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Theorem dmmulpq 7342
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 5934 . . 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 7312 . . . 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 4812 . . 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 7339 . . . . . . . . 9 |- ( x e. Q. -> E. w E. v x = [ <. w , v >. ] ~Q )
5 dmaddpqlem 7339 . . . . . . . . 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 1927 . . . . . . . 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 7322 . . . . . . . . . . . . . 14 |- ~Q e. _V
10 ecexg 6517 . . . . . . . . . . . . . 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 2738 . . . . . . . . . . . 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 1873 . . . . . . . . . . . 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 1594 . . . . . . . . . 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 1683 . . . . . . . . . 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 1594 . . . . . . . 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 1683 . . . . . . . 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 1899 . . . . 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 4056 . . 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 2201 . 2 |- dom .Q = { <. x , y >. | ( x e. Q. /\ y e. Q. ) }
28 df-xp 4617 . 2 |- ( Q. X. Q. ) = { <. x , y >. | ( x e. Q. /\ y e. Q. ) }
2927, 28eqtr4i 2194 1 |- dom .Q = ( Q. X. Q. )
Colors of variables: wff set class
Syntax hints:   /\ wa 103   = wceq 1348   E.wex 1485   e. wcel 2141   _Vcvv 2730   <.cop 3586   {copab 4049   X. cxp 4609   dom cdm 4611  (class class class)co 5853   {coprab 5854   [cec 6511   .pQ cmpq 7239   ~Q ceq 7241   Q.cnq 7242   .Q cmq 7245
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-iinf 4572
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-br 3990  df-opab 4051  df-iom 4575  df-xp 4617  df-cnv 4619  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-oprab 5857  df-ec 6515  df-qs 6519  df-ni 7266  df-enq 7309  df-nqqs 7310  df-mqqs 7312
This theorem is referenced by: (None)
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