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Theorem xpundi 4741
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
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-xp 4695 . 2  |-  ( A  X.  ( B  u.  C ) )  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  e.  ( B  u.  C
) ) }
2 df-xp 4695 . . . 4  |-  ( A  X.  B )  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  e.  B ) }
3 df-xp 4695 . . . 4  |-  ( A  X.  C )  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  e.  C ) }
42, 3uneq12i 3327 . . 3  |-  ( ( A  X.  B )  u.  ( A  X.  C ) )  =  ( { <. x ,  y >.  |  ( x  e.  A  /\  y  e.  B ) }  u.  { <. x ,  y >.  |  ( x  e.  A  /\  y  e.  C ) } )
5 elun 3316 . . . . . . 7  |-  ( y  e.  ( B  u.  C )  <->  ( y  e.  B  \/  y  e.  C ) )
65anbi2i 675 . . . . . 6  |-  ( ( x  e.  A  /\  y  e.  ( B  u.  C ) )  <->  ( x  e.  A  /\  (
y  e.  B  \/  y  e.  C )
) )
7 andi 837 . . . . . 6  |-  ( ( x  e.  A  /\  ( y  e.  B  \/  y  e.  C
) )  <->  ( (
x  e.  A  /\  y  e.  B )  \/  ( x  e.  A  /\  y  e.  C
) ) )
86, 7bitri 240 . . . . 5  |-  ( ( x  e.  A  /\  y  e.  ( B  u.  C ) )  <->  ( (
x  e.  A  /\  y  e.  B )  \/  ( x  e.  A  /\  y  e.  C
) ) )
98opabbii 4083 . . . 4  |-  { <. x ,  y >.  |  ( x  e.  A  /\  y  e.  ( B  u.  C ) ) }  =  { <. x ,  y >.  |  ( ( x  e.  A  /\  y  e.  B
)  \/  ( x  e.  A  /\  y  e.  C ) ) }
10 unopab 4095 . . . 4  |-  ( {
<. 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 ) ) }
119, 10eqtr4i 2306 . . 3  |-  { <. 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 ) } )
124, 11eqtr4i 2306 . 2  |-  ( ( A  X.  B )  u.  ( A  X.  C ) )  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  e.  ( B  u.  C
) ) }
131, 12eqtr4i 2306 1  |-  ( A  X.  ( B  u.  C ) )  =  ( ( A  X.  B )  u.  ( A  X.  C ) )
Colors of variables: wff set class
Syntax hints:    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684    u. cun 3150   {copab 4076    X. cxp 4687
This theorem is referenced by:  xpun  4747  xp2cda  7806  xpcdaen  7809  alephadd  8199
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-un 3157  df-opab 4078  df-xp 4695
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