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Theorem dmaddpqlem 7033
Description: Decomposition of a positive fraction into numerator and denominator. Lemma for dmaddpq 7035. (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 6384 . . 3  |-  ( x  e.  ( ( N. 
X.  N. ) /.  ~Q  )  ->  E. a  e.  ( N.  X.  N. )
x  =  [ a ]  ~Q  )
2 elxpi 4483 . . . . . . . 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 1544 . . . . . . . 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 334 . . . . . 6  |-  ( ( a  e.  ( N. 
X.  N. )  /\  x  =  [ a ]  ~Q  )  ->  ( E. w E. v  a  =  <. w ,  v >.  /\  x  =  [
a ]  ~Q  )
)
7 19.41vv 1838 . . . . . 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 6367 . . . . . . . 8  |-  ( a  =  <. w ,  v
>.  ->  [ a ]  ~Q  =  [ <. w ,  v >. ]  ~Q  )
1110adantr 271 . . . . . . 7  |-  ( ( a  =  <. w ,  v >.  /\  x  =  [ a ]  ~Q  )  ->  [ a ]  ~Q  =  [ <. w ,  v >. ]  ~Q  )
129, 11eqtrd 2127 . . . . . 6  |-  ( ( a  =  <. w ,  v >.  /\  x  =  [ a ]  ~Q  )  ->  x  =  [ <. w ,  v >. ]  ~Q  )
13122eximi 1544 . . . . 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 2497 . . 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 7004 . 2  |-  Q.  =  ( ( N.  X.  N. ) /.  ~Q  )
1816, 17eleq2s 2189 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 1296   E.wex 1433    e. wcel 1445   E.wrex 2371   <.cop 3469    X. cxp 4465   [cec 6330   /.cqs 6331   N.cnpi 6928    ~Q ceq 6935   Q.cnq 6936
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-v 2635  df-un 3017  df-in 3019  df-ss 3026  df-sn 3472  df-pr 3473  df-op 3475  df-br 3868  df-opab 3922  df-xp 4473  df-cnv 4475  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-ec 6334  df-qs 6338  df-nqqs 7004
This theorem is referenced by:  dmaddpq  7035  dmmulpq  7036
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