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Theorem dmmulpq 7397
Description: Domain of multiplication on positive fractions. (Contributed by NM, 24-Aug-1995.)
Assertion
Ref Expression
dmmulpq  |-  dom  .Q  =  ( Q.  X.  Q. )

Proof of Theorem dmmulpq
Dummy variables  x  y  z  v  w  u  f are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dmoprab 5972 . . 3  |-  dom  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e. 
Q.  /\  y  e.  Q. )  /\  E. w E. v E. u E. f ( ( x  =  [ <. w ,  v >. ]  ~Q  /\  y  =  [ <. u ,  f >. ]  ~Q  )  /\  z  =  [
( <. w ,  v
>.  .pQ  <. u ,  f
>. ) ]  ~Q  )
) }  =  { <. x ,  y >.  |  E. z ( ( x  e.  Q.  /\  y  e.  Q. )  /\  E. w E. v E. u E. f ( ( x  =  [ <. w ,  v >. ]  ~Q  /\  y  =  [ <. u ,  f
>. ]  ~Q  )  /\  z  =  [ ( <. w ,  v >.  .pQ  <. u ,  f
>. ) ]  ~Q  )
) }
2 df-mqqs 7367 . . . 4  |-  .Q  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e. 
Q.  /\  y  e.  Q. )  /\  E. w E. v E. u E. f ( ( x  =  [ <. w ,  v >. ]  ~Q  /\  y  =  [ <. u ,  f >. ]  ~Q  )  /\  z  =  [
( <. w ,  v
>.  .pQ  <. u ,  f
>. ) ]  ~Q  )
) }
32dmeqi 4843 . . 3  |-  dom  .Q  =  dom  { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  Q.  /\  y  e.  Q. )  /\  E. w E. v E. u E. f ( ( x  =  [ <. w ,  v >. ]  ~Q  /\  y  =  [ <. u ,  f
>. ]  ~Q  )  /\  z  =  [ ( <. w ,  v >.  .pQ  <. u ,  f
>. ) ]  ~Q  )
) }
4 dmaddpqlem 7394 . . . . . . . . 9  |-  ( x  e.  Q.  ->  E. w E. v  x  =  [ <. w ,  v
>. ]  ~Q  )
5 dmaddpqlem 7394 . . . . . . . . 9  |-  ( y  e.  Q.  ->  E. u E. f  y  =  [ <. u ,  f
>. ]  ~Q  )
64, 5anim12i 338 . . . . . . . 8  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  ( E. w E. v  x  =  [ <. w ,  v >. ]  ~Q  /\  E. u E. f  y  =  [ <. u ,  f
>. ]  ~Q  ) )
7 ee4anv 1946 . . . . . . . 8  |-  ( E. w E. v E. u E. f ( x  =  [ <. w ,  v >. ]  ~Q  /\  y  =  [ <. u ,  f >. ]  ~Q  ) 
<->  ( E. w E. v  x  =  [ <. w ,  v >. ]  ~Q  /\  E. u E. f  y  =  [ <. u ,  f
>. ]  ~Q  ) )
86, 7sylibr 134 . . . . . . 7  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  E. w E. v E. u E. f ( x  =  [ <. w ,  v >. ]  ~Q  /\  y  =  [ <. u ,  f >. ]  ~Q  ) )
9 enqex 7377 . . . . . . . . . . . . . 14  |-  ~Q  e.  _V
10 ecexg 6557 . . . . . . . . . . . . . 14  |-  (  ~Q  e.  _V  ->  [ ( <. w ,  v >.  .pQ  <. u ,  f
>. ) ]  ~Q  e.  _V )
119, 10ax-mp 5 . . . . . . . . . . . . 13  |-  [ (
<. w ,  v >.  .pQ  <. u ,  f
>. ) ]  ~Q  e.  _V
1211isseti 2760 . . . . . . . . . . . 12  |-  E. z 
z  =  [ (
<. w ,  v >.  .pQ  <. u ,  f
>. ) ]  ~Q
13 ax-ia3 108 . . . . . . . . . . . . 13  |-  ( ( x  =  [ <. w ,  v >. ]  ~Q  /\  y  =  [ <. u ,  f >. ]  ~Q  )  ->  ( z  =  [ ( <. w ,  v >.  .pQ  <. u ,  f >. ) ]  ~Q  ->  ( (
x  =  [ <. w ,  v >. ]  ~Q  /\  y  =  [ <. u ,  f >. ]  ~Q  )  /\  z  =  [
( <. w ,  v
>.  .pQ  <. u ,  f
>. ) ]  ~Q  )
) )
1413eximdv 1891 . . . . . . . . . . . 12  |-  ( ( x  =  [ <. w ,  v >. ]  ~Q  /\  y  =  [ <. u ,  f >. ]  ~Q  )  ->  ( E. z 
z  =  [ (
<. w ,  v >.  .pQ  <. u ,  f
>. ) ]  ~Q  ->  E. z ( ( x  =  [ <. w ,  v >. ]  ~Q  /\  y  =  [ <. u ,  f >. ]  ~Q  )  /\  z  =  [
( <. w ,  v
>.  .pQ  <. u ,  f
>. ) ]  ~Q  )
) )
1512, 14mpi 15 . . . . . . . . . . 11  |-  ( ( x  =  [ <. w ,  v >. ]  ~Q  /\  y  =  [ <. u ,  f >. ]  ~Q  )  ->  E. z ( ( x  =  [ <. w ,  v >. ]  ~Q  /\  y  =  [ <. u ,  f >. ]  ~Q  )  /\  z  =  [
( <. w ,  v
>.  .pQ  <. u ,  f
>. ) ]  ~Q  )
)
16152eximi 1612 . . . . . . . . . 10  |-  ( E. u E. f ( x  =  [ <. w ,  v >. ]  ~Q  /\  y  =  [ <. u ,  f >. ]  ~Q  )  ->  E. u E. f E. z ( ( x  =  [ <. w ,  v >. ]  ~Q  /\  y  =  [ <. u ,  f >. ]  ~Q  )  /\  z  =  [
( <. w ,  v
>.  .pQ  <. u ,  f
>. ) ]  ~Q  )
)
17 exrot3 1701 . . . . . . . . . 10  |-  ( E. z E. u E. f ( ( x  =  [ <. w ,  v >. ]  ~Q  /\  y  =  [ <. u ,  f >. ]  ~Q  )  /\  z  =  [
( <. w ,  v
>.  .pQ  <. u ,  f
>. ) ]  ~Q  )  <->  E. u E. f E. z ( ( x  =  [ <. w ,  v >. ]  ~Q  /\  y  =  [ <. u ,  f >. ]  ~Q  )  /\  z  =  [
( <. w ,  v
>.  .pQ  <. u ,  f
>. ) ]  ~Q  )
)
1816, 17sylibr 134 . . . . . . . . 9  |-  ( E. u E. f ( x  =  [ <. w ,  v >. ]  ~Q  /\  y  =  [ <. u ,  f >. ]  ~Q  )  ->  E. z E. u E. f ( ( x  =  [ <. w ,  v >. ]  ~Q  /\  y  =  [ <. u ,  f >. ]  ~Q  )  /\  z  =  [
( <. w ,  v
>.  .pQ  <. u ,  f
>. ) ]  ~Q  )
)
19182eximi 1612 . . . . . . . 8  |-  ( E. w E. v E. u E. f ( x  =  [ <. w ,  v >. ]  ~Q  /\  y  =  [ <. u ,  f >. ]  ~Q  )  ->  E. w E. v E. z E. u E. f ( ( x  =  [ <. w ,  v >. ]  ~Q  /\  y  =  [ <. u ,  f >. ]  ~Q  )  /\  z  =  [
( <. w ,  v
>.  .pQ  <. u ,  f
>. ) ]  ~Q  )
)
20 exrot3 1701 . . . . . . . 8  |-  ( E. z E. w E. v E. u E. f
( ( x  =  [ <. w ,  v
>. ]  ~Q  /\  y  =  [ <. u ,  f
>. ]  ~Q  )  /\  z  =  [ ( <. w ,  v >.  .pQ  <. u ,  f
>. ) ]  ~Q  )  <->  E. w E. v E. z E. u E. f ( ( x  =  [ <. w ,  v >. ]  ~Q  /\  y  =  [ <. u ,  f >. ]  ~Q  )  /\  z  =  [
( <. w ,  v
>.  .pQ  <. u ,  f
>. ) ]  ~Q  )
)
2119, 20sylibr 134 . . . . . . 7  |-  ( E. w E. v E. u E. f ( x  =  [ <. w ,  v >. ]  ~Q  /\  y  =  [ <. u ,  f >. ]  ~Q  )  ->  E. z E. w E. v E. u E. f ( ( x  =  [ <. w ,  v >. ]  ~Q  /\  y  =  [ <. u ,  f >. ]  ~Q  )  /\  z  =  [
( <. w ,  v
>.  .pQ  <. u ,  f
>. ) ]  ~Q  )
)
228, 21syl 14 . . . . . 6  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  E. z E. w E. v E. u E. f ( ( x  =  [ <. w ,  v >. ]  ~Q  /\  y  =  [ <. u ,  f >. ]  ~Q  )  /\  z  =  [
( <. w ,  v
>.  .pQ  <. u ,  f
>. ) ]  ~Q  )
)
2322pm4.71i 391 . . . . 5  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  <->  ( ( x  e.  Q.  /\  y  e.  Q. )  /\  E. z E. w E. v E. u E. f ( ( x  =  [ <. w ,  v >. ]  ~Q  /\  y  =  [ <. u ,  f >. ]  ~Q  )  /\  z  =  [
( <. w ,  v
>.  .pQ  <. u ,  f
>. ) ]  ~Q  )
) )
24 19.42v 1918 . . . . 5  |-  ( E. z ( ( x  e.  Q.  /\  y  e.  Q. )  /\  E. w E. v E. u E. f ( ( x  =  [ <. w ,  v >. ]  ~Q  /\  y  =  [ <. u ,  f >. ]  ~Q  )  /\  z  =  [
( <. w ,  v
>.  .pQ  <. u ,  f
>. ) ]  ~Q  )
)  <->  ( ( x  e.  Q.  /\  y  e.  Q. )  /\  E. z E. w E. v E. u E. f ( ( x  =  [ <. w ,  v >. ]  ~Q  /\  y  =  [ <. u ,  f
>. ]  ~Q  )  /\  z  =  [ ( <. w ,  v >.  .pQ  <. u ,  f
>. ) ]  ~Q  )
) )
2523, 24bitr4i 187 . . . 4  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  <->  E. z ( ( x  e.  Q.  /\  y  e.  Q. )  /\  E. w E. v E. u E. f ( ( x  =  [ <. w ,  v >. ]  ~Q  /\  y  =  [ <. u ,  f >. ]  ~Q  )  /\  z  =  [
( <. w ,  v
>.  .pQ  <. u ,  f
>. ) ]  ~Q  )
) )
2625opabbii 4085 . . 3  |-  { <. x ,  y >.  |  ( x  e.  Q.  /\  y  e.  Q. ) }  =  { <. x ,  y >.  |  E. z ( ( x  e.  Q.  /\  y  e.  Q. )  /\  E. w E. v E. u E. f ( ( x  =  [ <. w ,  v >. ]  ~Q  /\  y  =  [ <. u ,  f >. ]  ~Q  )  /\  z  =  [
( <. w ,  v
>.  .pQ  <. u ,  f
>. ) ]  ~Q  )
) }
271, 3, 263eqtr4i 2220 . 2  |-  dom  .Q  =  { <. x ,  y
>.  |  ( x  e.  Q.  /\  y  e. 
Q. ) }
28 df-xp 4647 . 2  |-  ( Q. 
X.  Q. )  =  { <. x ,  y >.  |  ( x  e. 
Q.  /\  y  e.  Q. ) }
2927, 28eqtr4i 2213 1  |-  dom  .Q  =  ( Q.  X.  Q. )
Colors of variables: wff set class
Syntax hints:    /\ wa 104    = wceq 1364   E.wex 1503    e. wcel 2160   _Vcvv 2752   <.cop 3610   {copab 4078    X. cxp 4639   dom cdm 4641  (class class class)co 5891   {coprab 5892   [cec 6551    .pQ cmpq 7294    ~Q ceq 7296   Q.cnq 7297    .Q cmq 7300
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-iinf 4602
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-br 4019  df-opab 4080  df-iom 4605  df-xp 4647  df-cnv 4649  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-oprab 5895  df-ec 6555  df-qs 6559  df-ni 7321  df-enq 7364  df-nqqs 7365  df-mqqs 7367
This theorem is referenced by: (None)
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