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Theorem xpundi 4320
Description: Distributive law for cross product over union. Theorem 103 of [Suppes] p. 52. (Contributed by NM, 12-Aug-2004.)
Assertion
Ref Expression
xpundi  |-  ( A  X.  ( B  u.  C ) )  =  ( ( A  X.  B )  u.  ( A  X.  C ) )

Proof of Theorem xpundi
StepHypRef Expression
1 elun 2935 . . . . . 6  |-  ( y  e.  ( B  u.  C )  <->  ( y  e.  B  \/  y  e.  C ) )
21anbi2i 666 . . . . 5  |-  ( ( x  e.  A  /\  y  e.  ( B  u.  C ) )  <->  ( x  e.  A  /\  (
y  e.  B  \/  y  e.  C )
) )
3 andi 796 . . . . 5  |-  ( ( x  e.  A  /\  ( y  e.  B  \/  y  e.  C
) )  <->  ( (
x  e.  A  /\  y  e.  B )  \/  ( x  e.  A  /\  y  e.  C
) ) )
42, 3bitri 238 . . . 4  |-  ( ( x  e.  A  /\  y  e.  ( B  u.  C ) )  <->  ( (
x  e.  A  /\  y  e.  B )  \/  ( x  e.  A  /\  y  e.  C
) ) )
54opabbii 3642 . . 3  |-  { <. x ,  y >.  |  ( x  e.  A  /\  y  e.  ( B  u.  C ) ) }  =  { <. x ,  y >.  |  ( ( x  e.  A  /\  y  e.  B
)  \/  ( x  e.  A  /\  y  e.  C ) ) }
6 unopab 3653 . . 3  |-  ( {
<. x ,  y >.  |  ( x  e.  A  /\  y  e.  B ) }  u.  {
<. x ,  y >.  |  ( x  e.  A  /\  y  e.  C ) } )  =  { <. x ,  y >.  |  ( ( x  e.  A  /\  y  e.  B
)  \/  ( x  e.  A  /\  y  e.  C ) ) }
75, 6eqtr4i 2084 . 2  |-  { <. x ,  y >.  |  ( x  e.  A  /\  y  e.  ( B  u.  C ) ) }  =  ( { <. x ,  y >.  |  ( x  e.  A  /\  y  e.  B ) }  u.  { <. x ,  y >.  |  ( x  e.  A  /\  y  e.  C ) } )
8 df-xp 4266 . 2  |-  ( A  X.  ( B  u.  C ) )  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  e.  ( B  u.  C
) ) }
9 df-xp 4266 . . 3  |-  ( A  X.  B )  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  e.  B ) }
10 df-xp 4266 . . 3  |-  ( A  X.  C )  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  e.  C ) }
119, 10uneq12i 2946 . 2  |-  ( ( A  X.  B )  u.  ( A  X.  C ) )  =  ( { <. x ,  y >.  |  ( x  e.  A  /\  y  e.  B ) }  u.  { <. x ,  y >.  |  ( x  e.  A  /\  y  e.  C ) } )
127, 8, 113eqtr4i 2091 1  |-  ( A  X.  ( B  u.  C ) )  =  ( ( A  X.  B )  u.  ( A  X.  C ) )
Colors of variables: wff set class
Syntax hints:    \/ wo 355    /\ wa 356    = wceq 1517    e. wcel 1519    u. cun 2788   {copab 3635    X. cxp 4250
This theorem is referenced by:  xpun  4326  xp2cda  7259  xpcdaen  7262  alephadd  7661
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-5 1439  ax-6 1440  ax-7 1441  ax-gen 1442  ax-8 1521  ax-11 1522  ax-17 1526  ax-12o 1559  ax-10 1573  ax-9 1579  ax-4 1586  ax-16 1772  ax-ext 2043
This theorem depends on definitions:  df-bi 175  df-or 357  df-an 358  df-ex 1444  df-sb 1733  df-clab 2049  df-cleq 2054  df-clel 2057  df-v 2474  df-un 2795  df-opab 3637  df-xp 4266
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