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Theorem nq0nn 7455
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 6601 . . 3  |-  ( A  e.  ( ( om 
X.  N. ) /. ~Q0  )  ->  E. a  e.  ( om  X.  N. ) A  =  [
a ] ~Q0  )
2 elxpi 4654 . . . . . . 7  |-  ( a  e.  ( om  X.  N. )  ->  E. w E. v ( a  = 
<. w ,  v >.  /\  ( w  e.  om  /\  v  e.  N. )
) )
32anim1i 340 . . . . . 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 1913 . . . . . 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 134 . . . . 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 528 . . . . . . 7  |-  ( ( ( a  =  <. w ,  v >.  /\  (
w  e.  om  /\  v  e.  N. )
)  /\  A  =  [ a ] ~Q0  )  ->  ( w  e.  om  /\  v  e. 
N. ) )
7 simpr 110 . . . . . . . 8  |-  ( ( ( a  =  <. w ,  v >.  /\  (
w  e.  om  /\  v  e.  N. )
)  /\  A  =  [ a ] ~Q0  )  ->  A  =  [ a ] ~Q0  )
8 eceq1 6584 . . . . . . . . 9  |-  ( a  =  <. w ,  v
>.  ->  [ a ] ~Q0  =  [ <. w ,  v
>. ] ~Q0  )
98ad2antrr 488 . . . . . . . 8  |-  ( ( ( a  =  <. w ,  v >.  /\  (
w  e.  om  /\  v  e.  N. )
)  /\  A  =  [ a ] ~Q0  )  ->  [ a ] ~Q0  =  [ <. w ,  v
>. ] ~Q0  )
107, 9eqtrd 2220 . . . . . . 7  |-  ( ( ( a  =  <. w ,  v >.  /\  (
w  e.  om  /\  v  e.  N. )
)  /\  A  =  [ a ] ~Q0  )  ->  A  =  [ <. w ,  v
>. ] ~Q0  )
116, 10jca 306 . . . . . 6  |-  ( ( ( a  =  <. w ,  v >.  /\  (
w  e.  om  /\  v  e.  N. )
)  /\  A  =  [ a ] ~Q0  )  ->  ( (
w  e.  om  /\  v  e.  N. )  /\  A  =  [ <. w ,  v >. ] ~Q0  ) )
12112eximi 1611 . . . . 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 2599 . . 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 7438 . 2  |- Q0  =  ( ( om 
X.  N. ) /. ~Q0  )
1715, 16eleq2s 2282 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 104    = wceq 1363   E.wex 1502    e. wcel 2158   E.wrex 2466   <.cop 3607   omcom 4601    X. cxp 4636   [cec 6547   /.cqs 6548   N.cnpi 7285   ~Q0 ceq0 7299  Q0cnq0 7300
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-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-un 3145  df-in 3147  df-ss 3154  df-sn 3610  df-pr 3611  df-op 3613  df-br 4016  df-opab 4077  df-xp 4644  df-cnv 4646  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-ec 6551  df-qs 6555  df-nq0 7438
This theorem is referenced by:  nqpnq0nq  7466  nq0m0r  7469  nq0a0  7470  nq02m  7478
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