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Theorem flhalf 11186
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 zre 10242 . . . . . . . 8  |-  ( N  e.  ZZ  ->  N  e.  RR )
2 peano2re 9195 . . . . . . . 8  |-  ( N  e.  RR  ->  ( N  +  1 )  e.  RR )
31, 2syl 16 . . . . . . 7  |-  ( N  e.  ZZ  ->  ( N  +  1 )  e.  RR )
43rehalfcld 10170 . . . . . 6  |-  ( N  e.  ZZ  ->  (
( N  +  1 )  /  2 )  e.  RR )
5 flltp1 11164 . . . . . 6  |-  ( ( ( N  +  1 )  /  2 )  e.  RR  ->  (
( N  +  1 )  /  2 )  <  ( ( |_
`  ( ( N  +  1 )  / 
2 ) )  +  1 ) )
64, 5syl 16 . . . . 5  |-  ( N  e.  ZZ  ->  (
( N  +  1 )  /  2 )  <  ( ( |_
`  ( ( N  +  1 )  / 
2 ) )  +  1 ) )
74flcld 11162 . . . . . . . 8  |-  ( N  e.  ZZ  ->  ( |_ `  ( ( N  +  1 )  / 
2 ) )  e.  ZZ )
87zred 10331 . . . . . . 7  |-  ( N  e.  ZZ  ->  ( |_ `  ( ( N  +  1 )  / 
2 ) )  e.  RR )
9 1re 9046 . . . . . . . 8  |-  1  e.  RR
109a1i 11 . . . . . . 7  |-  ( N  e.  ZZ  ->  1  e.  RR )
118, 10readdcld 9071 . . . . . 6  |-  ( N  e.  ZZ  ->  (
( |_ `  (
( N  +  1 )  /  2 ) )  +  1 )  e.  RR )
12 2rp 10573 . . . . . . 7  |-  2  e.  RR+
1312a1i 11 . . . . . 6  |-  ( N  e.  ZZ  ->  2  e.  RR+ )
143, 11, 13ltdivmuld 10651 . . . . 5  |-  ( N  e.  ZZ  ->  (
( ( N  + 
1 )  /  2
)  <  ( ( |_ `  ( ( N  +  1 )  / 
2 ) )  +  1 )  <->  ( N  +  1 )  < 
( 2  x.  (
( |_ `  (
( N  +  1 )  /  2 ) )  +  1 ) ) ) )
156, 14mpbid 202 . . . 4  |-  ( N  e.  ZZ  ->  ( N  +  1 )  <  ( 2  x.  ( ( |_ `  ( ( N  + 
1 )  /  2
) )  +  1 ) ) )
1610recnd 9070 . . . . . . 7  |-  ( N  e.  ZZ  ->  1  e.  CC )
17162timesd 10166 . . . . . 6  |-  ( N  e.  ZZ  ->  (
2  x.  1 )  =  ( 1  +  1 ) )
1817oveq2d 6056 . . . . 5  |-  ( N  e.  ZZ  ->  (
( 2  x.  ( |_ `  ( ( N  +  1 )  / 
2 ) ) )  +  ( 2  x.  1 ) )  =  ( ( 2  x.  ( |_ `  (
( N  +  1 )  /  2 ) ) )  +  ( 1  +  1 ) ) )
19 2cn 10026 . . . . . . 7  |-  2  e.  CC
2019a1i 11 . . . . . 6  |-  ( N  e.  ZZ  ->  2  e.  CC )
218recnd 9070 . . . . . 6  |-  ( N  e.  ZZ  ->  ( |_ `  ( ( N  +  1 )  / 
2 ) )  e.  CC )
2220, 21, 16adddid 9068 . . . . 5  |-  ( N  e.  ZZ  ->  (
2  x.  ( ( |_ `  ( ( N  +  1 )  /  2 ) )  +  1 ) )  =  ( ( 2  x.  ( |_ `  ( ( N  + 
1 )  /  2
) ) )  +  ( 2  x.  1 ) ) )
23 2re 10025 . . . . . . . . 9  |-  2  e.  RR
2423a1i 11 . . . . . . . 8  |-  ( N  e.  ZZ  ->  2  e.  RR )
2524, 8remulcld 9072 . . . . . . 7  |-  ( N  e.  ZZ  ->  (
2  x.  ( |_
`  ( ( N  +  1 )  / 
2 ) ) )  e.  RR )
2625recnd 9070 . . . . . 6  |-  ( N  e.  ZZ  ->  (
2  x.  ( |_
`  ( ( N  +  1 )  / 
2 ) ) )  e.  CC )
2726, 16, 16addassd 9066 . . . . 5  |-  ( N  e.  ZZ  ->  (
( ( 2  x.  ( |_ `  (
( N  +  1 )  /  2 ) ) )  +  1 )  +  1 )  =  ( ( 2  x.  ( |_ `  ( ( N  + 
1 )  /  2
) ) )  +  ( 1  +  1 ) ) )
2818, 22, 273eqtr4d 2446 . . . 4  |-  ( N  e.  ZZ  ->  (
2  x.  ( ( |_ `  ( ( N  +  1 )  /  2 ) )  +  1 ) )  =  ( ( ( 2  x.  ( |_
`  ( ( N  +  1 )  / 
2 ) ) )  +  1 )  +  1 ) )
2915, 28breqtrd 4196 . . 3  |-  ( N  e.  ZZ  ->  ( N  +  1 )  <  ( ( ( 2  x.  ( |_
`  ( ( N  +  1 )  / 
2 ) ) )  +  1 )  +  1 ) )
3025, 10readdcld 9071 . . . 4  |-  ( N  e.  ZZ  ->  (
( 2  x.  ( |_ `  ( ( N  +  1 )  / 
2 ) ) )  +  1 )  e.  RR )
311, 30, 10ltadd1d 9575 . . 3  |-  ( N  e.  ZZ  ->  ( N  <  ( ( 2  x.  ( |_ `  ( ( N  + 
1 )  /  2
) ) )  +  1 )  <->  ( N  +  1 )  < 
( ( ( 2  x.  ( |_ `  ( ( N  + 
1 )  /  2
) ) )  +  1 )  +  1 ) ) )
3229, 31mpbird 224 . 2  |-  ( N  e.  ZZ  ->  N  <  ( ( 2  x.  ( |_ `  (
( N  +  1 )  /  2 ) ) )  +  1 ) )
33 2z 10268 . . . . 5  |-  2  e.  ZZ
3433a1i 11 . . . 4  |-  ( N  e.  ZZ  ->  2  e.  ZZ )
3534, 7zmulcld 10337 . . 3  |-  ( N  e.  ZZ  ->  (
2  x.  ( |_
`  ( ( N  +  1 )  / 
2 ) ) )  e.  ZZ )
36 zleltp1 10282 . . 3  |-  ( ( N  e.  ZZ  /\  ( 2  x.  ( |_ `  ( ( N  +  1 )  / 
2 ) ) )  e.  ZZ )  -> 
( N  <_  (
2  x.  ( |_
`  ( ( N  +  1 )  / 
2 ) ) )  <-> 
N  <  ( (
2  x.  ( |_
`  ( ( N  +  1 )  / 
2 ) ) )  +  1 ) ) )
3735, 36mpdan 650 . 2  |-  ( N  e.  ZZ  ->  ( N  <_  ( 2  x.  ( |_ `  (
( N  +  1 )  /  2 ) ) )  <->  N  <  ( ( 2  x.  ( |_ `  ( ( N  +  1 )  / 
2 ) ) )  +  1 ) ) )
3832, 37mpbird 224 1  |-  ( N  e.  ZZ  ->  N  <_  ( 2  x.  ( |_ `  ( ( N  +  1 )  / 
2 ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    e. wcel 1721   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   CCcc 8944   RRcr 8945   1c1 8947    + caddc 8949    x. cmul 8951    < clt 9076    <_ cle 9077    / cdiv 9633   2c2 10005   ZZcz 10238   RR+crp 10568   |_cfl 11156
This theorem is referenced by:  ovolunlem1a  19345
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-fl 11157
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