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Theorem dmmulpq 7695
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 6134 . . 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 7665 . . . 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 4957 . . 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 7692 . . . . . . . . 9 |- ( x e. Q. -> E. w E. v x = [ <. w , v >. ] ~Q )
5 dmaddpqlem 7692 . . . . . . . . 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 1988 . . . . . . . 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 7675 . . . . . . . . . . . . . 14 |- ~Q e. _V
10 ecexg 6771 . . . . . . . . . . . . . 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 2822 . . . . . . . . . . . 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 1929 . . . . . . . . . . . 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 1650 . . . . . . . . . 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 1738 . . . . . . . . . 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 1650 . . . . . . . 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 1738 . . . . . . . 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 1956 . . . . 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 4177 . . 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 2263 . 2 |- dom .Q = { <. x , y >. | ( x e. Q. /\ y e. Q. ) }
28 df-xp 4755 . 2 |- ( Q. X. Q. ) = { <. x , y >. | ( x e. Q. /\ y e. Q. ) }
2927, 28eqtr4i 2256 1 |- dom .Q = ( Q. X. Q. )
Colors of variables: wff set class
Syntax hints:   /\ wa 104   = wceq 1398   E.wex 1541   e. wcel 2203   _Vcvv 2813   <.cop 3692   {copab 4170   X. cxp 4747   dom cdm 4749  (class class class)co 6050   {coprab 6051   [cec 6765   .pQ cmpq 7592   ~Q ceq 7594   Q.cnq 7595   .Q cmq 7598
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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-iinf 4710
This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-br 4110  df-opab 4172  df-iom 4713  df-xp 4755  df-cnv 4757  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-oprab 6054  df-ec 6769  df-qs 6773  df-ni 7619  df-enq 7662  df-nqqs 7663  df-mqqs 7665
This theorem is referenced by: (None)
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