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Theorem dfmpq2 6893
Description: Alternate definition of pre-multiplication on positive fractions. (Contributed by Jim Kingdon, 13-Sep-2019.)
Assertion
Ref Expression
dfmpq2  |-  .pQ  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  ( N.  X.  N. )  /\  y  e.  ( N.  X.  N. )
)  /\  E. w E. v E. u E. f ( ( x  =  <. w ,  v
>.  /\  y  =  <. u ,  f >. )  /\  z  =  <. ( w  .N  u ) ,  ( v  .N  f ) >. )
) }
Distinct variable group:    x, y, z, w, v, u, f

Proof of Theorem dfmpq2
StepHypRef Expression
1 df-mpt2 5639 . 2  |-  ( x  e.  ( N.  X.  N. ) ,  y  e.  ( N.  X.  N. )  |->  <. ( ( 1st `  x )  .N  ( 1st `  y ) ) ,  ( ( 2nd `  x )  .N  ( 2nd `  y ) )
>. )  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  ( N.  X.  N. )  /\  y  e.  ( N.  X.  N. )
)  /\  z  =  <. ( ( 1st `  x
)  .N  ( 1st `  y ) ) ,  ( ( 2nd `  x
)  .N  ( 2nd `  y ) ) >.
) }
2 df-mpq 6883 . 2  |-  .pQ  =  ( x  e.  ( N.  X.  N. ) ,  y  e.  ( N. 
X.  N. )  |->  <. (
( 1st `  x
)  .N  ( 1st `  y ) ) ,  ( ( 2nd `  x
)  .N  ( 2nd `  y ) ) >.
)
3 1st2nd2 5927 . . . . . . . . . 10  |-  ( x  e.  ( N.  X.  N. )  ->  x  = 
<. ( 1st `  x
) ,  ( 2nd `  x ) >. )
43eqeq1d 2096 . . . . . . . . 9  |-  ( x  e.  ( N.  X.  N. )  ->  ( x  =  <. w ,  v
>. 
<-> 
<. ( 1st `  x
) ,  ( 2nd `  x ) >.  =  <. w ,  v >. )
)
5 1st2nd2 5927 . . . . . . . . . 10  |-  ( y  e.  ( N.  X.  N. )  ->  y  = 
<. ( 1st `  y
) ,  ( 2nd `  y ) >. )
65eqeq1d 2096 . . . . . . . . 9  |-  ( y  e.  ( N.  X.  N. )  ->  ( y  =  <. u ,  f
>. 
<-> 
<. ( 1st `  y
) ,  ( 2nd `  y ) >.  =  <. u ,  f >. )
)
74, 6bi2anan9 573 . . . . . . . 8  |-  ( ( x  e.  ( N. 
X.  N. )  /\  y  e.  ( N.  X.  N. ) )  ->  (
( x  =  <. w ,  v >.  /\  y  =  <. u ,  f
>. )  <->  ( <. ( 1st `  x ) ,  ( 2nd `  x
) >.  =  <. w ,  v >.  /\  <. ( 1st `  y ) ,  ( 2nd `  y
) >.  =  <. u ,  f >. )
) )
87anbi1d 453 . . . . . . 7  |-  ( ( x  e.  ( N. 
X.  N. )  /\  y  e.  ( N.  X.  N. ) )  ->  (
( ( x  = 
<. w ,  v >.  /\  y  =  <. u ,  f >. )  /\  z  =  <. ( w  .N  u ) ,  ( v  .N  f ) >. )  <->  ( ( <. ( 1st `  x
) ,  ( 2nd `  x ) >.  =  <. w ,  v >.  /\  <. ( 1st `  y ) ,  ( 2nd `  y
) >.  =  <. u ,  f >. )  /\  z  =  <. ( w  .N  u ) ,  ( v  .N  f ) >. )
) )
98bicomd 139 . . . . . 6  |-  ( ( x  e.  ( N. 
X.  N. )  /\  y  e.  ( N.  X.  N. ) )  ->  (
( ( <. ( 1st `  x ) ,  ( 2nd `  x
) >.  =  <. w ,  v >.  /\  <. ( 1st `  y ) ,  ( 2nd `  y
) >.  =  <. u ,  f >. )  /\  z  =  <. ( w  .N  u ) ,  ( v  .N  f ) >. )  <->  ( ( x  =  <. w ,  v >.  /\  y  =  <. u ,  f
>. )  /\  z  =  <. ( w  .N  u ) ,  ( v  .N  f )
>. ) ) )
1094exbidv 1798 . . . . 5  |-  ( ( x  e.  ( N. 
X.  N. )  /\  y  e.  ( N.  X.  N. ) )  ->  ( E. w E. v E. u E. f ( ( <. ( 1st `  x
) ,  ( 2nd `  x ) >.  =  <. w ,  v >.  /\  <. ( 1st `  y ) ,  ( 2nd `  y
) >.  =  <. u ,  f >. )  /\  z  =  <. ( w  .N  u ) ,  ( v  .N  f ) >. )  <->  E. w E. v E. u E. f ( ( x  =  <. w ,  v >.  /\  y  =  <. u ,  f
>. )  /\  z  =  <. ( w  .N  u ) ,  ( v  .N  f )
>. ) ) )
11 xp1st 5918 . . . . . . 7  |-  ( x  e.  ( N.  X.  N. )  ->  ( 1st `  x )  e.  N. )
12 xp2nd 5919 . . . . . . 7  |-  ( x  e.  ( N.  X.  N. )  ->  ( 2nd `  x )  e.  N. )
1311, 12jca 300 . . . . . 6  |-  ( x  e.  ( N.  X.  N. )  ->  ( ( 1st `  x )  e.  N.  /\  ( 2nd `  x )  e. 
N. ) )
14 xp1st 5918 . . . . . . 7  |-  ( y  e.  ( N.  X.  N. )  ->  ( 1st `  y )  e.  N. )
15 xp2nd 5919 . . . . . . 7  |-  ( y  e.  ( N.  X.  N. )  ->  ( 2nd `  y )  e.  N. )
1614, 15jca 300 . . . . . 6  |-  ( y  e.  ( N.  X.  N. )  ->  ( ( 1st `  y )  e.  N.  /\  ( 2nd `  y )  e. 
N. ) )
17 simpll 496 . . . . . . . . . 10  |-  ( ( ( w  =  ( 1st `  x )  /\  v  =  ( 2nd `  x ) )  /\  ( u  =  ( 1st `  y
)  /\  f  =  ( 2nd `  y ) ) )  ->  w  =  ( 1st `  x
) )
18 simprl 498 . . . . . . . . . 10  |-  ( ( ( w  =  ( 1st `  x )  /\  v  =  ( 2nd `  x ) )  /\  ( u  =  ( 1st `  y
)  /\  f  =  ( 2nd `  y ) ) )  ->  u  =  ( 1st `  y
) )
1917, 18oveq12d 5652 . . . . . . . . 9  |-  ( ( ( w  =  ( 1st `  x )  /\  v  =  ( 2nd `  x ) )  /\  ( u  =  ( 1st `  y
)  /\  f  =  ( 2nd `  y ) ) )  ->  (
w  .N  u )  =  ( ( 1st `  x )  .N  ( 1st `  y ) ) )
20 simplr 497 . . . . . . . . . 10  |-  ( ( ( w  =  ( 1st `  x )  /\  v  =  ( 2nd `  x ) )  /\  ( u  =  ( 1st `  y
)  /\  f  =  ( 2nd `  y ) ) )  ->  v  =  ( 2nd `  x
) )
21 simprr 499 . . . . . . . . . 10  |-  ( ( ( w  =  ( 1st `  x )  /\  v  =  ( 2nd `  x ) )  /\  ( u  =  ( 1st `  y
)  /\  f  =  ( 2nd `  y ) ) )  ->  f  =  ( 2nd `  y
) )
2220, 21oveq12d 5652 . . . . . . . . 9  |-  ( ( ( w  =  ( 1st `  x )  /\  v  =  ( 2nd `  x ) )  /\  ( u  =  ( 1st `  y
)  /\  f  =  ( 2nd `  y ) ) )  ->  (
v  .N  f )  =  ( ( 2nd `  x )  .N  ( 2nd `  y ) ) )
2319, 22opeq12d 3625 . . . . . . . 8  |-  ( ( ( w  =  ( 1st `  x )  /\  v  =  ( 2nd `  x ) )  /\  ( u  =  ( 1st `  y
)  /\  f  =  ( 2nd `  y ) ) )  ->  <. (
w  .N  u ) ,  ( v  .N  f ) >.  =  <. ( ( 1st `  x
)  .N  ( 1st `  y ) ) ,  ( ( 2nd `  x
)  .N  ( 2nd `  y ) ) >.
)
2423eqeq2d 2099 . . . . . . 7  |-  ( ( ( w  =  ( 1st `  x )  /\  v  =  ( 2nd `  x ) )  /\  ( u  =  ( 1st `  y
)  /\  f  =  ( 2nd `  y ) ) )  ->  (
z  =  <. (
w  .N  u ) ,  ( v  .N  f ) >.  <->  z  =  <. ( ( 1st `  x
)  .N  ( 1st `  y ) ) ,  ( ( 2nd `  x
)  .N  ( 2nd `  y ) ) >.
) )
2524copsex4g 4065 . . . . . 6  |-  ( ( ( ( 1st `  x
)  e.  N.  /\  ( 2nd `  x )  e.  N. )  /\  ( ( 1st `  y
)  e.  N.  /\  ( 2nd `  y )  e.  N. ) )  ->  ( E. w E. v E. u E. f ( ( <.
( 1st `  x
) ,  ( 2nd `  x ) >.  =  <. w ,  v >.  /\  <. ( 1st `  y ) ,  ( 2nd `  y
) >.  =  <. u ,  f >. )  /\  z  =  <. ( w  .N  u ) ,  ( v  .N  f ) >. )  <->  z  =  <. ( ( 1st `  x )  .N  ( 1st `  y ) ) ,  ( ( 2nd `  x )  .N  ( 2nd `  y ) )
>. ) )
2613, 16, 25syl2an 283 . . . . 5  |-  ( ( x  e.  ( N. 
X.  N. )  /\  y  e.  ( N.  X.  N. ) )  ->  ( E. w E. v E. u E. f ( ( <. ( 1st `  x
) ,  ( 2nd `  x ) >.  =  <. w ,  v >.  /\  <. ( 1st `  y ) ,  ( 2nd `  y
) >.  =  <. u ,  f >. )  /\  z  =  <. ( w  .N  u ) ,  ( v  .N  f ) >. )  <->  z  =  <. ( ( 1st `  x )  .N  ( 1st `  y ) ) ,  ( ( 2nd `  x )  .N  ( 2nd `  y ) )
>. ) )
2710, 26bitr3d 188 . . . 4  |-  ( ( x  e.  ( N. 
X.  N. )  /\  y  e.  ( N.  X.  N. ) )  ->  ( E. w E. v E. u E. f ( ( x  =  <. w ,  v >.  /\  y  =  <. u ,  f
>. )  /\  z  =  <. ( w  .N  u ) ,  ( v  .N  f )
>. )  <->  z  =  <. ( ( 1st `  x
)  .N  ( 1st `  y ) ) ,  ( ( 2nd `  x
)  .N  ( 2nd `  y ) ) >.
) )
2827pm5.32i 442 . . 3  |-  ( ( ( x  e.  ( N.  X.  N. )  /\  y  e.  ( N.  X.  N. ) )  /\  E. w E. v E. u E. f
( ( x  = 
<. w ,  v >.  /\  y  =  <. u ,  f >. )  /\  z  =  <. ( w  .N  u ) ,  ( v  .N  f ) >. )
)  <->  ( ( x  e.  ( N.  X.  N. )  /\  y  e.  ( N.  X.  N. ) )  /\  z  =  <. ( ( 1st `  x )  .N  ( 1st `  y ) ) ,  ( ( 2nd `  x )  .N  ( 2nd `  y ) )
>. ) )
2928oprabbii 5686 . 2  |-  { <. <.
x ,  y >. ,  z >.  |  ( ( x  e.  ( N.  X.  N. )  /\  y  e.  ( N.  X.  N. ) )  /\  E. w E. v E. u E. f
( ( x  = 
<. w ,  v >.  /\  y  =  <. u ,  f >. )  /\  z  =  <. ( w  .N  u ) ,  ( v  .N  f ) >. )
) }  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  ( N.  X.  N. )  /\  y  e.  ( N.  X.  N. )
)  /\  z  =  <. ( ( 1st `  x
)  .N  ( 1st `  y ) ) ,  ( ( 2nd `  x
)  .N  ( 2nd `  y ) ) >.
) }
301, 2, 293eqtr4i 2118 1  |-  .pQ  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  ( N.  X.  N. )  /\  y  e.  ( N.  X.  N. )
)  /\  E. w E. v E. u E. f ( ( x  =  <. w ,  v
>.  /\  y  =  <. u ,  f >. )  /\  z  =  <. ( w  .N  u ) ,  ( v  .N  f ) >. )
) }
Colors of variables: wff set class
Syntax hints:    /\ wa 102    <-> wb 103    = wceq 1289   E.wex 1426    e. wcel 1438   <.cop 3444    X. cxp 4426   ` cfv 5002  (class class class)co 5634   {coprab 5635    |-> cmpt2 5636   1stc1st 5891   2ndc2nd 5892   N.cnpi 6810    .N cmi 6812    .pQ cmpq 6815
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2839  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-mpt 3893  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-iota 4967  df-fun 5004  df-fv 5010  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-mpq 6883
This theorem is referenced by:  mulpipqqs  6911
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