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Theorem rebtwnz 10312
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
Dummy variable  y is distinct from all other variables.

Proof of Theorem rebtwnz
StepHypRef Expression
1 renegcl 9107 . . 3  |-  ( A  e.  RR  ->  -u A  e.  RR )
2 zbtwnre 10311 . . 3  |-  ( -u A  e.  RR  ->  E! y  e.  ZZ  ( -u A  <_  y  /\  y  <  ( -u A  +  1 ) ) )
31, 2syl 17 . 2  |-  ( A  e.  RR  ->  E! y  e.  ZZ  ( -u A  <_  y  /\  y  <  ( -u A  +  1 ) ) )
4 znegcl 10052 . . . 4  |-  ( x  e.  ZZ  ->  -u x  e.  ZZ )
5 znegcl 10052 . . . . 5  |-  ( y  e.  ZZ  ->  -u y  e.  ZZ )
6 zcn 10026 . . . . . 6  |-  ( y  e.  ZZ  ->  y  e.  CC )
7 zcn 10026 . . . . . 6  |-  ( x  e.  ZZ  ->  x  e.  CC )
8 negcon2 9097 . . . . . 6  |-  ( ( y  e.  CC  /\  x  e.  CC )  ->  ( y  =  -u x 
<->  x  =  -u y
) )
96, 7, 8syl2an 465 . . . . 5  |-  ( ( y  e.  ZZ  /\  x  e.  ZZ )  ->  ( y  =  -u x 
<->  x  =  -u y
) )
105, 9reuhyp 4563 . . . 4  |-  ( y  e.  ZZ  ->  E! x  e.  ZZ  y  =  -u x )
11 breq2 4030 . . . . 5  |-  ( y  =  -u x  ->  ( -u A  <_  y  <->  -u A  <_  -u x ) )
12 breq1 4029 . . . . 5  |-  ( y  =  -u x  ->  (
y  <  ( -u A  +  1 )  <->  -u x  < 
( -u A  +  1 ) ) )
1311, 12anbi12d 693 . . . 4  |-  ( y  =  -u x  ->  (
( -u A  <_  y  /\  y  <  ( -u A  +  1 ) )  <->  ( -u A  <_ 
-u x  /\  -u x  <  ( -u A  + 
1 ) ) ) )
144, 10, 13reuxfr 4561 . . 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 10025 . . . . . 6  |-  ( x  e.  ZZ  ->  x  e.  RR )
16 leneg 9274 . . . . . . . 8  |-  ( ( x  e.  RR  /\  A  e.  RR )  ->  ( x  <_  A  <->  -u A  <_  -u x ) )
1716ancoms 441 . . . . . . 7  |-  ( ( A  e.  RR  /\  x  e.  RR )  ->  ( x  <_  A  <->  -u A  <_  -u x ) )
18 peano2rem 9110 . . . . . . . . 9  |-  ( A  e.  RR  ->  ( A  -  1 )  e.  RR )
19 ltneg 9271 . . . . . . . . 9  |-  ( ( ( A  -  1 )  e.  RR  /\  x  e.  RR )  ->  ( ( A  - 
1 )  <  x  <->  -u x  <  -u ( A  -  1 ) ) )
2018, 19sylan 459 . . . . . . . 8  |-  ( ( A  e.  RR  /\  x  e.  RR )  ->  ( ( A  - 
1 )  <  x  <->  -u x  <  -u ( A  -  1 ) ) )
21 1re 8834 . . . . . . . . 9  |-  1  e.  RR
22 ltsubadd 9241 . . . . . . . . 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 8824 . . . . . . . . . . 11  |-  ( A  e.  RR  ->  A  e.  CC )
25 ax-1cn 8792 . . . . . . . . . . 11  |-  1  e.  CC
26 negsubdi 9100 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  1  e.  CC )  -> 
-u ( A  - 
1 )  =  (
-u A  +  1 ) )
2724, 25, 26sylancl 645 . . . . . . . . . 10  |-  ( A  e.  RR  ->  -u ( A  -  1 )  =  ( -u A  +  1 ) )
2827adantr 453 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  x  e.  RR )  -> 
-u ( A  - 
1 )  =  (
-u A  +  1 ) )
2928breq2d 4038 . . . . . . . 8  |-  ( ( A  e.  RR  /\  x  e.  RR )  ->  ( -u x  <  -u ( A  -  1 )  <->  -u x  <  ( -u A  +  1 ) ) )
3020, 23, 293bitr3d 276 . . . . . . 7  |-  ( ( A  e.  RR  /\  x  e.  RR )  ->  ( A  <  (
x  +  1 )  <->  -u x  <  ( -u A  +  1 ) ) )
3117, 30anbi12d 693 . . . . . 6  |-  ( ( A  e.  RR  /\  x  e.  RR )  ->  ( ( x  <_  A  /\  A  <  (
x  +  1 ) )  <->  ( -u A  <_ 
-u x  /\  -u x  <  ( -u A  + 
1 ) ) ) )
3215, 31sylan2 462 . . . . 5  |-  ( ( A  e.  RR  /\  x  e.  ZZ )  ->  ( ( x  <_  A  /\  A  <  (
x  +  1 ) )  <->  ( -u A  <_ 
-u x  /\  -u x  <  ( -u A  + 
1 ) ) ) )
3332bicomd 194 . . . 4  |-  ( ( A  e.  RR  /\  x  e.  ZZ )  ->  ( ( -u A  <_ 
-u x  /\  -u x  <  ( -u A  + 
1 ) )  <->  ( x  <_  A  /\  A  < 
( x  +  1 ) ) ) )
3433reubidva 2726 . . 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 250 . 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 203 1  |-  ( A  e.  RR  ->  E! x  e.  ZZ  (
x  <_  A  /\  A  <  ( x  + 
1 ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    = wceq 1625    e. wcel 1687   E!wreu 2548   class class class wbr 4026  (class class class)co 5821   CCcc 8732   RRcr 8733   1c1 8735    + caddc 8737    < clt 8864    <_ cle 8865    - cmin 9034   -ucneg 9035   ZZcz 10021
This theorem is referenced by:  flcl  10923  fllelt  10925  flbi  10942
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1638  ax-8 1646  ax-13 1689  ax-14 1691  ax-6 1706  ax-7 1711  ax-11 1718  ax-12 1870  ax-ext 2267  ax-sep 4144  ax-nul 4152  ax-pow 4189  ax-pr 4215  ax-un 4513  ax-cnex 8790  ax-resscn 8791  ax-1cn 8792  ax-icn 8793  ax-addcl 8794  ax-addrcl 8795  ax-mulcl 8796  ax-mulrcl 8797  ax-mulcom 8798  ax-addass 8799  ax-mulass 8800  ax-distr 8801  ax-i2m1 8802  ax-1ne0 8803  ax-1rid 8804  ax-rnegex 8805  ax-rrecex 8806  ax-cnre 8807  ax-pre-lttri 8808  ax-pre-lttrn 8809  ax-pre-ltadd 8810  ax-pre-mulgt0 8811  ax-pre-sup 8812
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1531  df-nf 1534  df-sb 1633  df-eu 2150  df-mo 2151  df-clab 2273  df-cleq 2279  df-clel 2282  df-nfc 2411  df-ne 2451  df-nel 2452  df-ral 2551  df-rex 2552  df-reu 2553  df-rmo 2554  df-rab 2555  df-v 2793  df-sbc 2995  df-csb 3085  df-dif 3158  df-un 3160  df-in 3162  df-ss 3169  df-pss 3171  df-nul 3459  df-if 3569  df-pw 3630  df-sn 3649  df-pr 3650  df-tp 3651  df-op 3652  df-uni 3831  df-iun 3910  df-br 4027  df-opab 4081  df-mpt 4082  df-tr 4117  df-eprel 4306  df-id 4310  df-po 4315  df-so 4316  df-fr 4353  df-we 4355  df-ord 4396  df-on 4397  df-lim 4398  df-suc 4399  df-om 4658  df-xp 4696  df-rel 4697  df-cnv 4698  df-co 4699  df-dm 4700  df-rn 4701  df-res 4702  df-ima 4703  df-fun 5225  df-fn 5226  df-f 5227  df-f1 5228  df-fo 5229  df-f1o 5230  df-fv 5231  df-ov 5824  df-oprab 5825  df-mpt2 5826  df-iota 6254  df-riota 6301  df-recs 6385  df-rdg 6420  df-er 6657  df-en 6861  df-dom 6862  df-sdom 6863  df-sup 7191  df-pnf 8866  df-mnf 8867  df-xr 8868  df-ltxr 8869  df-le 8870  df-sub 9036  df-neg 9037  df-nn 9744  df-n0 9963  df-z 10022  df-uz 10228
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