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Theorem qdenre 11106
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 10160. (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 9568 . . . . 5  |-  ( B  e.  RR+  ->  B  e.  RR )
32adantl 275 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  ->  B  e.  RR )
41, 3resubcld 8257 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( A  -  B
)  e.  RR )
51, 3readdcld 7908 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( A  +  B
)  e.  RR )
6 ltsubrp 9598 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( A  -  B
)  <  A )
7 ltaddrp 9599 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  ->  A  <  ( A  +  B ) )
84, 1, 5, 6, 7lttrd 8002 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( A  -  B
)  <  ( A  +  B ) )
94, 5, 83jca 1162 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( ( A  -  B )  e.  RR  /\  ( A  +  B
)  e.  RR  /\  ( A  -  B
)  <  ( A  +  B ) ) )
10 qbtwnre 10160 . . 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 9535 . . . . . 6  |-  ( x  e.  QQ  ->  x  e.  RR )
1211adantl 275 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR+ )  /\  x  e.  QQ )  ->  x  e.  RR )
13 simpll 519 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR+ )  /\  x  e.  QQ )  ->  A  e.  RR )
142ad2antlr 481 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR+ )  /\  x  e.  QQ )  ->  B  e.  RR )
15 absdiflt 10996 . . . . . 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 1220 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR+ )  /\  x  e.  QQ )  ->  ( ( ( A  -  B )  <  x  /\  x  <  ( A  +  B
) )  ->  ( abs `  ( x  -  A ) )  < 
B ) )
1817reximdva 2559 . . 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 963    e. wcel 2128   E.wrex 2436   class class class wbr 3966   ` cfv 5171  (class class class)co 5825   RRcr 7732    + caddc 7736    < clt 7913    - cmin 8047   QQcq 9529   RR+crp 9561   abscabs 10901
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4080  ax-sep 4083  ax-nul 4091  ax-pow 4136  ax-pr 4170  ax-un 4394  ax-setind 4497  ax-iinf 4548  ax-cnex 7824  ax-resscn 7825  ax-1cn 7826  ax-1re 7827  ax-icn 7828  ax-addcl 7829  ax-addrcl 7830  ax-mulcl 7831  ax-mulrcl 7832  ax-addcom 7833  ax-mulcom 7834  ax-addass 7835  ax-mulass 7836  ax-distr 7837  ax-i2m1 7838  ax-0lt1 7839  ax-1rid 7840  ax-0id 7841  ax-rnegex 7842  ax-precex 7843  ax-cnre 7844  ax-pre-ltirr 7845  ax-pre-ltwlin 7846  ax-pre-lttrn 7847  ax-pre-apti 7848  ax-pre-ltadd 7849  ax-pre-mulgt0 7850  ax-pre-mulext 7851  ax-arch 7852  ax-caucvg 7853
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-if 3506  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3774  df-int 3809  df-iun 3852  df-br 3967  df-opab 4027  df-mpt 4028  df-tr 4064  df-id 4254  df-po 4257  df-iso 4258  df-iord 4327  df-on 4329  df-ilim 4330  df-suc 4332  df-iom 4551  df-xp 4593  df-rel 4594  df-cnv 4595  df-co 4596  df-dm 4597  df-rn 4598  df-res 4599  df-ima 4600  df-iota 5136  df-fun 5173  df-fn 5174  df-f 5175  df-f1 5176  df-fo 5177  df-f1o 5178  df-fv 5179  df-riota 5781  df-ov 5828  df-oprab 5829  df-mpo 5830  df-1st 6089  df-2nd 6090  df-recs 6253  df-frec 6339  df-pnf 7915  df-mnf 7916  df-xr 7917  df-ltxr 7918  df-le 7919  df-sub 8049  df-neg 8050  df-reap 8451  df-ap 8458  df-div 8547  df-inn 8835  df-2 8893  df-3 8894  df-4 8895  df-n0 9092  df-z 9169  df-uz 9441  df-q 9530  df-rp 9562  df-seqfrec 10349  df-exp 10423  df-cj 10746  df-re 10747  df-im 10748  df-rsqrt 10902  df-abs 10903
This theorem is referenced by:  qdencn  13640
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