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Theorem dmaddpqlem 6533
Description: Decomposition of a positive fraction into numerator and denominator. Lemma for dmaddpq 6535. (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 6189 . . 3  |-  ( x  e.  ( ( N. 
X.  N. ) /.  ~Q  )  ->  E. a  e.  ( N.  X.  N. )
x  =  [ a ]  ~Q  )
2 elxpi 4389 . . . . . . . 8  |-  ( a  e.  ( N.  X.  N. )  ->  E. w E. v ( a  = 
<. w ,  v >.  /\  ( w  e.  N.  /\  v  e.  N. )
) )
3 simpl 106 . . . . . . . . 9  |-  ( ( a  =  <. w ,  v >.  /\  (
w  e.  N.  /\  v  e.  N. )
)  ->  a  =  <. w ,  v >.
)
432eximi 1508 . . . . . . . 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 327 . . . . . 6  |-  ( ( a  e.  ( N. 
X.  N. )  /\  x  =  [ a ]  ~Q  )  ->  ( E. w E. v  a  =  <. w ,  v >.  /\  x  =  [
a ]  ~Q  )
)
7 19.41vv 1799 . . . . . 6  |-  ( E. w E. v ( a  =  <. w ,  v >.  /\  x  =  [ a ]  ~Q  ) 
<->  ( E. w E. v  a  =  <. w ,  v >.  /\  x  =  [ a ]  ~Q  ) )
86, 7sylibr 141 . . . . 5  |-  ( ( a  e.  ( N. 
X.  N. )  /\  x  =  [ a ]  ~Q  )  ->  E. w E. v
( a  =  <. w ,  v >.  /\  x  =  [ a ]  ~Q  ) )
9 simpr 107 . . . . . . 7  |-  ( ( a  =  <. w ,  v >.  /\  x  =  [ a ]  ~Q  )  ->  x  =  [
a ]  ~Q  )
10 eceq1 6172 . . . . . . . 8  |-  ( a  =  <. w ,  v
>.  ->  [ a ]  ~Q  =  [ <. w ,  v >. ]  ~Q  )
1110adantr 265 . . . . . . 7  |-  ( ( a  =  <. w ,  v >.  /\  x  =  [ a ]  ~Q  )  ->  [ a ]  ~Q  =  [ <. w ,  v >. ]  ~Q  )
129, 11eqtrd 2088 . . . . . 6  |-  ( ( a  =  <. w ,  v >.  /\  x  =  [ a ]  ~Q  )  ->  x  =  [ <. w ,  v >. ]  ~Q  )
13122eximi 1508 . . . . 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 2445 . . 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 6504 . 2  |-  Q.  =  ( ( N.  X.  N. ) /.  ~Q  )
1816, 17eleq2s 2148 1  |-  ( x  e.  Q.  ->  E. w E. v  x  =  [ <. w ,  v
>. ]  ~Q  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    = wceq 1259   E.wex 1397    e. wcel 1409   E.wrex 2324   <.cop 3406    X. cxp 4371   [cec 6135   /.cqs 6136   N.cnpi 6428    ~Q ceq 6435   Q.cnq 6436
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-opab 3847  df-xp 4379  df-cnv 4381  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-ec 6139  df-qs 6143  df-nqqs 6504
This theorem is referenced by:  dmaddpq  6535  dmmulpq  6536
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