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Theorem xpundir 4483
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 4434 . 2  |-  ( ( A  u.  B )  X.  C )  =  { <. x ,  y
>.  |  ( x  e.  ( A  u.  B
)  /\  y  e.  C ) }
2 df-xp 4434 . . . 4  |-  ( A  X.  C )  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  e.  C ) }
3 df-xp 4434 . . . 4  |-  ( B  X.  C )  =  { <. x ,  y
>.  |  ( x  e.  B  /\  y  e.  C ) }
42, 3uneq12i 3150 . . 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 3139 . . . . . . 7  |-  ( x  e.  ( A  u.  B )  <->  ( x  e.  A  \/  x  e.  B ) )
65anbi1i 446 . . . . . 6  |-  ( ( x  e.  ( A  u.  B )  /\  y  e.  C )  <->  ( ( x  e.  A  \/  x  e.  B
)  /\  y  e.  C ) )
7 andir 768 . . . . . 6  |-  ( ( ( x  e.  A  \/  x  e.  B
)  /\  y  e.  C )  <->  ( (
x  e.  A  /\  y  e.  C )  \/  ( x  e.  B  /\  y  e.  C
) ) )
86, 7bitri 182 . . . . 5  |-  ( ( x  e.  ( A  u.  B )  /\  y  e.  C )  <->  ( ( x  e.  A  /\  y  e.  C
)  \/  ( x  e.  B  /\  y  e.  C ) ) )
98opabbii 3897 . . . 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 3909 . . . 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 2111 . . 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 2111 . 2  |-  ( ( A  X.  C )  u.  ( B  X.  C ) )  =  { <. x ,  y
>.  |  ( x  e.  ( A  u.  B
)  /\  y  e.  C ) }
131, 12eqtr4i 2111 1  |-  ( ( A  u.  B )  X.  C )  =  ( ( A  X.  C )  u.  ( B  X.  C ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    \/ wo 664    = wceq 1289    e. wcel 1438    u. cun 2995   {copab 3890    X. cxp 4426
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3001  df-opab 3892  df-xp 4434
This theorem is referenced by:  xpun  4487  resundi  4714  xpfi  6619  hashxp  10199
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