MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fpar Structured version   Unicode version

Theorem fpar 6909
Description: Merge two functions in parallel. Use as the second argument of a composition with a (2-place) operation to build compound operations such as  z  =  ( ( sqr `  x
)  +  ( abs `  y ) ). (Contributed by NM, 17-Sep-2007.) (Proof shortened by Mario Carneiro, 28-Apr-2015.)
Hypothesis
Ref Expression
fpar.1  |-  H  =  ( ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  ( F  o.  ( 1st  |`  ( _V  X.  _V ) ) ) )  i^i  ( `' ( 2nd  |`  ( _V  X.  _V ) )  o.  ( G  o.  ( 2nd  |`  ( _V  X.  _V ) ) ) ) )
Assertion
Ref Expression
fpar  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  H  =  ( x  e.  A ,  y  e.  B  |->  <. ( F `  x ) ,  ( G `  y ) >. )
)
Distinct variable groups:    x, y, A    x, B, y    x, F, y    x, G, y
Allowed substitution hints:    H( x, y)

Proof of Theorem fpar
StepHypRef Expression
1 fparlem3 6907 . . 3  |-  ( F  Fn  A  ->  ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  ( F  o.  ( 1st  |`  ( _V  X.  _V ) ) ) )  =  U_ x  e.  A  ( ( { x }  X.  _V )  X.  ( { ( F `  x ) }  X.  _V ) ) )
2 fparlem4 6908 . . 3  |-  ( G  Fn  B  ->  ( `' ( 2nd  |`  ( _V  X.  _V ) )  o.  ( G  o.  ( 2nd  |`  ( _V  X.  _V ) ) ) )  =  U_ y  e.  B  ( ( _V  X.  { y } )  X.  ( _V 
X.  { ( G `
 y ) } ) ) )
31, 2ineqan12d 3667 . 2  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  ( ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  ( F  o.  ( 1st  |`  ( _V  X.  _V ) ) ) )  i^i  ( `' ( 2nd  |`  ( _V  X.  _V ) )  o.  ( G  o.  ( 2nd  |`  ( _V  X.  _V ) ) ) ) )  =  ( U_ x  e.  A  (
( { x }  X.  _V )  X.  ( { ( F `  x ) }  X.  _V ) )  i^i  U_ y  e.  B  (
( _V  X.  {
y } )  X.  ( _V  X.  {
( G `  y
) } ) ) ) )
4 fpar.1 . 2  |-  H  =  ( ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  ( F  o.  ( 1st  |`  ( _V  X.  _V ) ) ) )  i^i  ( `' ( 2nd  |`  ( _V  X.  _V ) )  o.  ( G  o.  ( 2nd  |`  ( _V  X.  _V ) ) ) ) )
5 opex 4683 . . . 4  |-  <. ( F `  x ) ,  ( G `  y ) >.  e.  _V
65dfmpt2 6895 . . 3  |-  ( x  e.  A ,  y  e.  B  |->  <. ( F `  x ) ,  ( G `  y ) >. )  =  U_ x  e.  A  U_ y  e.  B  { <. <. x ,  y
>. ,  <. ( F `
 x ) ,  ( G `  y
) >. >. }
7 inxp 4984 . . . . . . . 8  |-  ( ( ( { x }  X.  _V )  X.  ( { ( F `  x ) }  X.  _V ) )  i^i  (
( _V  X.  {
y } )  X.  ( _V  X.  {
( G `  y
) } ) ) )  =  ( ( ( { x }  X.  _V )  i^i  ( _V  X.  { y } ) )  X.  (
( { ( F `
 x ) }  X.  _V )  i^i  ( _V  X.  {
( G `  y
) } ) ) )
8 inxp 4984 . . . . . . . . . 10  |-  ( ( { x }  X.  _V )  i^i  ( _V  X.  { y } ) )  =  ( ( { x }  i^i  _V )  X.  ( _V  i^i  { y } ) )
9 inv1 3790 . . . . . . . . . . 11  |-  ( { x }  i^i  _V )  =  { x }
10 incom 3656 . . . . . . . . . . . 12  |-  ( _V 
i^i  { y } )  =  ( { y }  i^i  _V )
11 inv1 3790 . . . . . . . . . . . 12  |-  ( { y }  i^i  _V )  =  { y }
1210, 11eqtri 2452 . . . . . . . . . . 11  |-  ( _V 
i^i  { y } )  =  { y }
139, 12xpeq12i 4873 . . . . . . . . . 10  |-  ( ( { x }  i^i  _V )  X.  ( _V 
i^i  { y } ) )  =  ( { x }  X.  {
y } )
14 vex 3085 . . . . . . . . . . 11  |-  x  e. 
_V
15 vex 3085 . . . . . . . . . . 11  |-  y  e. 
_V
1614, 15xpsn 6079 . . . . . . . . . 10  |-  ( { x }  X.  {
y } )  =  { <. x ,  y
>. }
178, 13, 163eqtri 2456 . . . . . . . . 9  |-  ( ( { x }  X.  _V )  i^i  ( _V  X.  { y } ) )  =  { <. x ,  y >. }
18 inxp 4984 . . . . . . . . . 10  |-  ( ( { ( F `  x ) }  X.  _V )  i^i  ( _V  X.  { ( G `
 y ) } ) )  =  ( ( { ( F `
 x ) }  i^i  _V )  X.  ( _V  i^i  {
( G `  y
) } ) )
19 inv1 3790 . . . . . . . . . . 11  |-  ( { ( F `  x
) }  i^i  _V )  =  { ( F `  x ) }
20 incom 3656 . . . . . . . . . . . 12  |-  ( _V 
i^i  { ( G `  y ) } )  =  ( { ( G `  y ) }  i^i  _V )
21 inv1 3790 . . . . . . . . . . . 12  |-  ( { ( G `  y
) }  i^i  _V )  =  { ( G `  y ) }
2220, 21eqtri 2452 . . . . . . . . . . 11  |-  ( _V 
i^i  { ( G `  y ) } )  =  { ( G `
 y ) }
2319, 22xpeq12i 4873 . . . . . . . . . 10  |-  ( ( { ( F `  x ) }  i^i  _V )  X.  ( _V 
i^i  { ( G `  y ) } ) )  =  ( { ( F `  x
) }  X.  {
( G `  y
) } )
24 fvex 5889 . . . . . . . . . . 11  |-  ( F `
 x )  e. 
_V
25 fvex 5889 . . . . . . . . . . 11  |-  ( G `
 y )  e. 
_V
2624, 25xpsn 6079 . . . . . . . . . 10  |-  ( { ( F `  x
) }  X.  {
( G `  y
) } )  =  { <. ( F `  x ) ,  ( G `  y )
>. }
2718, 23, 263eqtri 2456 . . . . . . . . 9  |-  ( ( { ( F `  x ) }  X.  _V )  i^i  ( _V  X.  { ( G `
 y ) } ) )  =  { <. ( F `  x
) ,  ( G `
 y ) >. }
2817, 27xpeq12i 4873 . . . . . . . 8  |-  ( ( ( { x }  X.  _V )  i^i  ( _V  X.  { y } ) )  X.  (
( { ( F `
 x ) }  X.  _V )  i^i  ( _V  X.  {
( G `  y
) } ) ) )  =  ( {
<. x ,  y >. }  X.  { <. ( F `  x ) ,  ( G `  y ) >. } )
29 opex 4683 . . . . . . . . 9  |-  <. x ,  y >.  e.  _V
3029, 5xpsn 6079 . . . . . . . 8  |-  ( {
<. x ,  y >. }  X.  { <. ( F `  x ) ,  ( G `  y ) >. } )  =  { <. <. x ,  y >. ,  <. ( F `  x ) ,  ( G `  y ) >. >. }
317, 28, 303eqtri 2456 . . . . . . 7  |-  ( ( ( { x }  X.  _V )  X.  ( { ( F `  x ) }  X.  _V ) )  i^i  (
( _V  X.  {
y } )  X.  ( _V  X.  {
( G `  y
) } ) ) )  =  { <. <.
x ,  y >. ,  <. ( F `  x ) ,  ( G `  y )
>. >. }
3231a1i 11 . . . . . 6  |-  ( y  e.  B  ->  (
( ( { x }  X.  _V )  X.  ( { ( F `
 x ) }  X.  _V ) )  i^i  ( ( _V 
X.  { y } )  X.  ( _V 
X.  { ( G `
 y ) } ) ) )  =  { <. <. x ,  y
>. ,  <. ( F `
 x ) ,  ( G `  y
) >. >. } )
3332iuneq2i 4316 . . . . 5  |-  U_ y  e.  B  ( (
( { x }  X.  _V )  X.  ( { ( F `  x ) }  X.  _V ) )  i^i  (
( _V  X.  {
y } )  X.  ( _V  X.  {
( G `  y
) } ) ) )  =  U_ y  e.  B  { <. <. x ,  y >. ,  <. ( F `  x ) ,  ( G `  y ) >. >. }
3433a1i 11 . . . 4  |-  ( x  e.  A  ->  U_ y  e.  B  ( (
( { x }  X.  _V )  X.  ( { ( F `  x ) }  X.  _V ) )  i^i  (
( _V  X.  {
y } )  X.  ( _V  X.  {
( G `  y
) } ) ) )  =  U_ y  e.  B  { <. <. x ,  y >. ,  <. ( F `  x ) ,  ( G `  y ) >. >. } )
3534iuneq2i 4316 . . 3  |-  U_ x  e.  A  U_ y  e.  B  ( ( ( { x }  X.  _V )  X.  ( { ( F `  x ) }  X.  _V ) )  i^i  (
( _V  X.  {
y } )  X.  ( _V  X.  {
( G `  y
) } ) ) )  =  U_ x  e.  A  U_ y  e.  B  { <. <. x ,  y >. ,  <. ( F `  x ) ,  ( G `  y ) >. >. }
36 2iunin 4365 . . 3  |-  U_ x  e.  A  U_ y  e.  B  ( ( ( { x }  X.  _V )  X.  ( { ( F `  x ) }  X.  _V ) )  i^i  (
( _V  X.  {
y } )  X.  ( _V  X.  {
( G `  y
) } ) ) )  =  ( U_ x  e.  A  (
( { x }  X.  _V )  X.  ( { ( F `  x ) }  X.  _V ) )  i^i  U_ y  e.  B  (
( _V  X.  {
y } )  X.  ( _V  X.  {
( G `  y
) } ) ) )
376, 35, 363eqtr2i 2458 . 2  |-  ( x  e.  A ,  y  e.  B  |->  <. ( F `  x ) ,  ( G `  y ) >. )  =  ( U_ x  e.  A  ( ( { x }  X.  _V )  X.  ( { ( F `  x ) }  X.  _V ) )  i^i  U_ y  e.  B  (
( _V  X.  {
y } )  X.  ( _V  X.  {
( G `  y
) } ) ) )
383, 4, 373eqtr4g 2489 1  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  H  =  ( x  e.  A ,  y  e.  B  |->  <. ( F `  x ) ,  ( G `  y ) >. )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    = wceq 1438    e. wcel 1869   _Vcvv 3082    i^i cin 3436   {csn 3997   <.cop 4003   U_ciun 4297    X. cxp 4849   `'ccnv 4850    |` cres 4853    o. ccom 4855    Fn wfn 5594   ` cfv 5599    |-> cmpt2 6305   1stc1st 6803   2ndc2nd 6804
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 985  df-tru 1441  df-fal 1444  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-reu 2783  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-if 3911  df-sn 3998  df-pr 4000  df-op 4004  df-uni 4218  df-iun 4299  df-br 4422  df-opab 4481  df-mpt 4482  df-id 4766  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-oprab 6307  df-mpt2 6308  df-1st 6805  df-2nd 6806
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator