syl3anc 1184 . 2  |-  ( A  e.  RR  ->  (
( abs `  (
( |_ `  ( A  +  ( 1  /  2 ) ) )  -  A ) )  <_  ( 1  /  2 )  <->  ( ( A  -  ( 1  /  2 ) )  <_  ( |_ `  ( A  +  (
1  /  2 ) ) )  /\  ( |_ `  ( A  +  ( 1  /  2
) ) )  <_ 
( A  +  ( 1  /  2 ) ) ) ) )
3326, 28, 32mpbir2and 889 1  |-  ( A  e.  RR  ->  ( abs `  ( ( |_
`  ( A  +  ( 1  /  2
) ) )  -  A ) )  <_ 
( 1  /  2
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    e. wcel 1725   class class class wbr 4212   ` cfv 5454  (class class class)co 6081   CCcc 8988   RRcr 8989   1c1 8991    + caddc 8993    x. cmul 8995    <_ cle 9121    - cmin 9291    / cdiv 9677   2c2 10049   |_cfl 11201   abscabs 12039
This theorem is referenced by:  absrdbnd  12145  cntotbnd  26505
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-sup 7446  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-n0 10222  df-z 10283  df-uz 10489  df-rp 10613  df-fl 11202  df-seq 11324  df-exp 11383  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041
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