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Theorem dmaddpq 6535
Description: Domain of addition on positive fractions. (Contributed by NM, 24-Aug-1995.)
Assertion
Ref Expression
dmaddpq  |-  dom  +Q  =  ( Q.  X.  Q. )

Proof of Theorem dmaddpq
Dummy variables  x  y  z  v  w  u  f are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dmoprab 5613 . . 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-plqqs 6505 . . . 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 4564 . . 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 6533 . . . . . . . . 9  |-  ( x  e.  Q.  ->  E. w E. v  x  =  [ <. w ,  v
>. ]  ~Q  )
5 dmaddpqlem 6533 . . . . . . . . 9  |-  ( y  e.  Q.  ->  E. u E. f  y  =  [ <. u ,  f
>. ]  ~Q  )
64, 5anim12i 325 . . . . . . . 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 1825 . . . . . . . 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 141 . . . . . . 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 6516 . . . . . . . . . . . . . 14  |-  ~Q  e.  _V
10 ecexg 6141 . . . . . . . . . . . . . 14  |-  (  ~Q  e.  _V  ->  [ ( <. w ,  v >.  +pQ  <. u ,  f
>. ) ]  ~Q  e.  _V )
119, 10ax-mp 7 . . . . . . . . . . . . 13  |-  [ (
<. w ,  v >.  +pQ  <. u ,  f
>. ) ]  ~Q  e.  _V
1211isseti 2580 . . . . . . . . . . . 12  |-  E. z 
z  =  [ (
<. w ,  v >.  +pQ  <. u ,  f
>. ) ]  ~Q
13 ax-ia3 105 . . . . . . . . . . . . 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 1776 . . . . . . . . . . . 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 1508 . . . . . . . . . 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 1596 . . . . . . . . . 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 141 . . . . . . . . 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 1508 . . . . . . . 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 1596 . . . . . . . 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 141 . . . . . . 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 377 . . . . 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 1802 . . . . 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 180 . . . 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 3852 . . 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 2086 . 2  |-  dom  +Q  =  { <. x ,  y
>.  |  ( x  e.  Q.  /\  y  e. 
Q. ) }
28 df-xp 4379 . 2  |-  ( Q. 
X.  Q. )  =  { <. x ,  y >.  |  ( x  e. 
Q.  /\  y  e.  Q. ) }
2927, 28eqtr4i 2079 1  |-  dom  +Q  =  ( Q.  X.  Q. )
Colors of variables: wff set class
Syntax hints:    /\ wa 101    = wceq 1259   E.wex 1397    e. wcel 1409   _Vcvv 2574   <.cop 3406   {copab 3845    X. cxp 4371   dom cdm 4373  (class class class)co 5540   {coprab 5541   [cec 6135    +pQ cplpq 6432    ~Q ceq 6435   Q.cnq 6436    +Q cplq 6438
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-in1 554  ax-in2 555  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-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-iinf 4339
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-br 3793  df-opab 3847  df-iom 4342  df-xp 4379  df-cnv 4381  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-oprab 5544  df-ec 6139  df-qs 6143  df-ni 6460  df-enq 6503  df-nqqs 6504  df-plqqs 6505
This theorem is referenced by: (None)
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