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Theorem rebtwnz 10315
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 9110 . . 3  |-  ( A  e.  RR  ->  -u A  e.  RR )
2 zbtwnre 10314 . . 3  |-  ( -u A  e.  RR  ->  E! y  e.  ZZ  ( -u A  <_  y  /\  y  <  ( -u A  +  1 ) ) )
31, 2syl 15 . 2  |-  ( A  e.  RR  ->  E! y  e.  ZZ  ( -u A  <_  y  /\  y  <  ( -u A  +  1 ) ) )
4 znegcl 10055 . . . 4  |-  ( x  e.  ZZ  ->  -u x  e.  ZZ )
5 znegcl 10055 . . . . 5  |-  ( y  e.  ZZ  ->  -u y  e.  ZZ )
6 zcn 10029 . . . . . 6  |-  ( y  e.  ZZ  ->  y  e.  CC )
7 zcn 10029 . . . . . 6  |-  ( x  e.  ZZ  ->  x  e.  CC )
8 negcon2 9100 . . . . . 6  |-  ( ( y  e.  CC  /\  x  e.  CC )  ->  ( y  =  -u x 
<->  x  =  -u y
) )
96, 7, 8syl2an 463 . . . . 5  |-  ( ( y  e.  ZZ  /\  x  e.  ZZ )  ->  ( y  =  -u x 
<->  x  =  -u y
) )
105, 9reuhyp 4562 . . . 4  |-  ( y  e.  ZZ  ->  E! x  e.  ZZ  y  =  -u x )
11 breq2 4027 . . . . 5  |-  ( y  =  -u x  ->  ( -u A  <_  y  <->  -u A  <_  -u x ) )
12 breq1 4026 . . . . 5  |-  ( y  =  -u x  ->  (
y  <  ( -u A  +  1 )  <->  -u x  < 
( -u A  +  1 ) ) )
1311, 12anbi12d 691 . . . 4  |-  ( y  =  -u x  ->  (
( -u A  <_  y  /\  y  <  ( -u A  +  1 ) )  <->  ( -u A  <_ 
-u x  /\  -u x  <  ( -u A  + 
1 ) ) ) )
144, 10, 13reuxfr 4560 . . 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 10028 . . . . . 6  |-  ( x  e.  ZZ  ->  x  e.  RR )
16 leneg 9277 . . . . . . . 8  |-  ( ( x  e.  RR  /\  A  e.  RR )  ->  ( x  <_  A  <->  -u A  <_  -u x ) )
1716ancoms 439 . . . . . . 7  |-  ( ( A  e.  RR  /\  x  e.  RR )  ->  ( x  <_  A  <->  -u A  <_  -u x ) )
18 peano2rem 9113 . . . . . . . . 9  |-  ( A  e.  RR  ->  ( A  -  1 )  e.  RR )
19 ltneg 9274 . . . . . . . . 9  |-  ( ( ( A  -  1 )  e.  RR  /\  x  e.  RR )  ->  ( ( A  - 
1 )  <  x  <->  -u x  <  -u ( A  -  1 ) ) )
2018, 19sylan 457 . . . . . . . 8  |-  ( ( A  e.  RR  /\  x  e.  RR )  ->  ( ( A  - 
1 )  <  x  <->  -u x  <  -u ( A  -  1 ) ) )
21 1re 8837 . . . . . . . . 9  |-  1  e.  RR
22 ltsubadd 9244 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  1  e.  RR  /\  x  e.  RR )  ->  (
( A  -  1 )  <  x  <->  A  <  ( x  +  1 ) ) )
2321, 22mp3an2 1265 . . . . . . . 8  |-  ( ( A  e.  RR  /\  x  e.  RR )  ->  ( ( A  - 
1 )  <  x  <->  A  <  ( x  + 
1 ) ) )
24 recn 8827 . . . . . . . . . . 11  |-  ( A  e.  RR  ->  A  e.  CC )
25 ax-1cn 8795 . . . . . . . . . . 11  |-  1  e.  CC
26 negsubdi 9103 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  1  e.  CC )  -> 
-u ( A  - 
1 )  =  (
-u A  +  1 ) )
2724, 25, 26sylancl 643 . . . . . . . . . 10  |-  ( A  e.  RR  ->  -u ( A  -  1 )  =  ( -u A  +  1 ) )
2827adantr 451 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  x  e.  RR )  -> 
-u ( A  - 
1 )  =  (
-u A  +  1 ) )
2928breq2d 4035 . . . . . . . 8  |-  ( ( A  e.  RR  /\  x  e.  RR )  ->  ( -u x  <  -u ( A  -  1 )  <->  -u x  <  ( -u A  +  1 ) ) )
3020, 23, 293bitr3d 274 . . . . . . 7  |-  ( ( A  e.  RR  /\  x  e.  RR )  ->  ( A  <  (
x  +  1 )  <->  -u x  <  ( -u A  +  1 ) ) )
3117, 30anbi12d 691 . . . . . 6  |-  ( ( A  e.  RR  /\  x  e.  RR )  ->  ( ( x  <_  A  /\  A  <  (
x  +  1 ) )  <->  ( -u A  <_ 
-u x  /\  -u x  <  ( -u A  + 
1 ) ) ) )
3215, 31sylan2 460 . . . . 5  |-  ( ( A  e.  RR  /\  x  e.  ZZ )  ->  ( ( x  <_  A  /\  A  <  (
x  +  1 ) )  <->  ( -u A  <_ 
-u x  /\  -u x  <  ( -u A  + 
1 ) ) ) )
3332bicomd 192 . . . 4  |-  ( ( A  e.  RR  /\  x  e.  ZZ )  ->  ( ( -u A  <_ 
-u x  /\  -u x  <  ( -u A  + 
1 ) )  <->  ( x  <_  A  /\  A  < 
( x  +  1 ) ) ) )
3433reubidva 2723 . . 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 248 . 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 201 1  |-  ( A  e.  RR  ->  E! x  e.  ZZ  (
x  <_  A  /\  A  <  ( x  + 
1 ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   E!wreu 2545   class class class wbr 4023  (class class class)co 5858   CCcc 8735   RRcr 8736   1c1 8738    + caddc 8740    < clt 8867    <_ cle 8868    - cmin 9037   -ucneg 9038   ZZcz 10024
This theorem is referenced by:  flcl  10927  fllelt  10929  flbi  10946
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231
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