MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  flhalf Unicode version

Theorem flhalf 11043
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 10117 . . . . . . . 8  |-  ( N  e.  ZZ  ->  N  e.  RR )
2 peano2re 9072 . . . . . . . 8  |-  ( N  e.  RR  ->  ( N  +  1 )  e.  RR )
31, 2syl 15 . . . . . . 7  |-  ( N  e.  ZZ  ->  ( N  +  1 )  e.  RR )
43rehalfcld 10047 . . . . . 6  |-  ( N  e.  ZZ  ->  (
( N  +  1 )  /  2 )  e.  RR )
5 flltp1 11021 . . . . . 6  |-  ( ( ( N  +  1 )  /  2 )  e.  RR  ->  (
( N  +  1 )  /  2 )  <  ( ( |_
`  ( ( N  +  1 )  / 
2 ) )  +  1 ) )
64, 5syl 15 . . . . 5  |-  ( N  e.  ZZ  ->  (
( N  +  1 )  /  2 )  <  ( ( |_
`  ( ( N  +  1 )  / 
2 ) )  +  1 ) )
74flcld 11019 . . . . . . . 8  |-  ( N  e.  ZZ  ->  ( |_ `  ( ( N  +  1 )  / 
2 ) )  e.  ZZ )
87zred 10206 . . . . . . 7  |-  ( N  e.  ZZ  ->  ( |_ `  ( ( N  +  1 )  / 
2 ) )  e.  RR )
9 1re 8924 . . . . . . . 8  |-  1  e.  RR
109a1i 10 . . . . . . 7  |-  ( N  e.  ZZ  ->  1  e.  RR )
118, 10readdcld 8949 . . . . . 6  |-  ( N  e.  ZZ  ->  (
( |_ `  (
( N  +  1 )  /  2 ) )  +  1 )  e.  RR )
12 2rp 10448 . . . . . . 7  |-  2  e.  RR+
1312a1i 10 . . . . . 6  |-  ( N  e.  ZZ  ->  2  e.  RR+ )
143, 11, 13ltdivmuld 10526 . . . . 5  |-  ( N  e.  ZZ  ->  (
( ( N  + 
1 )  /  2
)  <  ( ( |_ `  ( ( N  +  1 )  / 
2 ) )  +  1 )  <->  ( N  +  1 )  < 
( 2  x.  (
( |_ `  (
( N  +  1 )  /  2 ) )  +  1 ) ) ) )
156, 14mpbid 201 . . . 4  |-  ( N  e.  ZZ  ->  ( N  +  1 )  <  ( 2  x.  ( ( |_ `  ( ( N  + 
1 )  /  2
) )  +  1 ) ) )
1610recnd 8948 . . . . . . 7  |-  ( N  e.  ZZ  ->  1  e.  CC )
17162timesd 10043 . . . . . 6  |-  ( N  e.  ZZ  ->  (
2  x.  1 )  =  ( 1  +  1 ) )
1817oveq2d 5958 . . . . 5  |-  ( N  e.  ZZ  ->  (
( 2  x.  ( |_ `  ( ( N  +  1 )  / 
2 ) ) )  +  ( 2  x.  1 ) )  =  ( ( 2  x.  ( |_ `  (
( N  +  1 )  /  2 ) ) )  +  ( 1  +  1 ) ) )
19 2cn 9903 . . . . . . 7  |-  2  e.  CC
2019a1i 10 . . . . . 6  |-  ( N  e.  ZZ  ->  2  e.  CC )
218recnd 8948 . . . . . 6  |-  ( N  e.  ZZ  ->  ( |_ `  ( ( N  +  1 )  / 
2 ) )  e.  CC )
2220, 21, 16adddid 8946 . . . . 5  |-  ( N  e.  ZZ  ->  (
2  x.  ( ( |_ `  ( ( N  +  1 )  /  2 ) )  +  1 ) )  =  ( ( 2  x.  ( |_ `  ( ( N  + 
1 )  /  2
) ) )  +  ( 2  x.  1 ) ) )
23 2re 9902 . . . . . . . . 9  |-  2  e.  RR
2423a1i 10 . . . . . . . 8  |-  ( N  e.  ZZ  ->  2  e.  RR )
2524, 8remulcld 8950 . . . . . . 7  |-  ( N  e.  ZZ  ->  (
2  x.  ( |_
`  ( ( N  +  1 )  / 
2 ) ) )  e.  RR )
2625recnd 8948 . . . . . 6  |-  ( N  e.  ZZ  ->  (
2  x.  ( |_
`  ( ( N  +  1 )  / 
2 ) ) )  e.  CC )
2726, 16, 16addassd 8944 . . . . 5  |-  ( N  e.  ZZ  ->  (
( ( 2  x.  ( |_ `  (
( N  +  1 )  /  2 ) ) )  +  1 )  +  1 )  =  ( ( 2  x.  ( |_ `  ( ( N  + 
1 )  /  2
) ) )  +  ( 1  +  1 ) ) )
2818, 22, 273eqtr4d 2400 . . . 4  |-  ( N  e.  ZZ  ->  (
2  x.  ( ( |_ `  ( ( N  +  1 )  /  2 ) )  +  1 ) )  =  ( ( ( 2  x.  ( |_
`  ( ( N  +  1 )  / 
2 ) ) )  +  1 )  +  1 ) )
2915, 28breqtrd 4126 . . 3  |-  ( N  e.  ZZ  ->  ( N  +  1 )  <  ( ( ( 2  x.  ( |_
`  ( ( N  +  1 )  / 
2 ) ) )  +  1 )  +  1 ) )
3025, 10readdcld 8949 . . . 4  |-  ( N  e.  ZZ  ->  (
( 2  x.  ( |_ `  ( ( N  +  1 )  / 
2 ) ) )  +  1 )  e.  RR )
311, 30, 10ltadd1d 9452 . . 3  |-  ( N  e.  ZZ  ->  ( N  <  ( ( 2  x.  ( |_ `  ( ( N  + 
1 )  /  2
) ) )  +  1 )  <->  ( N  +  1 )  < 
( ( ( 2  x.  ( |_ `  ( ( N  + 
1 )  /  2
) ) )  +  1 )  +  1 ) ) )
3229, 31mpbird 223 . 2  |-  ( N  e.  ZZ  ->  N  <  ( ( 2  x.  ( |_ `  (
( N  +  1 )  /  2 ) ) )  +  1 ) )
33 2z 10143 . . . . 5  |-  2  e.  ZZ
3433a1i 10 . . . 4  |-  ( N  e.  ZZ  ->  2  e.  ZZ )
3534, 7zmulcld 10212 . . 3  |-  ( N  e.  ZZ  ->  (
2  x.  ( |_
`  ( ( N  +  1 )  / 
2 ) ) )  e.  ZZ )
36 zleltp1 10157 . . 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 649 . 2  |-  ( N  e.  ZZ  ->  ( N  <_  ( 2  x.  ( |_ `  (
( N  +  1 )  /  2 ) ) )  <->  N  <  ( ( 2  x.  ( |_ `  ( ( N  +  1 )  / 
2 ) ) )  +  1 ) ) )
3832, 37mpbird 223 1  |-  ( N  e.  ZZ  ->  N  <_  ( 2  x.  ( |_ `  ( ( N  +  1 )  / 
2 ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    e. wcel 1710   class class class wbr 4102   ` cfv 5334  (class class class)co 5942   CCcc 8822   RRcr 8823   1c1 8825    + caddc 8827    x. cmul 8829    < clt 8954    <_ cle 8955    / cdiv 9510   2c2 9882   ZZcz 10113   RR+crp 10443   |_cfl 11013
This theorem is referenced by:  ovolunlem1a  18953
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-cnex 8880  ax-resscn 8881  ax-1cn 8882  ax-icn 8883  ax-addcl 8884  ax-addrcl 8885  ax-mulcl 8886  ax-mulrcl 8887  ax-mulcom 8888  ax-addass 8889  ax-mulass 8890  ax-distr 8891  ax-i2m1 8892  ax-1ne0 8893  ax-1rid 8894  ax-rnegex 8895  ax-rrecex 8896  ax-cnre 8897  ax-pre-lttri 8898  ax-pre-lttrn 8899  ax-pre-ltadd 8900  ax-pre-mulgt0 8901  ax-pre-sup 8902
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-we 4433  df-ord 4474  df-on 4475  df-lim 4476  df-suc 4477  df-om 4736  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-riota 6388  df-recs 6472  df-rdg 6507  df-er 6744  df-en 6949  df-dom 6950  df-sdom 6951  df-sup 7281  df-pnf 8956  df-mnf 8957  df-xr 8958  df-ltxr 8959  df-le 8960  df-sub 9126  df-neg 9127  df-div 9511  df-nn 9834  df-2 9891  df-n0 10055  df-z 10114  df-uz 10320  df-rp 10444  df-fl 11014
  Copyright terms: Public domain W3C validator