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Theorem nq0nn 6746
Description: Decomposition of a non-negative 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 6245 . . 3  |-  ( A  e.  ( ( om 
X.  N. ) /. ~Q0  )  ->  E. a  e.  ( om  X.  N. ) A  =  [
a ] ~Q0  )
2 elxpi 4407 . . . . . . 7  |-  ( a  e.  ( om  X.  N. )  ->  E. w E. v ( a  = 
<. w ,  v >.  /\  ( w  e.  om  /\  v  e.  N. )
) )
32anim1i 333 . . . . . 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 1826 . . . . . 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 132 . . . . 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 497 . . . . . . 7  |-  ( ( ( a  =  <. w ,  v >.  /\  (
w  e.  om  /\  v  e.  N. )
)  /\  A  =  [ a ] ~Q0  )  ->  ( w  e.  om  /\  v  e. 
N. ) )
7 simpr 108 . . . . . . . 8  |-  ( ( ( a  =  <. w ,  v >.  /\  (
w  e.  om  /\  v  e.  N. )
)  /\  A  =  [ a ] ~Q0  )  ->  A  =  [ a ] ~Q0  )
8 eceq1 6228 . . . . . . . . 9  |-  ( a  =  <. w ,  v
>.  ->  [ a ] ~Q0  =  [ <. w ,  v
>. ] ~Q0  )
98ad2antrr 472 . . . . . . . 8  |-  ( ( ( a  =  <. w ,  v >.  /\  (
w  e.  om  /\  v  e.  N. )
)  /\  A  =  [ a ] ~Q0  )  ->  [ a ] ~Q0  =  [ <. w ,  v
>. ] ~Q0  )
107, 9eqtrd 2115 . . . . . . 7  |-  ( ( ( a  =  <. w ,  v >.  /\  (
w  e.  om  /\  v  e.  N. )
)  /\  A  =  [ a ] ~Q0  )  ->  A  =  [ <. w ,  v
>. ] ~Q0  )
116, 10jca 300 . . . . . 6  |-  ( ( ( a  =  <. w ,  v >.  /\  (
w  e.  om  /\  v  e.  N. )
)  /\  A  =  [ a ] ~Q0  )  ->  ( (
w  e.  om  /\  v  e.  N. )  /\  A  =  [ <. w ,  v >. ] ~Q0  ) )
12112eximi 1533 . . . . 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 2477 . . 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 6729 . 2  |- Q0  =  ( ( om 
X.  N. ) /. ~Q0  )
1715, 16eleq2s 2177 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 102    = wceq 1285   E.wex 1422    e. wcel 1434   E.wrex 2354   <.cop 3419   omcom 4359    X. cxp 4389   [cec 6191   /.cqs 6192   N.cnpi 6576   ~Q0 ceq0 6590  Q0cnq0 6591
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-sn 3422  df-pr 3423  df-op 3425  df-br 3806  df-opab 3860  df-xp 4397  df-cnv 4399  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-ec 6195  df-qs 6199  df-nq0 6729
This theorem is referenced by:  nqpnq0nq  6757  nq0m0r  6760  nq0a0  6761  nq02m  6769
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