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Theorem dmaddpqlem 7208
Description: Decomposition of a positive fraction into numerator and denominator. Lemma for dmaddpq 7210. (Contributed by Jim Kingdon, 15-Sep-2019.)
Assertion
Ref Expression
dmaddpqlem  |-  ( x  e.  Q.  ->  E. w E. v  x  =  [ <. w ,  v
>. ]  ~Q  )
Distinct variable group:    w, v, x

Proof of Theorem dmaddpqlem
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 elqsi 6488 . . 3  |-  ( x  e.  ( ( N. 
X.  N. ) /.  ~Q  )  ->  E. a  e.  ( N.  X.  N. )
x  =  [ a ]  ~Q  )
2 elxpi 4562 . . . . . . . 8  |-  ( a  e.  ( N.  X.  N. )  ->  E. w E. v ( a  = 
<. w ,  v >.  /\  ( w  e.  N.  /\  v  e.  N. )
) )
3 simpl 108 . . . . . . . . 9  |-  ( ( a  =  <. w ,  v >.  /\  (
w  e.  N.  /\  v  e.  N. )
)  ->  a  =  <. w ,  v >.
)
432eximi 1581 . . . . . . . 8  |-  ( E. w E. v ( a  =  <. w ,  v >.  /\  (
w  e.  N.  /\  v  e.  N. )
)  ->  E. w E. v  a  =  <. w ,  v >.
)
52, 4syl 14 . . . . . . 7  |-  ( a  e.  ( N.  X.  N. )  ->  E. w E. v  a  =  <. w ,  v >.
)
65anim1i 338 . . . . . 6  |-  ( ( a  e.  ( N. 
X.  N. )  /\  x  =  [ a ]  ~Q  )  ->  ( E. w E. v  a  =  <. w ,  v >.  /\  x  =  [
a ]  ~Q  )
)
7 19.41vv 1876 . . . . . 6  |-  ( E. w E. v ( a  =  <. w ,  v >.  /\  x  =  [ a ]  ~Q  ) 
<->  ( E. w E. v  a  =  <. w ,  v >.  /\  x  =  [ a ]  ~Q  ) )
86, 7sylibr 133 . . . . 5  |-  ( ( a  e.  ( N. 
X.  N. )  /\  x  =  [ a ]  ~Q  )  ->  E. w E. v
( a  =  <. w ,  v >.  /\  x  =  [ a ]  ~Q  ) )
9 simpr 109 . . . . . . 7  |-  ( ( a  =  <. w ,  v >.  /\  x  =  [ a ]  ~Q  )  ->  x  =  [
a ]  ~Q  )
10 eceq1 6471 . . . . . . . 8  |-  ( a  =  <. w ,  v
>.  ->  [ a ]  ~Q  =  [ <. w ,  v >. ]  ~Q  )
1110adantr 274 . . . . . . 7  |-  ( ( a  =  <. w ,  v >.  /\  x  =  [ a ]  ~Q  )  ->  [ a ]  ~Q  =  [ <. w ,  v >. ]  ~Q  )
129, 11eqtrd 2173 . . . . . 6  |-  ( ( a  =  <. w ,  v >.  /\  x  =  [ a ]  ~Q  )  ->  x  =  [ <. w ,  v >. ]  ~Q  )
13122eximi 1581 . . . . 5  |-  ( E. w E. v ( a  =  <. w ,  v >.  /\  x  =  [ a ]  ~Q  )  ->  E. w E. v  x  =  [ <. w ,  v >. ]  ~Q  )
148, 13syl 14 . . . 4  |-  ( ( a  e.  ( N. 
X.  N. )  /\  x  =  [ a ]  ~Q  )  ->  E. w E. v  x  =  [ <. w ,  v >. ]  ~Q  )
1514rexlimiva 2547 . . 3  |-  ( E. a  e.  ( N. 
X.  N. ) x  =  [ a ]  ~Q  ->  E. w E. v  x  =  [ <. w ,  v >. ]  ~Q  )
161, 15syl 14 . 2  |-  ( x  e.  ( ( N. 
X.  N. ) /.  ~Q  )  ->  E. w E. v  x  =  [ <. w ,  v >. ]  ~Q  )
17 df-nqqs 7179 . 2  |-  Q.  =  ( ( N.  X.  N. ) /.  ~Q  )
1816, 17eleq2s 2235 1  |-  ( x  e.  Q.  ->  E. w E. v  x  =  [ <. w ,  v
>. ]  ~Q  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1332   E.wex 1469    e. wcel 1481   E.wrex 2418   <.cop 3534    X. cxp 4544   [cec 6434   /.cqs 6435   N.cnpi 7103    ~Q ceq 7110   Q.cnq 7111
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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3079  df-in 3081  df-ss 3088  df-sn 3537  df-pr 3538  df-op 3540  df-br 3937  df-opab 3997  df-xp 4552  df-cnv 4554  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-ec 6438  df-qs 6442  df-nqqs 7179
This theorem is referenced by:  dmaddpq  7210  dmmulpq  7211
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