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Theorem flhalf 11194
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 10250 . . . . . . . 8  |-  ( N  e.  ZZ  ->  N  e.  RR )
2 peano2re 9203 . . . . . . . 8  |-  ( N  e.  RR  ->  ( N  +  1 )  e.  RR )
31, 2syl 16 . . . . . . 7  |-  ( N  e.  ZZ  ->  ( N  +  1 )  e.  RR )
43rehalfcld 10178 . . . . . 6  |-  ( N  e.  ZZ  ->  (
( N  +  1 )  /  2 )  e.  RR )
5 flltp1 11172 . . . . . 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 11170 . . . . . . . 8  |-  ( N  e.  ZZ  ->  ( |_ `  ( ( N  +  1 )  / 
2 ) )  e.  ZZ )
87zred 10339 . . . . . . 7  |-  ( N  e.  ZZ  ->  ( |_ `  ( ( N  +  1 )  / 
2 ) )  e.  RR )
9 1re 9054 . . . . . . . 8  |-  1  e.  RR
109a1i 11 . . . . . . 7  |-  ( N  e.  ZZ  ->  1  e.  RR )
118, 10readdcld 9079 . . . . . 6  |-  ( N  e.  ZZ  ->  (
( |_ `  (
( N  +  1 )  /  2 ) )  +  1 )  e.  RR )
12 2rp 10581 . . . . . . 7  |-  2  e.  RR+
1312a1i 11 . . . . . 6  |-  ( N  e.  ZZ  ->  2  e.  RR+ )
143, 11, 13ltdivmuld 10659 . . . . 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 9078 . . . . . . 7  |-  ( N  e.  ZZ  ->  1  e.  CC )
17162timesd 10174 . . . . . 6  |-  ( N  e.  ZZ  ->  (
2  x.  1 )  =  ( 1  +  1 ) )
1817oveq2d 6064 . . . . 5  |-  ( N  e.  ZZ  ->  (
( 2  x.  ( |_ `  ( ( N  +  1 )  / 
2 ) ) )  +  ( 2  x.  1 ) )  =  ( ( 2  x.  ( |_ `  (
( N  +  1 )  /  2 ) ) )  +  ( 1  +  1 ) ) )
19 2cn 10034 . . . . . . 7  |-  2  e.  CC
2019a1i 11 . . . . . 6  |-  ( N  e.  ZZ  ->  2  e.  CC )
218recnd 9078 . . . . . 6  |-  ( N  e.  ZZ  ->  ( |_ `  ( ( N  +  1 )  / 
2 ) )  e.  CC )
2220, 21, 16adddid 9076 . . . . 5  |-  ( N  e.  ZZ  ->  (
2  x.  ( ( |_ `  ( ( N  +  1 )  /  2 ) )  +  1 ) )  =  ( ( 2  x.  ( |_ `  ( ( N  + 
1 )  /  2
) ) )  +  ( 2  x.  1 ) ) )
23 2re 10033 . . . . . . . . 9  |-  2  e.  RR
2423a1i 11 . . . . . . . 8  |-  ( N  e.  ZZ  ->  2  e.  RR )
2524, 8remulcld 9080 . . . . . . 7  |-  ( N  e.  ZZ  ->  (
2  x.  ( |_
`  ( ( N  +  1 )  / 
2 ) ) )  e.  RR )
2625recnd 9078 . . . . . 6  |-  ( N  e.  ZZ  ->  (
2  x.  ( |_
`  ( ( N  +  1 )  / 
2 ) ) )  e.  CC )
2726, 16, 16addassd 9074 . . . . 5  |-  ( N  e.  ZZ  ->  (
( ( 2  x.  ( |_ `  (
( N  +  1 )  /  2 ) ) )  +  1 )  +  1 )  =  ( ( 2  x.  ( |_ `  ( ( N  + 
1 )  /  2
) ) )  +  ( 1  +  1 ) ) )
2818, 22, 273eqtr4d 2454 . . . 4  |-  ( N  e.  ZZ  ->  (
2  x.  ( ( |_ `  ( ( N  +  1 )  /  2 ) )  +  1 ) )  =  ( ( ( 2  x.  ( |_
`  ( ( N  +  1 )  / 
2 ) ) )  +  1 )  +  1 ) )
2915, 28breqtrd 4204 . . 3  |-  ( N  e.  ZZ  ->  ( N  +  1 )  <  ( ( ( 2  x.  ( |_
`  ( ( N  +  1 )  / 
2 ) ) )  +  1 )  +  1 ) )
3025, 10readdcld 9079 . . . 4  |-  ( N  e.  ZZ  ->  (
( 2  x.  ( |_ `  ( ( N  +  1 )  / 
2 ) ) )  +  1 )  e.  RR )
311, 30, 10ltadd1d 9583 . . 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 10276 . . . . 5  |-  2  e.  ZZ
3433a1i 11 . . . 4  |-  ( N  e.  ZZ  ->  2  e.  ZZ )
3534, 7zmulcld 10345 . . 3  |-  ( N  e.  ZZ  ->  (
2  x.  ( |_
`  ( ( N  +  1 )  / 
2 ) ) )  e.  ZZ )
36 zleltp1 10290 . . 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 4180   ` cfv 5421  (class class class)co 6048   CCcc 8952   RRcr 8953   1c1 8955    + caddc 8957    x. cmul 8959    < clt 9084    <_ cle 9085    / cdiv 9641   2c2 10013   ZZcz 10246   RR+crp 10576   |_cfl 11164
This theorem is referenced by:  ovolunlem1a  19353
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 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031  ax-pre-sup 9032
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-riota 6516  df-recs 6600  df-rdg 6635  df-er 6872  df-en 7077  df-dom 7078  df-sdom 7079  df-sup 7412  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-div 9642  df-nn 9965  df-2 10022  df-n0 10186  df-z 10247  df-uz 10453  df-rp 10577  df-fl 11165
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