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Theorem nq0nn 7364
Description: Decomposition of a nonnegative fraction into numerator and denominator. (Contributed by Jim Kingdon, 24-Nov-2019.)
Assertion
Ref Expression
nq0nn  |-  ( A  e. Q0  ->  E. w E. v
( ( w  e. 
om  /\  v  e.  N. )  /\  A  =  [ <. w ,  v
>. ] ~Q0  ) )
Distinct variable group:    v, A, w

Proof of Theorem nq0nn
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 elqsi 6534 . . 3  |-  ( A  e.  ( ( om 
X.  N. ) /. ~Q0  )  ->  E. a  e.  ( om  X.  N. ) A  =  [
a ] ~Q0  )
2 elxpi 4604 . . . . . . 7  |-  ( a  e.  ( om  X.  N. )  ->  E. w E. v ( a  = 
<. w ,  v >.  /\  ( w  e.  om  /\  v  e.  N. )
) )
32anim1i 338 . . . . . 6  |-  ( ( a  e.  ( om 
X.  N. )  /\  A  =  [ a ] ~Q0  )  ->  ( E. w E. v ( a  =  <. w ,  v
>.  /\  ( w  e. 
om  /\  v  e.  N. ) )  /\  A  =  [ a ] ~Q0  ) )
4 19.41vv 1883 . . . . . 6  |-  ( E. w E. v ( ( a  =  <. w ,  v >.  /\  (
w  e.  om  /\  v  e.  N. )
)  /\  A  =  [ a ] ~Q0  )  <->  ( E. w E. v ( a  = 
<. w ,  v >.  /\  ( w  e.  om  /\  v  e.  N. )
)  /\  A  =  [ a ] ~Q0  ) )
53, 4sylibr 133 . . . . 5  |-  ( ( a  e.  ( om 
X.  N. )  /\  A  =  [ a ] ~Q0  )  ->  E. w E. v ( ( a  =  <. w ,  v
>.  /\  ( w  e. 
om  /\  v  e.  N. ) )  /\  A  =  [ a ] ~Q0  ) )
6 simplr 520 . . . . . . 7  |-  ( ( ( a  =  <. w ,  v >.  /\  (
w  e.  om  /\  v  e.  N. )
)  /\  A  =  [ a ] ~Q0  )  ->  ( w  e.  om  /\  v  e. 
N. ) )
7 simpr 109 . . . . . . . 8  |-  ( ( ( a  =  <. w ,  v >.  /\  (
w  e.  om  /\  v  e.  N. )
)  /\  A  =  [ a ] ~Q0  )  ->  A  =  [ a ] ~Q0  )
8 eceq1 6517 . . . . . . . . 9  |-  ( a  =  <. w ,  v
>.  ->  [ a ] ~Q0  =  [ <. w ,  v
>. ] ~Q0  )
98ad2antrr 480 . . . . . . . 8  |-  ( ( ( a  =  <. w ,  v >.  /\  (
w  e.  om  /\  v  e.  N. )
)  /\  A  =  [ a ] ~Q0  )  ->  [ a ] ~Q0  =  [ <. w ,  v
>. ] ~Q0  )
107, 9eqtrd 2190 . . . . . . 7  |-  ( ( ( a  =  <. w ,  v >.  /\  (
w  e.  om  /\  v  e.  N. )
)  /\  A  =  [ a ] ~Q0  )  ->  A  =  [ <. w ,  v
>. ] ~Q0  )
116, 10jca 304 . . . . . 6  |-  ( ( ( a  =  <. w ,  v >.  /\  (
w  e.  om  /\  v  e.  N. )
)  /\  A  =  [ a ] ~Q0  )  ->  ( (
w  e.  om  /\  v  e.  N. )  /\  A  =  [ <. w ,  v >. ] ~Q0  ) )
12112eximi 1581 . . . . 5  |-  ( E. w E. v ( ( a  =  <. w ,  v >.  /\  (
w  e.  om  /\  v  e.  N. )
)  /\  A  =  [ a ] ~Q0  )  ->  E. w E. v ( ( w  e.  om  /\  v  e.  N. )  /\  A  =  [ <. w ,  v
>. ] ~Q0  ) )
135, 12syl 14 . . . 4  |-  ( ( a  e.  ( om 
X.  N. )  /\  A  =  [ a ] ~Q0  )  ->  E. w E. v ( ( w  e.  om  /\  v  e.  N. )  /\  A  =  [ <. w ,  v
>. ] ~Q0  ) )
1413rexlimiva 2569 . . 3  |-  ( E. a  e.  ( om 
X.  N. ) A  =  [ a ] ~Q0  ->  E. w E. v
( ( w  e. 
om  /\  v  e.  N. )  /\  A  =  [ <. w ,  v
>. ] ~Q0  ) )
151, 14syl 14 . 2  |-  ( A  e.  ( ( om 
X.  N. ) /. ~Q0  )  ->  E. w E. v ( ( w  e.  om  /\  v  e.  N. )  /\  A  =  [ <. w ,  v
>. ] ~Q0  ) )
16 df-nq0 7347 . 2  |- Q0  =  ( ( om 
X.  N. ) /. ~Q0  )
1715, 16eleq2s 2252 1  |-  ( A  e. Q0  ->  E. w E. v
( ( w  e. 
om  /\  v  e.  N. )  /\  A  =  [ <. w ,  v
>. ] ~Q0  ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1335   E.wex 1472    e. wcel 2128   E.wrex 2436   <.cop 3564   omcom 4551    X. cxp 4586   [cec 6480   /.cqs 6481   N.cnpi 7194   ~Q0 ceq0 7208  Q0cnq0 7209
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-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-ext 2139
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-sn 3567  df-pr 3568  df-op 3570  df-br 3968  df-opab 4028  df-xp 4594  df-cnv 4596  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-ec 6484  df-qs 6488  df-nq0 7347
This theorem is referenced by:  nqpnq0nq  7375  nq0m0r  7378  nq0a0  7379  nq02m  7387
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