ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  rebtwn2z Unicode version

Theorem rebtwn2z 10000
Description: A real number can be bounded by integers above and below which are two apart.

The proof starts by finding two integers which are less than and greater than the given real number. Then this range can be shrunk by choosing an integer in between the endpoints of the range and then deciding which half of the range to keep based on weak linearity, and iterating until the range consists of integers which are two apart. (Contributed by Jim Kingdon, 13-Oct-2021.)

Assertion
Ref Expression
rebtwn2z  |-  ( A  e.  RR  ->  E. x  e.  ZZ  ( x  < 
A  /\  A  <  ( x  +  2 ) ) )
Distinct variable group:    x, A

Proof of Theorem rebtwn2z
Dummy variables  m  n  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 btwnz 9138 . . 3  |-  ( A  e.  RR  ->  ( E. m  e.  ZZ  m  <  A  /\  E. n  e.  ZZ  A  <  n ) )
2 reeanv 2577 . . 3  |-  ( E. m  e.  ZZ  E. n  e.  ZZ  (
m  <  A  /\  A  <  n )  <->  ( E. m  e.  ZZ  m  <  A  /\  E. n  e.  ZZ  A  <  n
) )
31, 2sylibr 133 . 2  |-  ( A  e.  RR  ->  E. m  e.  ZZ  E. n  e.  ZZ  ( m  < 
A  /\  A  <  n ) )
4 simpll 503 . . . . 5  |-  ( ( ( A  e.  RR  /\  ( m  e.  ZZ  /\  n  e.  ZZ ) )  /\  ( m  <  A  /\  A  <  n ) )  ->  A  e.  RR )
5 simplrl 509 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  ( m  e.  ZZ  /\  n  e.  ZZ ) )  /\  ( m  <  A  /\  A  <  n ) )  ->  m  e.  ZZ )
65zred 9141 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  ( m  e.  ZZ  /\  n  e.  ZZ ) )  /\  ( m  <  A  /\  A  <  n ) )  ->  m  e.  RR )
7 simplrr 510 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  ( m  e.  ZZ  /\  n  e.  ZZ ) )  /\  ( m  <  A  /\  A  <  n ) )  ->  n  e.  ZZ )
87zred 9141 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  ( m  e.  ZZ  /\  n  e.  ZZ ) )  /\  ( m  <  A  /\  A  <  n ) )  ->  n  e.  RR )
9 simprl 505 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  ( m  e.  ZZ  /\  n  e.  ZZ ) )  /\  ( m  <  A  /\  A  <  n ) )  ->  m  <  A )
10 simprr 506 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  ( m  e.  ZZ  /\  n  e.  ZZ ) )  /\  ( m  <  A  /\  A  <  n ) )  ->  A  <  n )
116, 4, 8, 9, 10lttrd 7856 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  ( m  e.  ZZ  /\  n  e.  ZZ ) )  /\  ( m  <  A  /\  A  <  n ) )  ->  m  <  n )
12 znnsub 9073 . . . . . . . 8  |-  ( ( m  e.  ZZ  /\  n  e.  ZZ )  ->  ( m  <  n  <->  ( n  -  m )  e.  NN ) )
1312ad2antlr 480 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  ( m  e.  ZZ  /\  n  e.  ZZ ) )  /\  ( m  <  A  /\  A  <  n ) )  -> 
( m  <  n  <->  ( n  -  m )  e.  NN ) )
1411, 13mpbid 146 . . . . . 6  |-  ( ( ( A  e.  RR  /\  ( m  e.  ZZ  /\  n  e.  ZZ ) )  /\  ( m  <  A  /\  A  <  n ) )  -> 
( n  -  m
)  e.  NN )
15 elnnuz 9330 . . . . . . . 8  |-  ( ( n  -  m )  e.  NN  <->  ( n  -  m )  e.  (
ZZ>= `  1 ) )
16 eluzp1p1 9319 . . . . . . . 8  |-  ( ( n  -  m )  e.  ( ZZ>= `  1
)  ->  ( (
n  -  m )  +  1 )  e.  ( ZZ>= `  ( 1  +  1 ) ) )
1715, 16sylbi 120 . . . . . . 7  |-  ( ( n  -  m )  e.  NN  ->  (
( n  -  m
)  +  1 )  e.  ( ZZ>= `  (
1  +  1 ) ) )
18 df-2 8747 . . . . . . . 8  |-  2  =  ( 1  +  1 )
1918fveq2i 5392 . . . . . . 7  |-  ( ZZ>= ` 
2 )  =  (
ZZ>= `  ( 1  +  1 ) )
2017, 19eleqtrrdi 2211 . . . . . 6  |-  ( ( n  -  m )  e.  NN  ->  (
( n  -  m
)  +  1 )  e.  ( ZZ>= `  2
) )
2114, 20syl 14 . . . . 5  |-  ( ( ( A  e.  RR  /\  ( m  e.  ZZ  /\  n  e.  ZZ ) )  /\  ( m  <  A  /\  A  <  n ) )  -> 
( ( n  -  m )  +  1 )  e.  ( ZZ>= ` 
2 ) )
225zcnd 9142 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  ( m  e.  ZZ  /\  n  e.  ZZ ) )  /\  ( m  <  A  /\  A  <  n ) )  ->  m  e.  CC )
237zcnd 9142 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  ( m  e.  ZZ  /\  n  e.  ZZ ) )  /\  ( m  <  A  /\  A  <  n ) )  ->  n  e.  CC )
2422, 23pncan3d 8044 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  ( m  e.  ZZ  /\  n  e.  ZZ ) )  /\  ( m  <  A  /\  A  <  n ) )  -> 
( m  +  ( n  -  m ) )  =  n )
2524, 8eqeltrd 2194 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  ( m  e.  ZZ  /\  n  e.  ZZ ) )  /\  ( m  <  A  /\  A  <  n ) )  -> 
( m  +  ( n  -  m ) )  e.  RR )
268, 6resubcld 8111 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  ( m  e.  ZZ  /\  n  e.  ZZ ) )  /\  ( m  <  A  /\  A  <  n ) )  -> 
( n  -  m
)  e.  RR )
27 1red 7749 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  ( m  e.  ZZ  /\  n  e.  ZZ ) )  /\  ( m  <  A  /\  A  <  n ) )  -> 
1  e.  RR )
2826, 27readdcld 7763 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  ( m  e.  ZZ  /\  n  e.  ZZ ) )  /\  ( m  <  A  /\  A  <  n ) )  -> 
( ( n  -  m )  +  1 )  e.  RR )
296, 28readdcld 7763 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  ( m  e.  ZZ  /\  n  e.  ZZ ) )  /\  ( m  <  A  /\  A  <  n ) )  -> 
( m  +  ( ( n  -  m
)  +  1 ) )  e.  RR )
3010, 24breqtrrd 3926 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  ( m  e.  ZZ  /\  n  e.  ZZ ) )  /\  ( m  <  A  /\  A  <  n ) )  ->  A  <  ( m  +  ( n  -  m
) ) )
3126ltp1d 8656 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  ( m  e.  ZZ  /\  n  e.  ZZ ) )  /\  ( m  <  A  /\  A  <  n ) )  -> 
( n  -  m
)  <  ( (
n  -  m )  +  1 ) )
3226, 28, 6, 31ltadd2dd 8152 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  ( m  e.  ZZ  /\  n  e.  ZZ ) )  /\  ( m  <  A  /\  A  <  n ) )  -> 
( m  +  ( n  -  m ) )  <  ( m  +  ( ( n  -  m )  +  1 ) ) )
334, 25, 29, 30, 32lttrd 7856 . . . . . 6  |-  ( ( ( A  e.  RR  /\  ( m  e.  ZZ  /\  n  e.  ZZ ) )  /\  ( m  <  A  /\  A  <  n ) )  ->  A  <  ( m  +  ( ( n  -  m )  +  1 ) ) )
34 breq1 3902 . . . . . . . 8  |-  ( y  =  m  ->  (
y  <  A  <->  m  <  A ) )
35 oveq1 5749 . . . . . . . . 9  |-  ( y  =  m  ->  (
y  +  ( ( n  -  m )  +  1 ) )  =  ( m  +  ( ( n  -  m )  +  1 ) ) )
3635breq2d 3911 . . . . . . . 8  |-  ( y  =  m  ->  ( A  <  ( y  +  ( ( n  -  m )  +  1 ) )  <->  A  <  ( m  +  ( ( n  -  m )  +  1 ) ) ) )
3734, 36anbi12d 464 . . . . . . 7  |-  ( y  =  m  ->  (
( y  <  A  /\  A  <  ( y  +  ( ( n  -  m )  +  1 ) ) )  <-> 
( m  <  A  /\  A  <  ( m  +  ( ( n  -  m )  +  1 ) ) ) ) )
3837rspcev 2763 . . . . . 6  |-  ( ( m  e.  ZZ  /\  ( m  <  A  /\  A  <  ( m  +  ( ( n  -  m )  +  1 ) ) ) )  ->  E. y  e.  ZZ  ( y  <  A  /\  A  <  ( y  +  ( ( n  -  m )  +  1 ) ) ) )
395, 9, 33, 38syl12anc 1199 . . . . 5  |-  ( ( ( A  e.  RR  /\  ( m  e.  ZZ  /\  n  e.  ZZ ) )  /\  ( m  <  A  /\  A  <  n ) )  ->  E. y  e.  ZZ  ( y  <  A  /\  A  <  ( y  +  ( ( n  -  m )  +  1 ) ) ) )
40 rebtwn2zlemshrink 9999 . . . . 5  |-  ( ( A  e.  RR  /\  ( ( n  -  m )  +  1 )  e.  ( ZZ>= ` 
2 )  /\  E. y  e.  ZZ  (
y  <  A  /\  A  <  ( y  +  ( ( n  -  m )  +  1 ) ) ) )  ->  E. x  e.  ZZ  ( x  <  A  /\  A  <  ( x  + 
2 ) ) )
414, 21, 39, 40syl3anc 1201 . . . 4  |-  ( ( ( A  e.  RR  /\  ( m  e.  ZZ  /\  n  e.  ZZ ) )  /\  ( m  <  A  /\  A  <  n ) )  ->  E. x  e.  ZZ  ( x  <  A  /\  A  <  ( x  + 
2 ) ) )
4241ex 114 . . 3  |-  ( ( A  e.  RR  /\  ( m  e.  ZZ  /\  n  e.  ZZ ) )  ->  ( (
m  <  A  /\  A  <  n )  ->  E. x  e.  ZZ  ( x  <  A  /\  A  <  ( x  + 
2 ) ) ) )
4342rexlimdvva 2534 . 2  |-  ( A  e.  RR  ->  ( E. m  e.  ZZ  E. n  e.  ZZ  (
m  <  A  /\  A  <  n )  ->  E. x  e.  ZZ  ( x  <  A  /\  A  <  ( x  + 
2 ) ) ) )
443, 43mpd 13 1  |-  ( A  e.  RR  ->  E. x  e.  ZZ  ( x  < 
A  /\  A  <  ( x  +  2 ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    e. wcel 1465   E.wrex 2394   class class class wbr 3899   ` cfv 5093  (class class class)co 5742   RRcr 7587   1c1 7589    + caddc 7591    < clt 7768    - cmin 7901   NNcn 8688   2c2 8739   ZZcz 9022   ZZ>=cuz 9294
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-1re 7682  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-addcom 7688  ax-addass 7690  ax-distr 7692  ax-i2m1 7693  ax-0lt1 7694  ax-0id 7696  ax-rnegex 7697  ax-cnre 7699  ax-pre-ltirr 7700  ax-pre-ltwlin 7701  ax-pre-lttrn 7702  ax-pre-ltadd 7704  ax-arch 7707
This theorem depends on definitions:  df-bi 116  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-pnf 7770  df-mnf 7771  df-xr 7772  df-ltxr 7773  df-le 7774  df-sub 7903  df-neg 7904  df-inn 8689  df-2 8747  df-n0 8946  df-z 9023  df-uz 9295
This theorem is referenced by:  qbtwnre  10002
  Copyright terms: Public domain W3C validator