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Theorem dmmulpq 7737
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 6159 . . 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 7707 . . . 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 4977 . . 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 7734 . . . . . . . . 9 |- ( x e. Q. -> E. w E. v x = [ <. w , v >. ] ~Q )
5 dmaddpqlem 7734 . . . . . . . . 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 1994 . . . . . . . 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 7717 . . . . . . . . . . . . . 14 |- ~Q e. _V
10 ecexg 6801 . . . . . . . . . . . . . 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 2830 . . . . . . . . . . . 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 1933 . . . . . . . . . . . 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 1654 . . . . . . . . . 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 1742 . . . . . . . . . 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 1654 . . . . . . . 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 1742 . . . . . . . 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 395 . . . . 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 1962 . . . . 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 4193 . . 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 2269 . 2 |- dom .Q = { <. x , y >. | ( x e. Q. /\ y e. Q. ) }
28 df-xp 4775 . 2 |- ( Q. X. Q. ) = { <. x , y >. | ( x e. Q. /\ y e. Q. ) }
2927, 28eqtr4i 2262 1 |- dom .Q = ( Q. X. Q. )
Colors of variables: wff set class
Syntax hints:   /\ wa 104   = wceq 1402   E.wex 1545   e. wcel 2209   _Vcvv 2821   <.cop 3708   {copab 4186   X. cxp 4767   dom cdm 4769  (class class class)co 6075   {coprab 6076   [cec 6795   .pQ cmpq 7634   ~Q ceq 7636   Q.cnq 7637   .Q cmq 7640
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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-iinf 4730
This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-iom 4733  df-xp 4775  df-cnv 4777  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-oprab 6079  df-ec 6799  df-qs 6803  df-ni 7661  df-enq 7704  df-nqqs 7705  df-mqqs 7707
This theorem is referenced by: (None)
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