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

Theorem qdenre 10967
Description: The rational numbers are dense in  RR: any real number can be approximated with arbitrary precision by a rational number. For order theoretic density, see qbtwnre 10027. (Contributed by BJ, 15-Oct-2021.)
Assertion
Ref Expression
qdenre  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  ->  E. x  e.  QQ  ( abs `  ( x  -  A ) )  <  B )
Distinct variable groups:    x, A    x, B

Proof of Theorem qdenre
StepHypRef Expression
1 simpl 108 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  ->  A  e.  RR )
2 rpre 9441 . . . . 5  |-  ( B  e.  RR+  ->  B  e.  RR )
32adantl 275 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  ->  B  e.  RR )
41, 3resubcld 8136 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( A  -  B
)  e.  RR )
51, 3readdcld 7788 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( A  +  B
)  e.  RR )
6 ltsubrp 9471 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( A  -  B
)  <  A )
7 ltaddrp 9472 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  ->  A  <  ( A  +  B ) )
84, 1, 5, 6, 7lttrd 7881 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( A  -  B
)  <  ( A  +  B ) )
94, 5, 83jca 1161 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( ( A  -  B )  e.  RR  /\  ( A  +  B
)  e.  RR  /\  ( A  -  B
)  <  ( A  +  B ) ) )
10 qbtwnre 10027 . . 3  |-  ( ( ( A  -  B
)  e.  RR  /\  ( A  +  B
)  e.  RR  /\  ( A  -  B
)  <  ( A  +  B ) )  ->  E. x  e.  QQ  ( ( A  -  B )  <  x  /\  x  <  ( A  +  B ) ) )
11 qre 9410 . . . . . 6  |-  ( x  e.  QQ  ->  x  e.  RR )
1211adantl 275 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR+ )  /\  x  e.  QQ )  ->  x  e.  RR )
13 simpll 518 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR+ )  /\  x  e.  QQ )  ->  A  e.  RR )
142ad2antlr 480 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR+ )  /\  x  e.  QQ )  ->  B  e.  RR )
15 absdiflt 10857 . . . . . 6  |-  ( ( x  e.  RR  /\  A  e.  RR  /\  B  e.  RR )  ->  (
( abs `  (
x  -  A ) )  <  B  <->  ( ( A  -  B )  <  x  /\  x  < 
( A  +  B
) ) ) )
1615biimprd 157 . . . . 5  |-  ( ( x  e.  RR  /\  A  e.  RR  /\  B  e.  RR )  ->  (
( ( A  -  B )  <  x  /\  x  <  ( A  +  B ) )  ->  ( abs `  (
x  -  A ) )  <  B ) )
1712, 13, 14, 16syl3anc 1216 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR+ )  /\  x  e.  QQ )  ->  ( ( ( A  -  B )  <  x  /\  x  <  ( A  +  B
) )  ->  ( abs `  ( x  -  A ) )  < 
B ) )
1817reximdva 2532 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( E. x  e.  QQ  ( ( A  -  B )  < 
x  /\  x  <  ( A  +  B ) )  ->  E. x  e.  QQ  ( abs `  (
x  -  A ) )  <  B ) )
1910, 18syl5 32 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( ( ( A  -  B )  e.  RR  /\  ( A  +  B )  e.  RR  /\  ( A  -  B )  < 
( A  +  B
) )  ->  E. x  e.  QQ  ( abs `  (
x  -  A ) )  <  B ) )
209, 19mpd 13 1  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  ->  E. x  e.  QQ  ( abs `  ( x  -  A ) )  <  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    /\ w3a 962    e. wcel 1480   E.wrex 2415   class class class wbr 3924   ` cfv 5118  (class class class)co 5767   RRcr 7612    + caddc 7616    < clt 7793    - cmin 7926   QQcq 9404   RR+crp 9434   abscabs 10762
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497  ax-cnex 7704  ax-resscn 7705  ax-1cn 7706  ax-1re 7707  ax-icn 7708  ax-addcl 7709  ax-addrcl 7710  ax-mulcl 7711  ax-mulrcl 7712  ax-addcom 7713  ax-mulcom 7714  ax-addass 7715  ax-mulass 7716  ax-distr 7717  ax-i2m1 7718  ax-0lt1 7719  ax-1rid 7720  ax-0id 7721  ax-rnegex 7722  ax-precex 7723  ax-cnre 7724  ax-pre-ltirr 7725  ax-pre-ltwlin 7726  ax-pre-lttrn 7727  ax-pre-apti 7728  ax-pre-ltadd 7729  ax-pre-mulgt0 7730  ax-pre-mulext 7731  ax-arch 7732  ax-caucvg 7733
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-reu 2421  df-rmo 2422  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-if 3470  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-id 4210  df-po 4213  df-iso 4214  df-iord 4283  df-on 4285  df-ilim 4286  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-riota 5723  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-frec 6281  df-pnf 7795  df-mnf 7796  df-xr 7797  df-ltxr 7798  df-le 7799  df-sub 7928  df-neg 7929  df-reap 8330  df-ap 8337  df-div 8426  df-inn 8714  df-2 8772  df-3 8773  df-4 8774  df-n0 8971  df-z 9048  df-uz 9320  df-q 9405  df-rp 9435  df-seqfrec 10212  df-exp 10286  df-cj 10607  df-re 10608  df-im 10609  df-rsqrt 10763  df-abs 10764
This theorem is referenced by:  qdencn  13211
  Copyright terms: Public domain W3C validator