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Theorem rebtwnz 10282
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 9078 . . 3  |-  ( A  e.  RR  ->  -u A  e.  RR )
2 zbtwnre 10281 . . 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 10022 . . . 4  |-  ( x  e.  ZZ  ->  -u x  e.  ZZ )
5 znegcl 10022 . . . . 5  |-  ( y  e.  ZZ  ->  -u y  e.  ZZ )
6 zcn 9996 . . . . . 6  |-  ( y  e.  ZZ  ->  y  e.  CC )
7 zcn 9996 . . . . . 6  |-  ( x  e.  ZZ  ->  x  e.  CC )
8 negcon2 9068 . . . . . 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 4534 . . . 4  |-  ( y  e.  ZZ  ->  E! x  e.  ZZ  y  =  -u x )
11 breq2 4001 . . . . 5  |-  ( y  =  -u x  ->  ( -u A  <_  y  <->  -u A  <_  -u x ) )
12 breq1 4000 . . . . 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 4532 . . 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 9995 . . . . . 6  |-  ( x  e.  ZZ  ->  x  e.  RR )
16 leneg 9245 . . . . . . . 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 9081 . . . . . . . . 9  |-  ( A  e.  RR  ->  ( A  -  1 )  e.  RR )
19 ltneg 9242 . . . . . . . . 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 8805 . . . . . . . . 9  |-  1  e.  RR
22 ltsubadd 9212 . . . . . . . . 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 8795 . . . . . . . . . . 11  |-  ( A  e.  RR  ->  A  e.  CC )
25 ax-1cn 8763 . . . . . . . . . . 11  |-  1  e.  CC
26 negsubdi 9071 . . . . . . . . . . 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 4009 . . . . . . . 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 2698 . . 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 2520   class class class wbr 3997  (class class class)co 5792   CCcc 8703   RRcr 8704   1c1 8706    + caddc 8708    < clt 8835    <_ cle 8836    - cmin 9005   -ucneg 9006   ZZcz 9991
This theorem is referenced by:  flcl  10893  fllelt  10895  flbi  10912
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 2239  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-cnex 8761  ax-resscn 8762  ax-1cn 8763  ax-icn 8764  ax-addcl 8765  ax-addrcl 8766  ax-mulcl 8767  ax-mulrcl 8768  ax-mulcom 8769  ax-addass 8770  ax-mulass 8771  ax-distr 8772  ax-i2m1 8773  ax-1ne0 8774  ax-1rid 8775  ax-rnegex 8776  ax-rrecex 8777  ax-cnre 8778  ax-pre-lttri 8779  ax-pre-lttrn 8780  ax-pre-ltadd 8781  ax-pre-mulgt0 8782  ax-pre-sup 8783
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 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-iota 6225  df-riota 6272  df-recs 6356  df-rdg 6391  df-er 6628  df-en 6832  df-dom 6833  df-sdom 6834  df-sup 7162  df-pnf 8837  df-mnf 8838  df-xr 8839  df-ltxr 8840  df-le 8841  df-sub 9007  df-neg 9008  df-n 9715  df-n0 9933  df-z 9992  df-uz 10198
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