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Theorem dmmulpq 7321
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 5923 . . 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 7291 . . . 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 4805 . . 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 7318 . . . . . . . . 9 |- ( x e. Q. -> E. w E. v x = [ <. w , v >. ] ~Q )
5 dmaddpqlem 7318 . . . . . . . . 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 1922 . . . . . . . 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 7301 . . . . . . . . . . . . . 14 |- ~Q e. _V
10 ecexg 6505 . . . . . . . . . . . . . 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 2734 . . . . . . . . . . . 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 1868 . . . . . . . . . . . 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 1589 . . . . . . . . . 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 1678 . . . . . . . . . 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 1589 . . . . . . . 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 1678 . . . . . . . 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 1894 . . . . 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 4049 . . 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 2196 . 2 |- dom .Q = { <. x , y >. | ( x e. Q. /\ y e. Q. ) }
28 df-xp 4610 . 2 |- ( Q. X. Q. ) = { <. x , y >. | ( x e. Q. /\ y e. Q. ) }
2927, 28eqtr4i 2189 1 |- dom .Q = ( Q. X. Q. )
Colors of variables: wff set class
Syntax hints:   /\ wa 103   = wceq 1343   E.wex 1480   e. wcel 2136   _Vcvv 2726   <.cop 3579   {copab 4042   X. cxp 4602   dom cdm 4604  (class class class)co 5842   {coprab 5843   [cec 6499   .pQ cmpq 7218   ~Q ceq 7220   Q.cnq 7221   .Q cmq 7224
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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-iinf 4565
This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-br 3983  df-opab 4044  df-iom 4568  df-xp 4610  df-cnv 4612  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-oprab 5846  df-ec 6503  df-qs 6507  df-ni 7245  df-enq 7288  df-nqqs 7289  df-mqqs 7291
This theorem is referenced by: (None)
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