ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  flhalf Unicode version

Theorem flhalf 9373
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 8457 . . . . . . 7  |-  ( N  e.  ZZ  ->  ( N  +  1 )  e.  ZZ )
2 2nn 8249 . . . . . . 7  |-  2  e.  NN
3 znq 8779 . . . . . . 7  |-  ( ( ( N  +  1 )  e.  ZZ  /\  2  e.  NN )  ->  ( ( N  + 
1 )  /  2
)  e.  QQ )
41, 2, 3sylancl 404 . . . . . 6  |-  ( N  e.  ZZ  ->  (
( N  +  1 )  /  2 )  e.  QQ )
5 flqltp1 9350 . . . . . 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 8425 . . . . . . 7  |-  ( N  e.  ZZ  ->  N  e.  RR )
8 peano2re 7300 . . . . . . 7  |-  ( N  e.  RR  ->  ( N  +  1 )  e.  RR )
97, 8syl 14 . . . . . 6  |-  ( N  e.  ZZ  ->  ( N  +  1 )  e.  RR )
104flqcld 9348 . . . . . . . 8  |-  ( N  e.  ZZ  ->  ( |_ `  ( ( N  +  1 )  / 
2 ) )  e.  ZZ )
1110zred 8539 . . . . . . 7  |-  ( N  e.  ZZ  ->  ( |_ `  ( ( N  +  1 )  / 
2 ) )  e.  RR )
12 1red 7185 . . . . . . 7  |-  ( N  e.  ZZ  ->  1  e.  RR )
1311, 12readdcld 7199 . . . . . 6  |-  ( N  e.  ZZ  ->  (
( |_ `  (
( N  +  1 )  /  2 ) )  +  1 )  e.  RR )
14 2rp 8809 . . . . . . 7  |-  2  e.  RR+
1514a1i 9 . . . . . 6  |-  ( N  e.  ZZ  ->  2  e.  RR+ )
169, 13, 15ltdivmuld 8895 . . . . 5  |-  ( N  e.  ZZ  ->  (
( ( N  + 
1 )  /  2
)  <  ( ( |_ `  ( ( N  +  1 )  / 
2 ) )  +  1 )  <->  ( N  +  1 )  < 
( 2  x.  (
( |_ `  (
( N  +  1 )  /  2 ) )  +  1 ) ) ) )
176, 16mpbid 145 . . . 4  |-  ( N  e.  ZZ  ->  ( N  +  1 )  <  ( 2  x.  ( ( |_ `  ( ( N  + 
1 )  /  2
) )  +  1 ) ) )
1812recnd 7198 . . . . . . 7  |-  ( N  e.  ZZ  ->  1  e.  CC )
19182timesd 8329 . . . . . 6  |-  ( N  e.  ZZ  ->  (
2  x.  1 )  =  ( 1  +  1 ) )
2019oveq2d 5553 . . . . 5  |-  ( N  e.  ZZ  ->  (
( 2  x.  ( |_ `  ( ( N  +  1 )  / 
2 ) ) )  +  ( 2  x.  1 ) )  =  ( ( 2  x.  ( |_ `  (
( N  +  1 )  /  2 ) ) )  +  ( 1  +  1 ) ) )
21 2cnd 8168 . . . . . 6  |-  ( N  e.  ZZ  ->  2  e.  CC )
2211recnd 7198 . . . . . 6  |-  ( N  e.  ZZ  ->  ( |_ `  ( ( N  +  1 )  / 
2 ) )  e.  CC )
2321, 22, 18adddid 7194 . . . . 5  |-  ( N  e.  ZZ  ->  (
2  x.  ( ( |_ `  ( ( N  +  1 )  /  2 ) )  +  1 ) )  =  ( ( 2  x.  ( |_ `  ( ( N  + 
1 )  /  2
) ) )  +  ( 2  x.  1 ) ) )
24 2re 8165 . . . . . . . . 9  |-  2  e.  RR
2524a1i 9 . . . . . . . 8  |-  ( N  e.  ZZ  ->  2  e.  RR )
2625, 11remulcld 7200 . . . . . . 7  |-  ( N  e.  ZZ  ->  (
2  x.  ( |_
`  ( ( N  +  1 )  / 
2 ) ) )  e.  RR )
2726recnd 7198 . . . . . 6  |-  ( N  e.  ZZ  ->  (
2  x.  ( |_
`  ( ( N  +  1 )  / 
2 ) ) )  e.  CC )
2827, 18, 18addassd 7192 . . . . 5  |-  ( N  e.  ZZ  ->  (
( ( 2  x.  ( |_ `  (
( N  +  1 )  /  2 ) ) )  +  1 )  +  1 )  =  ( ( 2  x.  ( |_ `  ( ( N  + 
1 )  /  2
) ) )  +  ( 1  +  1 ) ) )
2920, 23, 283eqtr4d 2124 . . . 4  |-  ( N  e.  ZZ  ->  (
2  x.  ( ( |_ `  ( ( N  +  1 )  /  2 ) )  +  1 ) )  =  ( ( ( 2  x.  ( |_
`  ( ( N  +  1 )  / 
2 ) ) )  +  1 )  +  1 ) )
3017, 29breqtrd 3811 . . 3  |-  ( N  e.  ZZ  ->  ( N  +  1 )  <  ( ( ( 2  x.  ( |_
`  ( ( N  +  1 )  / 
2 ) ) )  +  1 )  +  1 ) )
3126, 12readdcld 7199 . . . 4  |-  ( N  e.  ZZ  ->  (
( 2  x.  ( |_ `  ( ( N  +  1 )  / 
2 ) ) )  +  1 )  e.  RR )
327, 31, 12ltadd1d 7694 . . 3  |-  ( N  e.  ZZ  ->  ( N  <  ( ( 2  x.  ( |_ `  ( ( N  + 
1 )  /  2
) ) )  +  1 )  <->  ( N  +  1 )  < 
( ( ( 2  x.  ( |_ `  ( ( N  + 
1 )  /  2
) ) )  +  1 )  +  1 ) ) )
3330, 32mpbird 165 . 2  |-  ( N  e.  ZZ  ->  N  <  ( ( 2  x.  ( |_ `  (
( N  +  1 )  /  2 ) ) )  +  1 ) )
34 2z 8449 . . . . 5  |-  2  e.  ZZ
3534a1i 9 . . . 4  |-  ( N  e.  ZZ  ->  2  e.  ZZ )
3635, 10zmulcld 8545 . . 3  |-  ( N  e.  ZZ  ->  (
2  x.  ( |_
`  ( ( N  +  1 )  / 
2 ) ) )  e.  ZZ )
37 zleltp1 8476 . . 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 412 . 2  |-  ( N  e.  ZZ  ->  ( N  <_  ( 2  x.  ( |_ `  (
( N  +  1 )  /  2 ) ) )  <->  N  <  ( ( 2  x.  ( |_ `  ( ( N  +  1 )  / 
2 ) ) )  +  1 ) ) )
3933, 38mpbird 165 1  |-  ( N  e.  ZZ  ->  N  <_  ( 2  x.  ( |_ `  ( ( N  +  1 )  / 
2 ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    e. wcel 1434   class class class wbr 3787   ` cfv 4926  (class class class)co 5537   RRcr 7031   1c1 7033    + caddc 7035    x. cmul 7037    < clt 7204    <_ cle 7205    / cdiv 7816   NNcn 8095   2c2 8145   ZZcz 8421   QQcq 8774   RR+crp 8804   |_cfl 9339
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3898  ax-pow 3950  ax-pr 3966  ax-un 4190  ax-setind 4282  ax-cnex 7118  ax-resscn 7119  ax-1cn 7120  ax-1re 7121  ax-icn 7122  ax-addcl 7123  ax-addrcl 7124  ax-mulcl 7125  ax-mulrcl 7126  ax-addcom 7127  ax-mulcom 7128  ax-addass 7129  ax-mulass 7130  ax-distr 7131  ax-i2m1 7132  ax-0lt1 7133  ax-1rid 7134  ax-0id 7135  ax-rnegex 7136  ax-precex 7137  ax-cnre 7138  ax-pre-ltirr 7139  ax-pre-ltwlin 7140  ax-pre-lttrn 7141  ax-pre-apti 7142  ax-pre-ltadd 7143  ax-pre-mulgt0 7144  ax-pre-mulext 7145  ax-arch 7146
This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-nel 2341  df-ral 2354  df-rex 2355  df-reu 2356  df-rmo 2357  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3604  df-int 3639  df-iun 3682  df-br 3788  df-opab 3842  df-mpt 3843  df-id 4050  df-po 4053  df-iso 4054  df-xp 4371  df-rel 4372  df-cnv 4373  df-co 4374  df-dm 4375  df-rn 4376  df-res 4377  df-ima 4378  df-iota 4891  df-fun 4928  df-fn 4929  df-f 4930  df-fv 4934  df-riota 5493  df-ov 5540  df-oprab 5541  df-mpt2 5542  df-1st 5792  df-2nd 5793  df-pnf 7206  df-mnf 7207  df-xr 7208  df-ltxr 7209  df-le 7210  df-sub 7337  df-neg 7338  df-reap 7731  df-ap 7738  df-div 7817  df-inn 8096  df-2 8154  df-n0 8345  df-z 8422  df-q 8775  df-rp 8805  df-fl 9341
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator