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Theorem rddif 11840
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 9831 . . . . . . . . . 10  |-  2  e.  RR
2 2ne0 9845 . . . . . . . . . 10  |-  2  =/=  0
31, 2rereccli 9541 . . . . . . . . 9  |-  ( 1  /  2 )  e.  RR
43recni 8865 . . . . . . . 8  |-  ( 1  /  2 )  e.  CC
542timesi 9861 . . . . . . 7  |-  ( 2  x.  ( 1  / 
2 ) )  =  ( ( 1  / 
2 )  +  ( 1  /  2 ) )
6 2cn 9832 . . . . . . . 8  |-  2  e.  CC
76, 2recidi 9507 . . . . . . 7  |-  ( 2  x.  ( 1  / 
2 ) )  =  1
85, 7eqtr3i 2318 . . . . . 6  |-  ( ( 1  /  2 )  +  ( 1  / 
2 ) )  =  1
98oveq2i 5885 . . . . 5  |-  ( ( A  -  ( 1  /  2 ) )  +  ( ( 1  /  2 )  +  ( 1  /  2
) ) )  =  ( ( A  -  ( 1  /  2
) )  +  1 )
10 recn 8843 . . . . . 6  |-  ( A  e.  RR  ->  A  e.  CC )
114a1i 10 . . . . . 6  |-  ( A  e.  RR  ->  (
1  /  2 )  e.  CC )
1210, 11, 11nppcan3d 9200 . . . . 5  |-  ( A  e.  RR  ->  (
( A  -  (
1  /  2 ) )  +  ( ( 1  /  2 )  +  ( 1  / 
2 ) ) )  =  ( A  +  ( 1  /  2
) ) )
139, 12syl5eqr 2342 . . . 4  |-  ( A  e.  RR  ->  (
( A  -  (
1  /  2 ) )  +  1 )  =  ( A  +  ( 1  /  2
) ) )
14 readdcl 8836 . . . . . 6  |-  ( ( A  e.  RR  /\  ( 1  /  2
)  e.  RR )  ->  ( A  +  ( 1  /  2
) )  e.  RR )
153, 14mpan2 652 . . . . 5  |-  ( A  e.  RR  ->  ( A  +  ( 1  /  2 ) )  e.  RR )
16 fllep1 10949 . . . . 5  |-  ( ( A  +  ( 1  /  2 ) )  e.  RR  ->  ( A  +  ( 1  /  2 ) )  <_  ( ( |_
`  ( A  +  ( 1  /  2
) ) )  +  1 ) )
1715, 16syl 15 . . . 4  |-  ( A  e.  RR  ->  ( A  +  ( 1  /  2 ) )  <_  ( ( |_
`  ( A  +  ( 1  /  2
) ) )  +  1 ) )
1813, 17eqbrtrd 4059 . . 3  |-  ( A  e.  RR  ->  (
( A  -  (
1  /  2 ) )  +  1 )  <_  ( ( |_
`  ( A  +  ( 1  /  2
) ) )  +  1 ) )
19 resubcl 9127 . . . . 5  |-  ( ( A  e.  RR  /\  ( 1  /  2
)  e.  RR )  ->  ( A  -  ( 1  /  2
) )  e.  RR )
203, 19mpan2 652 . . . 4  |-  ( A  e.  RR  ->  ( A  -  ( 1  /  2 ) )  e.  RR )
21 reflcl 10944 . . . . 5  |-  ( ( A  +  ( 1  /  2 ) )  e.  RR  ->  ( |_ `  ( A  +  ( 1  /  2
) ) )  e.  RR )
2215, 21syl 15 . . . 4  |-  ( A  e.  RR  ->  ( |_ `  ( A  +  ( 1  /  2
) ) )  e.  RR )
23 1re 8853 . . . . 5  |-  1  e.  RR
2423a1i 10 . . . 4  |-  ( A  e.  RR  ->  1  e.  RR )
2520, 22, 24leadd1d 9382 . . 3  |-  ( A  e.  RR  ->  (
( A  -  (
1  /  2 ) )  <_  ( |_ `  ( A  +  ( 1  /  2 ) ) )  <->  ( ( A  -  ( 1  /  2 ) )  +  1 )  <_ 
( ( |_ `  ( A  +  (
1  /  2 ) ) )  +  1 ) ) )
2618, 25mpbird 223 . 2  |-  ( A  e.  RR  ->  ( A  -  ( 1  /  2 ) )  <_  ( |_ `  ( A  +  (
1  /  2 ) ) ) )
27 flle 10947 . . 3  |-  ( ( A  +  ( 1  /  2 ) )  e.  RR  ->  ( |_ `  ( A  +  ( 1  /  2
) ) )  <_ 
( A  +  ( 1  /  2 ) ) )
2815, 27syl 15 . 2  |-  ( A  e.  RR  ->  ( |_ `  ( A  +  ( 1  /  2
) ) )  <_ 
( A  +  ( 1  /  2 ) ) )
29 id 19 . . 3  |-  ( A  e.  RR  ->  A  e.  RR )
303a1i 10 . . 3  |-  ( A  e.  RR  ->  (
1  /  2 )  e.  RR )
31 absdifle 11818 . . 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 1182 . 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 888 1  |-  ( A  e.  RR  ->  ( abs `  ( ( |_
`  ( A  +  ( 1  /  2
) ) )  -  A ) )  <_ 
( 1  /  2
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    e. wcel 1696   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   CCcc 8751   RRcr 8752   1c1 8754    + caddc 8756    x. cmul 8758    <_ cle 8884    - cmin 9053    / cdiv 9439   2c2 9811   |_cfl 10940   abscabs 11735
This theorem is referenced by:  absrdbnd  11841  cntotbnd  26623
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-fl 10941  df-seq 11063  df-exp 11121  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737
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