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Theorem rebtwnz 10529
Description: There is a unique greatest integer less than or equal to a real number. Exercise 4 of [Apostol] p. 28. (Contributed by NM, 15-Nov-2004.)
Assertion
Ref Expression
rebtwnz  |-  ( A  e.  RR  ->  E! x  e.  ZZ  (
x  <_  A  /\  A  <  ( x  + 
1 ) ) )
Distinct variable group:    x, A

Proof of Theorem rebtwnz
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 renegcl 9320 . . 3  |-  ( A  e.  RR  ->  -u A  e.  RR )
2 zbtwnre 10528 . . 3  |-  ( -u A  e.  RR  ->  E! y  e.  ZZ  ( -u A  <_  y  /\  y  <  ( -u A  +  1 ) ) )
31, 2syl 16 . 2  |-  ( A  e.  RR  ->  E! y  e.  ZZ  ( -u A  <_  y  /\  y  <  ( -u A  +  1 ) ) )
4 znegcl 10269 . . . 4  |-  ( x  e.  ZZ  ->  -u x  e.  ZZ )
5 znegcl 10269 . . . . 5  |-  ( y  e.  ZZ  ->  -u y  e.  ZZ )
6 zcn 10243 . . . . . 6  |-  ( y  e.  ZZ  ->  y  e.  CC )
7 zcn 10243 . . . . . 6  |-  ( x  e.  ZZ  ->  x  e.  CC )
8 negcon2 9310 . . . . . 6  |-  ( ( y  e.  CC  /\  x  e.  CC )  ->  ( y  =  -u x 
<->  x  =  -u y
) )
96, 7, 8syl2an 464 . . . . 5  |-  ( ( y  e.  ZZ  /\  x  e.  ZZ )  ->  ( y  =  -u x 
<->  x  =  -u y
) )
105, 9reuhyp 4710 . . . 4  |-  ( y  e.  ZZ  ->  E! x  e.  ZZ  y  =  -u x )
11 breq2 4176 . . . . 5  |-  ( y  =  -u x  ->  ( -u A  <_  y  <->  -u A  <_  -u x ) )
12 breq1 4175 . . . . 5  |-  ( y  =  -u x  ->  (
y  <  ( -u A  +  1 )  <->  -u x  < 
( -u A  +  1 ) ) )
1311, 12anbi12d 692 . . . 4  |-  ( y  =  -u x  ->  (
( -u A  <_  y  /\  y  <  ( -u A  +  1 ) )  <->  ( -u A  <_ 
-u x  /\  -u x  <  ( -u A  + 
1 ) ) ) )
144, 10, 13reuxfr 4708 . . 3  |-  ( E! y  e.  ZZ  ( -u A  <_  y  /\  y  <  ( -u A  +  1 ) )  <-> 
E! x  e.  ZZ  ( -u A  <_  -u x  /\  -u x  <  ( -u A  +  1 ) ) )
15 zre 10242 . . . . . 6  |-  ( x  e.  ZZ  ->  x  e.  RR )
16 leneg 9487 . . . . . . . 8  |-  ( ( x  e.  RR  /\  A  e.  RR )  ->  ( x  <_  A  <->  -u A  <_  -u x ) )
1716ancoms 440 . . . . . . 7  |-  ( ( A  e.  RR  /\  x  e.  RR )  ->  ( x  <_  A  <->  -u A  <_  -u x ) )
18 peano2rem 9323 . . . . . . . . 9  |-  ( A  e.  RR  ->  ( A  -  1 )  e.  RR )
19 ltneg 9484 . . . . . . . . 9  |-  ( ( ( A  -  1 )  e.  RR  /\  x  e.  RR )  ->  ( ( A  - 
1 )  <  x  <->  -u x  <  -u ( A  -  1 ) ) )
2018, 19sylan 458 . . . . . . . 8  |-  ( ( A  e.  RR  /\  x  e.  RR )  ->  ( ( A  - 
1 )  <  x  <->  -u x  <  -u ( A  -  1 ) ) )
21 1re 9046 . . . . . . . . 9  |-  1  e.  RR
22 ltsubadd 9454 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  1  e.  RR  /\  x  e.  RR )  ->  (
( A  -  1 )  <  x  <->  A  <  ( x  +  1 ) ) )
2321, 22mp3an2 1267 . . . . . . . 8  |-  ( ( A  e.  RR  /\  x  e.  RR )  ->  ( ( A  - 
1 )  <  x  <->  A  <  ( x  + 
1 ) ) )
24 recn 9036 . . . . . . . . . . 11  |-  ( A  e.  RR  ->  A  e.  CC )
25 ax-1cn 9004 . . . . . . . . . . 11  |-  1  e.  CC
26 negsubdi 9313 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  1  e.  CC )  -> 
-u ( A  - 
1 )  =  (
-u A  +  1 ) )
2724, 25, 26sylancl 644 . . . . . . . . . 10  |-  ( A  e.  RR  ->  -u ( A  -  1 )  =  ( -u A  +  1 ) )
2827adantr 452 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  x  e.  RR )  -> 
-u ( A  - 
1 )  =  (
-u A  +  1 ) )
2928breq2d 4184 . . . . . . . 8  |-  ( ( A  e.  RR  /\  x  e.  RR )  ->  ( -u x  <  -u ( A  -  1 )  <->  -u x  <  ( -u A  +  1 ) ) )
3020, 23, 293bitr3d 275 . . . . . . 7  |-  ( ( A  e.  RR  /\  x  e.  RR )  ->  ( A  <  (
x  +  1 )  <->  -u x  <  ( -u A  +  1 ) ) )
3117, 30anbi12d 692 . . . . . 6  |-  ( ( A  e.  RR  /\  x  e.  RR )  ->  ( ( x  <_  A  /\  A  <  (
x  +  1 ) )  <->  ( -u A  <_ 
-u x  /\  -u x  <  ( -u A  + 
1 ) ) ) )
3215, 31sylan2 461 . . . . 5  |-  ( ( A  e.  RR  /\  x  e.  ZZ )  ->  ( ( x  <_  A  /\  A  <  (
x  +  1 ) )  <->  ( -u A  <_ 
-u x  /\  -u x  <  ( -u A  + 
1 ) ) ) )
3332bicomd 193 . . . 4  |-  ( ( A  e.  RR  /\  x  e.  ZZ )  ->  ( ( -u A  <_ 
-u x  /\  -u x  <  ( -u A  + 
1 ) )  <->  ( x  <_  A  /\  A  < 
( x  +  1 ) ) ) )
3433reubidva 2851 . . 3  |-  ( A  e.  RR  ->  ( E! x  e.  ZZ  ( -u A  <_  -u x  /\  -u x  <  ( -u A  +  1 ) )  <->  E! x  e.  ZZ  ( x  <_  A  /\  A  <  ( x  + 
1 ) ) ) )
3514, 34syl5bb 249 . 2  |-  ( A  e.  RR  ->  ( E! y  e.  ZZ  ( -u A  <_  y  /\  y  <  ( -u A  +  1 ) )  <->  E! x  e.  ZZ  ( x  <_  A  /\  A  <  ( x  + 
1 ) ) ) )
363, 35mpbid 202 1  |-  ( A  e.  RR  ->  E! x  e.  ZZ  (
x  <_  A  /\  A  <  ( x  + 
1 ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   E!wreu 2668   class class class wbr 4172  (class class class)co 6040   CCcc 8944   RRcr 8945   1c1 8947    + caddc 8949    < clt 9076    <_ cle 9077    - cmin 9247   -ucneg 9248   ZZcz 10238
This theorem is referenced by:  flcl  11159  fllelt  11161  flbi  11178  ltflcei  26140  lxflflp1  26142
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-nn 9957  df-n0 10178  df-z 10239  df-uz 10445
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