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Theorem dmmulpq 7342
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 5934 . . 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 7312 . . . 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 4812 . . 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 7339 . . . . . . . . 9  |-  ( x  e.  Q.  ->  E. w E. v  x  =  [ <. w ,  v
>. ]  ~Q  )
5 dmaddpqlem 7339 . . . . . . . . 9  |-  ( y  e.  Q.  ->  E. u E. f  y  =  [ <. u ,  f
>. ]  ~Q  )
64, 5anim12i 336 . . . . . . . 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 1927 . . . . . . . 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 133 . . . . . . 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 7322 . . . . . . . . . . . . . 14  |-  ~Q  e.  _V
10 ecexg 6517 . . . . . . . . . . . . . 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 2738 . . . . . . . . . . . 12  |-  E. z 
z  =  [ (
<. w ,  v >.  .pQ  <. u ,  f
>. ) ]  ~Q
13 ax-ia3 107 . . . . . . . . . . . . 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 1873 . . . . . . . . . . . 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 1594 . . . . . . . . . 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 1683 . . . . . . . . . 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 133 . . . . . . . . 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 1594 . . . . . . . 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 1683 . . . . . . . 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 133 . . . . . . 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 389 . . . . 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 1899 . . . . 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 186 . . . 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 4056 . . 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 2201 . 2  |-  dom  .Q  =  { <. x ,  y
>.  |  ( x  e.  Q.  /\  y  e. 
Q. ) }
28 df-xp 4617 . 2  |-  ( Q. 
X.  Q. )  =  { <. x ,  y >.  |  ( x  e. 
Q.  /\  y  e.  Q. ) }
2927, 28eqtr4i 2194 1  |-  dom  .Q  =  ( Q.  X.  Q. )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    = wceq 1348   E.wex 1485    e. wcel 2141   _Vcvv 2730   <.cop 3586   {copab 4049    X. cxp 4609   dom cdm 4611  (class class class)co 5853   {coprab 5854   [cec 6511    .pQ cmpq 7239    ~Q ceq 7241   Q.cnq 7242    .Q cmq 7245
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-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-iinf 4572
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-br 3990  df-opab 4051  df-iom 4575  df-xp 4617  df-cnv 4619  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-oprab 5857  df-ec 6515  df-qs 6519  df-ni 7266  df-enq 7309  df-nqqs 7310  df-mqqs 7312
This theorem is referenced by: (None)
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