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Theorem appdiv0nq 7513
Description: Approximate division for positive rationals. This can be thought of as a variation of appdivnq 7512 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 7361 . . 3  |-  ( B  e.  Q.  ->  E. x  e.  Q.  x  <Q  B )
21adantr 274 . 2  |-  ( ( B  e.  Q.  /\  C  e.  Q. )  ->  E. x  e.  Q.  x  <Q  B )
3 appdivnq 7512 . . . . 5  |-  ( ( x  <Q  B  /\  C  e.  Q. )  ->  E. m  e.  Q.  ( x  <Q  ( m  .Q  C )  /\  ( m  .Q  C
)  <Q  B ) )
4 simpr 109 . . . . . 6  |-  ( ( x  <Q  ( m  .Q  C )  /\  (
m  .Q  C ) 
<Q  B )  ->  (
m  .Q  C ) 
<Q  B )
54reximi 2567 . . . . 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 266 . . 3  |-  ( ( C  e.  Q.  /\  x  <Q  B )  ->  E. m  e.  Q.  ( m  .Q  C
)  <Q  B )
87ad2ant2l 505 . 2  |-  ( ( ( B  e.  Q.  /\  C  e.  Q. )  /\  ( x  e.  Q.  /\  x  <Q  B )
)  ->  E. m  e.  Q.  ( m  .Q  C )  <Q  B )
92, 8rexlimddv 2592 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 103    e. wcel 2141   E.wrex 2449   class class class wbr 3987  (class class class)co 5850   Q.cnq 7229    .Q cmq 7232    <Q cltq 7234
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-iinf 4570
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-tr 4086  df-eprel 4272  df-id 4276  df-po 4279  df-iso 4280  df-iord 4349  df-on 4351  df-suc 4354  df-iom 4573  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-ov 5853  df-oprab 5854  df-mpo 5855  df-1st 6116  df-2nd 6117  df-recs 6281  df-irdg 6346  df-1o 6392  df-oadd 6396  df-omul 6397  df-er 6509  df-ec 6511  df-qs 6515  df-ni 7253  df-pli 7254  df-mi 7255  df-lti 7256  df-plpq 7293  df-mpq 7294  df-enq 7296  df-nqqs 7297  df-plqqs 7298  df-mqqs 7299  df-1nqqs 7300  df-rq 7301  df-ltnqqs 7302
This theorem is referenced by:  prmuloc  7515
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