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Theorem xpundi 4553
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 4503 . 2  |-  ( A  X.  ( B  u.  C ) )  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  e.  ( B  u.  C
) ) }
2 df-xp 4503 . . . 4  |-  ( A  X.  B )  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  e.  B ) }
3 df-xp 4503 . . . 4  |-  ( A  X.  C )  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  e.  C ) }
42, 3uneq12i 3192 . . 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 3181 . . . . . . 7  |-  ( y  e.  ( B  u.  C )  <->  ( y  e.  B  \/  y  e.  C ) )
65anbi2i 450 . . . . . 6  |-  ( ( x  e.  A  /\  y  e.  ( B  u.  C ) )  <->  ( x  e.  A  /\  (
y  e.  B  \/  y  e.  C )
) )
7 andi 790 . . . . . 6  |-  ( ( x  e.  A  /\  ( y  e.  B  \/  y  e.  C
) )  <->  ( (
x  e.  A  /\  y  e.  B )  \/  ( x  e.  A  /\  y  e.  C
) ) )
86, 7bitri 183 . . . . 5  |-  ( ( x  e.  A  /\  y  e.  ( B  u.  C ) )  <->  ( (
x  e.  A  /\  y  e.  B )  \/  ( x  e.  A  /\  y  e.  C
) ) )
98opabbii 3953 . . . 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 3965 . . . 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 2136 . . 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 2136 . 2  |-  ( ( A  X.  B )  u.  ( A  X.  C ) )  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  e.  ( B  u.  C
) ) }
131, 12eqtr4i 2136 1  |-  ( A  X.  ( B  u.  C ) )  =  ( ( A  X.  B )  u.  ( A  X.  C ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    \/ wo 680    = wceq 1312    e. wcel 1461    u. cun 3033   {copab 3946    X. cxp 4495
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095
This theorem depends on definitions:  df-bi 116  df-tru 1315  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-v 2657  df-un 3039  df-opab 3948  df-xp 4503
This theorem is referenced by:  xpun  4558  djuassen  7017  xpdjuen  7018
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