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Theorem nq0nn 7404
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 6565 . . 3  |-  ( A  e.  ( ( om 
X.  N. ) /. ~Q0  )  ->  E. a  e.  ( om  X.  N. ) A  =  [
a ] ~Q0  )
2 elxpi 4627 . . . . . . 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 1896 . . . . . 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 525 . . . . . . 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 6548 . . . . . . . . 9  |-  ( a  =  <. w ,  v
>.  ->  [ a ] ~Q0  =  [ <. w ,  v
>. ] ~Q0  )
98ad2antrr 485 . . . . . . . 8  |-  ( ( ( a  =  <. w ,  v >.  /\  (
w  e.  om  /\  v  e.  N. )
)  /\  A  =  [ a ] ~Q0  )  ->  [ a ] ~Q0  =  [ <. w ,  v
>. ] ~Q0  )
107, 9eqtrd 2203 . . . . . . 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 1594 . . . . 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 2582 . . 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 7387 . 2  |- Q0  =  ( ( om 
X.  N. ) /. ~Q0  )
1715, 16eleq2s 2265 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 1348   E.wex 1485    e. wcel 2141   E.wrex 2449   <.cop 3586   omcom 4574    X. cxp 4609   [cec 6511   /.cqs 6512   N.cnpi 7234   ~Q0 ceq0 7248  Q0cnq0 7249
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 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-ext 2152
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-xp 4617  df-cnv 4619  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-ec 6515  df-qs 6519  df-nq0 7387
This theorem is referenced by:  nqpnq0nq  7415  nq0m0r  7418  nq0a0  7419  nq02m  7427
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