ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  dmmulpq Unicode version
Theorem dmmulpq 7378
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 5955 . . 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 7348 . . . 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 4828 . . 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 7375 . . . . . . . . 9 |- ( x e. Q. -> E. w E. v x = [ <. w , v >. ] ~Q )
5 dmaddpqlem 7375 . . . . . . . . 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 1934 . . . . . . . 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 7358 . . . . . . . . . . . . . 14 |- ~Q e. _V
10 ecexg 6538 . . . . . . . . . . . . . 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 2745 . . . . . . . . . . . 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 1880 . . . . . . . . . . . 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 1601 . . . . . . . . . 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 1690 . . . . . . . . . 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 1601 . . . . . . . 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 1690 . . . . . . . 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 1906 . . . . 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 4070 . . 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 2208 . 2 |- dom .Q = { <. x , y >. | ( x e. Q. /\ y e. Q. ) }
28 df-xp 4632 . 2 |- ( Q. X. Q. ) = { <. x , y >. | ( x e. Q. /\ y e. Q. ) }
2927, 28eqtr4i 2201 1 |- dom .Q = ( Q. X. Q. )
Colors of variables: wff set class
Syntax hints:   /\ wa 104   = wceq 1353   E.wex 1492   e. wcel 2148   _Vcvv 2737   <.cop 3595   {copab 4063   X. cxp 4624   dom cdm 4626  (class class class)co 5874   {coprab 5875   [cec 6532   .pQ cmpq 7275   ~Q ceq 7277   Q.cnq 7278   .Q cmq 7281
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-iinf 4587
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-br 4004  df-opab 4065  df-iom 4590  df-xp 4632  df-cnv 4634  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-oprab 5878  df-ec 6536  df-qs 6540  df-ni 7302  df-enq 7345  df-nqqs 7346  df-mqqs 7348
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator