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Theorem dfmpq2 6511
Description: Alternative 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 5545 . 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 6501 . 2  |-  .pQ  =  ( x  e.  ( N.  X.  N. ) ,  y  e.  ( N. 
X.  N. )  |->  <. (
( 1st `  x
)  .N  ( 1st `  y ) ) ,  ( ( 2nd `  x
)  .N  ( 2nd `  y ) ) >.
)
3 1st2nd2 5829 . . . . . . . . . 10  |-  ( x  e.  ( N.  X.  N. )  ->  x  = 
<. ( 1st `  x
) ,  ( 2nd `  x ) >. )
43eqeq1d 2064 . . . . . . . . 9  |-  ( x  e.  ( N.  X.  N. )  ->  ( x  =  <. w ,  v
>. 
<-> 
<. ( 1st `  x
) ,  ( 2nd `  x ) >.  =  <. w ,  v >. )
)
5 1st2nd2 5829 . . . . . . . . . 10  |-  ( y  e.  ( N.  X.  N. )  ->  y  = 
<. ( 1st `  y
) ,  ( 2nd `  y ) >. )
65eqeq1d 2064 . . . . . . . . 9  |-  ( y  e.  ( N.  X.  N. )  ->  ( y  =  <. u ,  f
>. 
<-> 
<. ( 1st `  y
) ,  ( 2nd `  y ) >.  =  <. u ,  f >. )
)
74, 6bi2anan9 548 . . . . . . . 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 446 . . . . . . 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 133 . . . . . 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 1766 . . . . 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 5820 . . . . . . 7  |-  ( x  e.  ( N.  X.  N. )  ->  ( 1st `  x )  e.  N. )
12 xp2nd 5821 . . . . . . 7  |-  ( x  e.  ( N.  X.  N. )  ->  ( 2nd `  x )  e.  N. )
1311, 12jca 294 . . . . . 6  |-  ( x  e.  ( N.  X.  N. )  ->  ( ( 1st `  x )  e.  N.  /\  ( 2nd `  x )  e. 
N. ) )
14 xp1st 5820 . . . . . . 7  |-  ( y  e.  ( N.  X.  N. )  ->  ( 1st `  y )  e.  N. )
15 xp2nd 5821 . . . . . . 7  |-  ( y  e.  ( N.  X.  N. )  ->  ( 2nd `  y )  e.  N. )
1614, 15jca 294 . . . . . 6  |-  ( y  e.  ( N.  X.  N. )  ->  ( ( 1st `  y )  e.  N.  /\  ( 2nd `  y )  e. 
N. ) )
17 simpll 489 . . . . . . . . . 10  |-  ( ( ( w  =  ( 1st `  x )  /\  v  =  ( 2nd `  x ) )  /\  ( u  =  ( 1st `  y
)  /\  f  =  ( 2nd `  y ) ) )  ->  w  =  ( 1st `  x
) )
18 simprl 491 . . . . . . . . . 10  |-  ( ( ( w  =  ( 1st `  x )  /\  v  =  ( 2nd `  x ) )  /\  ( u  =  ( 1st `  y
)  /\  f  =  ( 2nd `  y ) ) )  ->  u  =  ( 1st `  y
) )
1917, 18oveq12d 5558 . . . . . . . . 9  |-  ( ( ( w  =  ( 1st `  x )  /\  v  =  ( 2nd `  x ) )  /\  ( u  =  ( 1st `  y
)  /\  f  =  ( 2nd `  y ) ) )  ->  (
w  .N  u )  =  ( ( 1st `  x )  .N  ( 1st `  y ) ) )
20 simplr 490 . . . . . . . . . 10  |-  ( ( ( w  =  ( 1st `  x )  /\  v  =  ( 2nd `  x ) )  /\  ( u  =  ( 1st `  y
)  /\  f  =  ( 2nd `  y ) ) )  ->  v  =  ( 2nd `  x
) )
21 simprr 492 . . . . . . . . . 10  |-  ( ( ( w  =  ( 1st `  x )  /\  v  =  ( 2nd `  x ) )  /\  ( u  =  ( 1st `  y
)  /\  f  =  ( 2nd `  y ) ) )  ->  f  =  ( 2nd `  y
) )
2220, 21oveq12d 5558 . . . . . . . . 9  |-  ( ( ( w  =  ( 1st `  x )  /\  v  =  ( 2nd `  x ) )  /\  ( u  =  ( 1st `  y
)  /\  f  =  ( 2nd `  y ) ) )  ->  (
v  .N  f )  =  ( ( 2nd `  x )  .N  ( 2nd `  y ) ) )
2319, 22opeq12d 3585 . . . . . . . 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 2067 . . . . . . 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 4012 . . . . . 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 277 . . . . 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 183 . . . 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 435 . . 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 5588 . 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 2086 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 101    <-> wb 102    = wceq 1259   E.wex 1397    e. wcel 1409   <.cop 3406    X. cxp 4371   ` cfv 4930  (class class class)co 5540   {coprab 5541    |-> cmpt2 5542   1stc1st 5793   2ndc2nd 5794   N.cnpi 6428    .N cmi 6430    .pQ cmpq 6433
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-un 4198
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2788  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-mpt 3848  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-iota 4895  df-fun 4932  df-fv 4938  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-1st 5795  df-2nd 5796  df-mpq 6501
This theorem is referenced by:  mulpipqqs  6529
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