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Theorem dmaddpqlem 6475
Description: Decomposition of a positive fraction into numerator and denominator. Lemma for dmaddpq 6477. (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 6158 . . 3  |-  ( x  e.  ( ( N. 
X.  N. ) /.  ~Q  )  ->  E. a  e.  ( N.  X.  N. )
x  =  [ a ]  ~Q  )
2 elxpi 4361 . . . . . . . 8  |-  ( a  e.  ( N.  X.  N. )  ->  E. w E. v ( a  = 
<. w ,  v >.  /\  ( w  e.  N.  /\  v  e.  N. )
) )
3 simpl 102 . . . . . . . . 9  |-  ( ( a  =  <. w ,  v >.  /\  (
w  e.  N.  /\  v  e.  N. )
)  ->  a  =  <. w ,  v >.
)
432eximi 1492 . . . . . . . 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 323 . . . . . 6  |-  ( ( a  e.  ( N. 
X.  N. )  /\  x  =  [ a ]  ~Q  )  ->  ( E. w E. v  a  =  <. w ,  v >.  /\  x  =  [
a ]  ~Q  )
)
7 19.41vv 1783 . . . . . 6  |-  ( E. w E. v ( a  =  <. w ,  v >.  /\  x  =  [ a ]  ~Q  ) 
<->  ( E. w E. v  a  =  <. w ,  v >.  /\  x  =  [ a ]  ~Q  ) )
86, 7sylibr 137 . . . . 5  |-  ( ( a  e.  ( N. 
X.  N. )  /\  x  =  [ a ]  ~Q  )  ->  E. w E. v
( a  =  <. w ,  v >.  /\  x  =  [ a ]  ~Q  ) )
9 simpr 103 . . . . . . 7  |-  ( ( a  =  <. w ,  v >.  /\  x  =  [ a ]  ~Q  )  ->  x  =  [
a ]  ~Q  )
10 eceq1 6141 . . . . . . . 8  |-  ( a  =  <. w ,  v
>.  ->  [ a ]  ~Q  =  [ <. w ,  v >. ]  ~Q  )
1110adantr 261 . . . . . . 7  |-  ( ( a  =  <. w ,  v >.  /\  x  =  [ a ]  ~Q  )  ->  [ a ]  ~Q  =  [ <. w ,  v >. ]  ~Q  )
129, 11eqtrd 2072 . . . . . 6  |-  ( ( a  =  <. w ,  v >.  /\  x  =  [ a ]  ~Q  )  ->  x  =  [ <. w ,  v >. ]  ~Q  )
13122eximi 1492 . . . . 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 2428 . . 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 6446 . 2  |-  Q.  =  ( ( N.  X.  N. ) /.  ~Q  )
1816, 17eleq2s 2132 1  |-  ( x  e.  Q.  ->  E. w E. v  x  =  [ <. w ,  v
>. ]  ~Q  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    = wceq 1243   E.wex 1381    e. wcel 1393   E.wrex 2307   <.cop 3378    X. cxp 4343   [cec 6104   /.cqs 6105   N.cnpi 6370    ~Q ceq 6377   Q.cnq 6378
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-cnv 4353  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-ec 6108  df-qs 6112  df-nqqs 6446
This theorem is referenced by:  dmaddpq  6477  dmmulpq  6478
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