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Theorem absrdbnd 13137
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 10754 . . . . . . . 8  |-  ( 1  /  2 )  e.  RR
2 readdcl 9575 . . . . . . . 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 11901 . . . . . . 7  |-  ( ( A  +  ( 1  /  2 ) )  e.  RR  ->  ( |_ `  ( A  +  ( 1  /  2
) ) )  e.  RR )
53, 4syl 16 . . . . . 6  |-  ( A  e.  RR  ->  ( |_ `  ( A  +  ( 1  /  2
) ) )  e.  RR )
65recnd 9622 . . . . 5  |-  ( A  e.  RR  ->  ( |_ `  ( A  +  ( 1  /  2
) ) )  e.  CC )
7 abscl 13074 . . . . 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 9582 . . . . 5  |-  ( A  e.  RR  ->  A  e.  CC )
10 abscl 13074 . . . . 5  |-  ( A  e.  CC  ->  ( abs `  A )  e.  RR )
119, 10syl 16 . . . 4  |-  ( A  e.  RR  ->  ( abs `  A )  e.  RR )
12 1re 9595 . . . . 5  |-  1  e.  RR
1312a1i 11 . . . 4  |-  ( A  e.  RR  ->  1  e.  RR )
148, 11resubcld 9987 . . . . 5  |-  ( A  e.  RR  ->  (
( abs `  ( |_ `  ( A  +  ( 1  /  2
) ) ) )  -  ( abs `  A
) )  e.  RR )
15 resubcl 9883 . . . . . . . 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 9622 . . . . . 6  |-  ( A  e.  RR  ->  (
( |_ `  ( A  +  ( 1  /  2 ) ) )  -  A )  e.  CC )
18 abscl 13074 . . . . . 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 13128 . . . . . 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 13136 . . . . . 6  |-  ( A  e.  RR  ->  ( abs `  ( ( |_
`  ( A  +  ( 1  /  2
) ) )  -  A ) )  <_ 
( 1  /  2
) )
24 halflt1 10757 . . . . . . . 8  |-  ( 1  /  2 )  <  1
251, 12, 24ltleii 9707 . . . . . . 7  |-  ( 1  /  2 )  <_ 
1
2625a1i 11 . . . . . 6  |-  ( A  e.  RR  ->  (
1  /  2 )  <_  1 )
2719, 22, 13, 23, 26letrd 9738 . . . . 5  |-  ( A  e.  RR  ->  ( abs `  ( ( |_
`  ( A  +  ( 1  /  2
) ) )  -  A ) )  <_ 
1 )
2814, 19, 13, 21, 27letrd 9738 . . . 4  |-  ( A  e.  RR  ->  (
( abs `  ( |_ `  ( A  +  ( 1  /  2
) ) ) )  -  ( abs `  A
) )  <_  1
)
298, 11, 13, 28subled 10155 . . 3  |-  ( A  e.  RR  ->  (
( abs `  ( |_ `  ( A  +  ( 1  /  2
) ) ) )  -  1 )  <_ 
( abs `  A
) )
303flcld 11903 . . . . . . 7  |-  ( A  e.  RR  ->  ( |_ `  ( A  +  ( 1  /  2
) ) )  e.  ZZ )
31 nn0abscl 13108 . . . . . . 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 10964 . . . . 5  |-  ( A  e.  RR  ->  ( abs `  ( |_ `  ( A  +  (
1  /  2 ) ) ) )  e.  ZZ )
34 peano2zm 10906 . . . . 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 11910 . . . 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 11901 . . . 4  |-  ( ( abs `  A )  e.  RR  ->  ( |_ `  ( abs `  A
) )  e.  RR )
4011, 39syl 16 . . 3  |-  ( A  e.  RR  ->  ( |_ `  ( abs `  A
) )  e.  RR )
418, 13, 40lesubaddd 10149 . 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 1767   class class class wbr 4447   ` cfv 5588  (class class class)co 6284   CCcc 9490   RRcr 9491   1c1 9493    + caddc 9495    <_ cle 9629    - cmin 9805    / cdiv 10206   2c2 10585   NN0cn0 10795   ZZcz 10864   |_cfl 11895   abscabs 13030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-pre-sup 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-2nd 6785  df-recs 7042  df-rdg 7076  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-sup 7901  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-div 10207  df-nn 10537  df-2 10594  df-3 10595  df-n0 10796  df-z 10865  df-uz 11083  df-rp 11221  df-fl 11897  df-seq 12076  df-exp 12135  df-cj 12895  df-re 12896  df-im 12897  df-sqrt 13031  df-abs 13032
This theorem is referenced by: (None)
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