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Theorem flhalf 10015
Description: Ordering relation for the floor of half of an integer. (Contributed by NM, 1-Jan-2006.) (Proof shortened by Mario Carneiro, 7-Jun-2016.)
Assertion
Ref Expression
flhalf  |-  ( N  e.  ZZ  ->  N  <_  ( 2  x.  ( |_ `  ( ( N  +  1 )  / 
2 ) ) ) )

Proof of Theorem flhalf
StepHypRef Expression
1 peano2z 9041 . . . . . . 7  |-  ( N  e.  ZZ  ->  ( N  +  1 )  e.  ZZ )
2 2nn 8832 . . . . . . 7  |-  2  e.  NN
3 znq 9365 . . . . . . 7  |-  ( ( ( N  +  1 )  e.  ZZ  /\  2  e.  NN )  ->  ( ( N  + 
1 )  /  2
)  e.  QQ )
41, 2, 3sylancl 407 . . . . . 6  |-  ( N  e.  ZZ  ->  (
( N  +  1 )  /  2 )  e.  QQ )
5 flqltp1 9992 . . . . . 6  |-  ( ( ( N  +  1 )  /  2 )  e.  QQ  ->  (
( N  +  1 )  /  2 )  <  ( ( |_
`  ( ( N  +  1 )  / 
2 ) )  +  1 ) )
64, 5syl 14 . . . . 5  |-  ( N  e.  ZZ  ->  (
( N  +  1 )  /  2 )  <  ( ( |_
`  ( ( N  +  1 )  / 
2 ) )  +  1 ) )
7 zre 9009 . . . . . . 7  |-  ( N  e.  ZZ  ->  N  e.  RR )
8 peano2re 7862 . . . . . . 7  |-  ( N  e.  RR  ->  ( N  +  1 )  e.  RR )
97, 8syl 14 . . . . . 6  |-  ( N  e.  ZZ  ->  ( N  +  1 )  e.  RR )
104flqcld 9990 . . . . . . . 8  |-  ( N  e.  ZZ  ->  ( |_ `  ( ( N  +  1 )  / 
2 ) )  e.  ZZ )
1110zred 9124 . . . . . . 7  |-  ( N  e.  ZZ  ->  ( |_ `  ( ( N  +  1 )  / 
2 ) )  e.  RR )
12 1red 7745 . . . . . . 7  |-  ( N  e.  ZZ  ->  1  e.  RR )
1311, 12readdcld 7759 . . . . . 6  |-  ( N  e.  ZZ  ->  (
( |_ `  (
( N  +  1 )  /  2 ) )  +  1 )  e.  RR )
14 2rp 9395 . . . . . . 7  |-  2  e.  RR+
1514a1i 9 . . . . . 6  |-  ( N  e.  ZZ  ->  2  e.  RR+ )
169, 13, 15ltdivmuld 9481 . . . . 5  |-  ( N  e.  ZZ  ->  (
( ( N  + 
1 )  /  2
)  <  ( ( |_ `  ( ( N  +  1 )  / 
2 ) )  +  1 )  <->  ( N  +  1 )  < 
( 2  x.  (
( |_ `  (
( N  +  1 )  /  2 ) )  +  1 ) ) ) )
176, 16mpbid 146 . . . 4  |-  ( N  e.  ZZ  ->  ( N  +  1 )  <  ( 2  x.  ( ( |_ `  ( ( N  + 
1 )  /  2
) )  +  1 ) ) )
1812recnd 7758 . . . . . . 7  |-  ( N  e.  ZZ  ->  1  e.  CC )
19182timesd 8913 . . . . . 6  |-  ( N  e.  ZZ  ->  (
2  x.  1 )  =  ( 1  +  1 ) )
2019oveq2d 5756 . . . . 5  |-  ( N  e.  ZZ  ->  (
( 2  x.  ( |_ `  ( ( N  +  1 )  / 
2 ) ) )  +  ( 2  x.  1 ) )  =  ( ( 2  x.  ( |_ `  (
( N  +  1 )  /  2 ) ) )  +  ( 1  +  1 ) ) )
21 2cnd 8750 . . . . . 6  |-  ( N  e.  ZZ  ->  2  e.  CC )
2211recnd 7758 . . . . . 6  |-  ( N  e.  ZZ  ->  ( |_ `  ( ( N  +  1 )  / 
2 ) )  e.  CC )
2321, 22, 18adddid 7754 . . . . 5  |-  ( N  e.  ZZ  ->  (
2  x.  ( ( |_ `  ( ( N  +  1 )  /  2 ) )  +  1 ) )  =  ( ( 2  x.  ( |_ `  ( ( N  + 
1 )  /  2
) ) )  +  ( 2  x.  1 ) ) )
24 2re 8747 . . . . . . . . 9  |-  2  e.  RR
2524a1i 9 . . . . . . . 8  |-  ( N  e.  ZZ  ->  2  e.  RR )
2625, 11remulcld 7760 . . . . . . 7  |-  ( N  e.  ZZ  ->  (
2  x.  ( |_
`  ( ( N  +  1 )  / 
2 ) ) )  e.  RR )
2726recnd 7758 . . . . . 6  |-  ( N  e.  ZZ  ->  (
2  x.  ( |_
`  ( ( N  +  1 )  / 
2 ) ) )  e.  CC )
2827, 18, 18addassd 7752 . . . . 5  |-  ( N  e.  ZZ  ->  (
( ( 2  x.  ( |_ `  (
( N  +  1 )  /  2 ) ) )  +  1 )  +  1 )  =  ( ( 2  x.  ( |_ `  ( ( N  + 
1 )  /  2
) ) )  +  ( 1  +  1 ) ) )
2920, 23, 283eqtr4d 2158 . . . 4  |-  ( N  e.  ZZ  ->  (
2  x.  ( ( |_ `  ( ( N  +  1 )  /  2 ) )  +  1 ) )  =  ( ( ( 2  x.  ( |_
`  ( ( N  +  1 )  / 
2 ) ) )  +  1 )  +  1 ) )
3017, 29breqtrd 3922 . . 3  |-  ( N  e.  ZZ  ->  ( N  +  1 )  <  ( ( ( 2  x.  ( |_
`  ( ( N  +  1 )  / 
2 ) ) )  +  1 )  +  1 ) )
3126, 12readdcld 7759 . . . 4  |-  ( N  e.  ZZ  ->  (
( 2  x.  ( |_ `  ( ( N  +  1 )  / 
2 ) ) )  +  1 )  e.  RR )
327, 31, 12ltadd1d 8263 . . 3  |-  ( N  e.  ZZ  ->  ( N  <  ( ( 2  x.  ( |_ `  ( ( N  + 
1 )  /  2
) ) )  +  1 )  <->  ( N  +  1 )  < 
( ( ( 2  x.  ( |_ `  ( ( N  + 
1 )  /  2
) ) )  +  1 )  +  1 ) ) )
3330, 32mpbird 166 . 2  |-  ( N  e.  ZZ  ->  N  <  ( ( 2  x.  ( |_ `  (
( N  +  1 )  /  2 ) ) )  +  1 ) )
34 2z 9033 . . . . 5  |-  2  e.  ZZ
3534a1i 9 . . . 4  |-  ( N  e.  ZZ  ->  2  e.  ZZ )
3635, 10zmulcld 9130 . . 3  |-  ( N  e.  ZZ  ->  (
2  x.  ( |_
`  ( ( N  +  1 )  / 
2 ) ) )  e.  ZZ )
37 zleltp1 9060 . . 3  |-  ( ( N  e.  ZZ  /\  ( 2  x.  ( |_ `  ( ( N  +  1 )  / 
2 ) ) )  e.  ZZ )  -> 
( N  <_  (
2  x.  ( |_
`  ( ( N  +  1 )  / 
2 ) ) )  <-> 
N  <  ( (
2  x.  ( |_
`  ( ( N  +  1 )  / 
2 ) ) )  +  1 ) ) )
3836, 37mpdan 415 . 2  |-  ( N  e.  ZZ  ->  ( N  <_  ( 2  x.  ( |_ `  (
( N  +  1 )  /  2 ) ) )  <->  N  <  ( ( 2  x.  ( |_ `  ( ( N  +  1 )  / 
2 ) ) )  +  1 ) ) )
3933, 38mpbird 166 1  |-  ( N  e.  ZZ  ->  N  <_  ( 2  x.  ( |_ `  ( ( N  +  1 )  / 
2 ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    e. wcel 1463   class class class wbr 3897   ` cfv 5091  (class class class)co 5740   RRcr 7583   1c1 7585    + caddc 7587    x. cmul 7589    < clt 7764    <_ cle 7765    / cdiv 8392   NNcn 8677   2c2 8728   ZZcz 9005   QQcq 9360   RR+crp 9390   |_cfl 9981
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-cnex 7675  ax-resscn 7676  ax-1cn 7677  ax-1re 7678  ax-icn 7679  ax-addcl 7680  ax-addrcl 7681  ax-mulcl 7682  ax-mulrcl 7683  ax-addcom 7684  ax-mulcom 7685  ax-addass 7686  ax-mulass 7687  ax-distr 7688  ax-i2m1 7689  ax-0lt1 7690  ax-1rid 7691  ax-0id 7692  ax-rnegex 7693  ax-precex 7694  ax-cnre 7695  ax-pre-ltirr 7696  ax-pre-ltwlin 7697  ax-pre-lttrn 7698  ax-pre-apti 7699  ax-pre-ltadd 7700  ax-pre-mulgt0 7701  ax-pre-mulext 7702  ax-arch 7703
This theorem depends on definitions:  df-bi 116  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-nel 2379  df-ral 2396  df-rex 2397  df-reu 2398  df-rmo 2399  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-id 4183  df-po 4186  df-iso 4187  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-fv 5099  df-riota 5696  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-pnf 7766  df-mnf 7767  df-xr 7768  df-ltxr 7769  df-le 7770  df-sub 7899  df-neg 7900  df-reap 8300  df-ap 8307  df-div 8393  df-inn 8678  df-2 8736  df-n0 8929  df-z 9006  df-q 9361  df-rp 9391  df-fl 9983
This theorem is referenced by: (None)
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