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Theorem dmaddpqlem 7153
Description: Decomposition of a positive fraction into numerator and denominator. Lemma for dmaddpq 7155. (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 6449 . . 3  |-  ( x  e.  ( ( N. 
X.  N. ) /.  ~Q  )  ->  E. a  e.  ( N.  X.  N. )
x  =  [ a ]  ~Q  )
2 elxpi 4525 . . . . . . . 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 1565 . . . . . . . 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 1859 . . . . . 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 6432 . . . . . . . 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 2150 . . . . . 6  |-  ( ( a  =  <. w ,  v >.  /\  x  =  [ a ]  ~Q  )  ->  x  =  [ <. w ,  v >. ]  ~Q  )
13122eximi 1565 . . . . 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 2521 . . 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 7124 . 2  |-  Q.  =  ( ( N.  X.  N. ) /.  ~Q  )
1816, 17eleq2s 2212 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 1316   E.wex 1453    e. wcel 1465   E.wrex 2394   <.cop 3500    X. cxp 4507   [cec 6395   /.cqs 6396   N.cnpi 7048    ~Q ceq 7055   Q.cnq 7056
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-xp 4515  df-cnv 4517  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-ec 6399  df-qs 6403  df-nqqs 7124
This theorem is referenced by:  dmaddpq  7155  dmmulpq  7156
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