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Theorem rebtwnz 10247
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
StepHypRef Expression
1 renegcl 9043 . . 3  |-  ( A  e.  RR  ->  -u A  e.  RR )
2 zbtwnre 10246 . . 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 9987 . . . 4  |-  ( x  e.  ZZ  ->  -u x  e.  ZZ )
5 znegcl 9987 . . . . 5  |-  ( y  e.  ZZ  ->  -u y  e.  ZZ )
6 zcn 9961 . . . . . 6  |-  ( y  e.  ZZ  ->  y  e.  CC )
7 zcn 9961 . . . . . 6  |-  ( x  e.  ZZ  ->  x  e.  CC )
8 negcon2 9033 . . . . . 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 4499 . . . 4  |-  ( y  e.  ZZ  ->  E! x  e.  ZZ  y  =  -u x )
11 breq2 3967 . . . . 5  |-  ( y  =  -u x  ->  ( -u A  <_  y  <->  -u A  <_  -u x ) )
12 breq1 3966 . . . . 5  |-  ( y  =  -u x  ->  (
y  <  ( -u A  +  1 )  <->  -u x  < 
( -u A  +  1 ) ) )
1311, 12anbi12d 694 . . . 4  |-  ( y  =  -u x  ->  (
( -u A  <_  y  /\  y  <  ( -u A  +  1 ) )  <->  ( -u A  <_ 
-u x  /\  -u x  <  ( -u A  + 
1 ) ) ) )
144, 10, 13reuxfr 4497 . . 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 9960 . . . . . 6  |-  ( x  e.  ZZ  ->  x  e.  RR )
16 leneg 9210 . . . . . . . 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 9046 . . . . . . . . 9  |-  ( A  e.  RR  ->  ( A  -  1 )  e.  RR )
19 ltneg 9207 . . . . . . . . 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 8770 . . . . . . . . 9  |-  1  e.  RR
22 ltsubadd 9177 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  1  e.  RR  /\  x  e.  RR )  ->  (
( A  -  1 )  <  x  <->  A  <  ( x  +  1 ) ) )
2321, 22mp3an2 1270 . . . . . . . 8  |-  ( ( A  e.  RR  /\  x  e.  RR )  ->  ( ( A  - 
1 )  <  x  <->  A  <  ( x  + 
1 ) ) )
24 recn 8760 . . . . . . . . . . 11  |-  ( A  e.  RR  ->  A  e.  CC )
25 ax-1cn 8728 . . . . . . . . . . 11  |-  1  e.  CC
26 negsubdi 9036 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  1  e.  CC )  -> 
-u ( A  - 
1 )  =  (
-u A  +  1 ) )
2724, 25, 26sylancl 646 . . . . . . . . . 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 3975 . . . . . . . 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 694 . . . . . 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 2691 . . 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 1619    e. wcel 1621   E!wreu 2518   class class class wbr 3963  (class class class)co 5757   CCcc 8668   RRcr 8669   1c1 8671    + caddc 8673    < clt 8800    <_ cle 8801    - cmin 8970   -ucneg 8971   ZZcz 9956
This theorem is referenced by:  flcl  10858  fllelt  10860  flbi  10877
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449  ax-cnex 8726  ax-resscn 8727  ax-1cn 8728  ax-icn 8729  ax-addcl 8730  ax-addrcl 8731  ax-mulcl 8732  ax-mulrcl 8733  ax-mulcom 8734  ax-addass 8735  ax-mulass 8736  ax-distr 8737  ax-i2m1 8738  ax-1ne0 8739  ax-1rid 8740  ax-rnegex 8741  ax-rrecex 8742  ax-cnre 8743  ax-pre-lttri 8744  ax-pre-lttrn 8745  ax-pre-ltadd 8746  ax-pre-mulgt0 8747  ax-pre-sup 8748
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-uni 3769  df-iun 3848  df-br 3964  df-opab 4018  df-mpt 4019  df-tr 4054  df-eprel 4242  df-id 4246  df-po 4251  df-so 4252  df-fr 4289  df-we 4291  df-ord 4332  df-on 4333  df-lim 4334  df-suc 4335  df-om 4594  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-iota 6190  df-riota 6237  df-recs 6321  df-rdg 6356  df-er 6593  df-en 6797  df-dom 6798  df-sdom 6799  df-sup 7127  df-pnf 8802  df-mnf 8803  df-xr 8804  df-ltxr 8805  df-le 8806  df-sub 8972  df-neg 8973  df-n 9680  df-n0 9898  df-z 9957  df-uz 10163
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