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

Theorem xpundir 4758
Description: Distributive law for cross product over union. Similar to Theorem 103 of [Suppes] p. 52. (Contributed by NM, 30-Sep-2002.)
Assertion
Ref Expression
xpundir  |-  ( ( A  u.  B )  X.  C )  =  ( ( A  X.  C )  u.  ( B  X.  C ) )

Proof of Theorem xpundir
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-xp 4711 . 2  |-  ( ( A  u.  B )  X.  C )  =  { <. x ,  y
>.  |  ( x  e.  ( A  u.  B
)  /\  y  e.  C ) }
2 df-xp 4711 . . . 4  |-  ( A  X.  C )  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  e.  C ) }
3 df-xp 4711 . . . 4  |-  ( B  X.  C )  =  { <. x ,  y
>.  |  ( x  e.  B  /\  y  e.  C ) }
42, 3uneq12i 3340 . . 3  |-  ( ( A  X.  C )  u.  ( B  X.  C ) )  =  ( { <. x ,  y >.  |  ( x  e.  A  /\  y  e.  C ) }  u.  { <. x ,  y >.  |  ( x  e.  B  /\  y  e.  C ) } )
5 elun 3329 . . . . . . 7  |-  ( x  e.  ( A  u.  B )  <->  ( x  e.  A  \/  x  e.  B ) )
65anbi1i 676 . . . . . 6  |-  ( ( x  e.  ( A  u.  B )  /\  y  e.  C )  <->  ( ( x  e.  A  \/  x  e.  B
)  /\  y  e.  C ) )
7 andir 838 . . . . . 6  |-  ( ( ( x  e.  A  \/  x  e.  B
)  /\  y  e.  C )  <->  ( (
x  e.  A  /\  y  e.  C )  \/  ( x  e.  B  /\  y  e.  C
) ) )
86, 7bitri 240 . . . . 5  |-  ( ( x  e.  ( A  u.  B )  /\  y  e.  C )  <->  ( ( x  e.  A  /\  y  e.  C
)  \/  ( x  e.  B  /\  y  e.  C ) ) )
98opabbii 4099 . . . 4  |-  { <. x ,  y >.  |  ( x  e.  ( A  u.  B )  /\  y  e.  C ) }  =  { <. x ,  y >.  |  ( ( x  e.  A  /\  y  e.  C
)  \/  ( x  e.  B  /\  y  e.  C ) ) }
10 unopab 4111 . . . 4  |-  ( {
<. x ,  y >.  |  ( x  e.  A  /\  y  e.  C ) }  u.  {
<. x ,  y >.  |  ( x  e.  B  /\  y  e.  C ) } )  =  { <. x ,  y >.  |  ( ( x  e.  A  /\  y  e.  C
)  \/  ( x  e.  B  /\  y  e.  C ) ) }
119, 10eqtr4i 2319 . . 3  |-  { <. x ,  y >.  |  ( x  e.  ( A  u.  B )  /\  y  e.  C ) }  =  ( { <. x ,  y >.  |  ( x  e.  A  /\  y  e.  C ) }  u.  {
<. x ,  y >.  |  ( x  e.  B  /\  y  e.  C ) } )
124, 11eqtr4i 2319 . 2  |-  ( ( A  X.  C )  u.  ( B  X.  C ) )  =  { <. x ,  y
>.  |  ( x  e.  ( A  u.  B
)  /\  y  e.  C ) }
131, 12eqtr4i 2319 1  |-  ( ( A  u.  B )  X.  C )  =  ( ( A  X.  C )  u.  ( B  X.  C ) )
Colors of variables: wff set class
Syntax hints:    \/ wo 357    /\ wa 358    = wceq 1632    e. wcel 1696    u. cun 3163   {copab 4092    X. cxp 4703
This theorem is referenced by:  xpun  4763  resundi  4985  xpfi  7144  cdaassen  7824  hashxplem  11401  cnmpt2pc  18442  pwssplit4  27294
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-un 3170  df-opab 4094  df-xp 4711
  Copyright terms: Public domain W3C validator