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Theorem appdiv0nq 7594
Description: Approximate division for positive rationals. This can be thought of as a variation of appdivnq 7593 in which  A is zero, although it can be stated and proved in terms of positive rationals alone, without zero as such. (Contributed by Jim Kingdon, 9-Dec-2019.)
Assertion
Ref Expression
appdiv0nq  |-  ( ( B  e.  Q.  /\  C  e.  Q. )  ->  E. m  e.  Q.  ( m  .Q  C
)  <Q  B )
Distinct variable groups:    B, m    C, m

Proof of Theorem appdiv0nq
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 nsmallnqq 7442 . . 3  |-  ( B  e.  Q.  ->  E. x  e.  Q.  x  <Q  B )
21adantr 276 . 2  |-  ( ( B  e.  Q.  /\  C  e.  Q. )  ->  E. x  e.  Q.  x  <Q  B )
3 appdivnq 7593 . . . . 5  |-  ( ( x  <Q  B  /\  C  e.  Q. )  ->  E. m  e.  Q.  ( x  <Q  ( m  .Q  C )  /\  ( m  .Q  C
)  <Q  B ) )
4 simpr 110 . . . . . 6  |-  ( ( x  <Q  ( m  .Q  C )  /\  (
m  .Q  C ) 
<Q  B )  ->  (
m  .Q  C ) 
<Q  B )
54reximi 2587 . . . . 5  |-  ( E. m  e.  Q.  (
x  <Q  ( m  .Q  C )  /\  (
m  .Q  C ) 
<Q  B )  ->  E. m  e.  Q.  ( m  .Q  C )  <Q  B )
63, 5syl 14 . . . 4  |-  ( ( x  <Q  B  /\  C  e.  Q. )  ->  E. m  e.  Q.  ( m  .Q  C
)  <Q  B )
76ancoms 268 . . 3  |-  ( ( C  e.  Q.  /\  x  <Q  B )  ->  E. m  e.  Q.  ( m  .Q  C
)  <Q  B )
87ad2ant2l 508 . 2  |-  ( ( ( B  e.  Q.  /\  C  e.  Q. )  /\  ( x  e.  Q.  /\  x  <Q  B )
)  ->  E. m  e.  Q.  ( m  .Q  C )  <Q  B )
92, 8rexlimddv 2612 1  |-  ( ( B  e.  Q.  /\  C  e.  Q. )  ->  E. m  e.  Q.  ( m  .Q  C
)  <Q  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    e. wcel 2160   E.wrex 2469   class class class wbr 4018  (class class class)co 5897   Q.cnq 7310    .Q cmq 7313    <Q cltq 7315
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-eprel 4307  df-id 4311  df-po 4314  df-iso 4315  df-iord 4384  df-on 4386  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-ov 5900  df-oprab 5901  df-mpo 5902  df-1st 6166  df-2nd 6167  df-recs 6331  df-irdg 6396  df-1o 6442  df-oadd 6446  df-omul 6447  df-er 6560  df-ec 6562  df-qs 6566  df-ni 7334  df-pli 7335  df-mi 7336  df-lti 7337  df-plpq 7374  df-mpq 7375  df-enq 7377  df-nqqs 7378  df-plqqs 7379  df-mqqs 7380  df-1nqqs 7381  df-rq 7382  df-ltnqqs 7383
This theorem is referenced by:  prmuloc  7596
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