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Theorem xpundi 4342
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 2955 . . . . . 6  |-  ( y  e.  ( B  u.  C
)  <->  ( y  e.  B  \/  y  e.  C ) )
21anbi2i 671 . . . . 5  |-  ( (
x  e.  A  /\  y  e.  ( B  u.  C ) )  <->  ( x  e.  A  /\  (
y  e.  B  \/  y  e.  C )
) )
3 andi 801 . . . . 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 3660 . . 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 3671 . . 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 2105 . 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 4289 . 2  |-  ( A  X.  ( B  u.  C
) )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  e.  ( B  u.  C
) ) }
9 df-xp 4289 . . 3  |-  ( A  X.  B )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  e.  B ) }
10 df-xp 4289 . . 3  |-  ( A  X.  C )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  e.  C ) }
119, 10uneq12i 2966 . 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 2112 1  |-  ( A  X.  ( B  u.  C
) )  =  ( ( A  X.  B
)  u.  ( A  X.  C ) )
Colors of variables: wff set class
Syntax hints:    \/ wo 356    /\ wa 357    = wceq 1536    e. wcel 1538    u. cun 2808   {copab 3653    X. cxp 4273
This theorem is referenced by:  xpun  4348  xp2cda  7249  xpcdaen  7252  alephadd  7651
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-5 1451  ax-6 1452  ax-7 1453  ax-gen 1454  ax-8 1540  ax-11 1541  ax-17 1545  ax-12o 1578  ax-10 1592  ax-9 1598  ax-4 1606  ax-16 1793  ax-ext 2064
This theorem depends on definitions:  df-bi 175  df-or 358  df-an 359  df-ex 1456  df-sb 1754  df-clab 2070  df-cleq 2075  df-clel 2078  df-v 2494  df-un 2815  df-opab 3655  df-xp 4289
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