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Theorem xpundir 4566
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 4515 . 2  |-  ( ( A  u.  B )  X.  C )  =  { <. x ,  y
>.  |  ( x  e.  ( A  u.  B
)  /\  y  e.  C ) }
2 df-xp 4515 . . . 4  |-  ( A  X.  C )  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  e.  C ) }
3 df-xp 4515 . . . 4  |-  ( B  X.  C )  =  { <. x ,  y
>.  |  ( x  e.  B  /\  y  e.  C ) }
42, 3uneq12i 3198 . . 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 3187 . . . . . . 7  |-  ( x  e.  ( A  u.  B )  <->  ( x  e.  A  \/  x  e.  B ) )
65anbi1i 453 . . . . . 6  |-  ( ( x  e.  ( A  u.  B )  /\  y  e.  C )  <->  ( ( x  e.  A  \/  x  e.  B
)  /\  y  e.  C ) )
7 andir 793 . . . . . 6  |-  ( ( ( x  e.  A  \/  x  e.  B
)  /\  y  e.  C )  <->  ( (
x  e.  A  /\  y  e.  C )  \/  ( x  e.  B  /\  y  e.  C
) ) )
86, 7bitri 183 . . . . 5  |-  ( ( x  e.  ( A  u.  B )  /\  y  e.  C )  <->  ( ( x  e.  A  /\  y  e.  C
)  \/  ( x  e.  B  /\  y  e.  C ) ) )
98opabbii 3965 . . . 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 3977 . . . 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 2141 . . 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 2141 . 2  |-  ( ( A  X.  C )  u.  ( B  X.  C ) )  =  { <. x ,  y
>.  |  ( x  e.  ( A  u.  B
)  /\  y  e.  C ) }
131, 12eqtr4i 2141 1  |-  ( ( A  u.  B )  X.  C )  =  ( ( A  X.  C )  u.  ( B  X.  C ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    \/ wo 682    = wceq 1316    e. wcel 1465    u. cun 3039   {copab 3958    X. cxp 4507
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-un 3045  df-opab 3960  df-xp 4515
This theorem is referenced by:  xpun  4570  resundi  4802  xpfi  6786  xp2dju  7039  hashxp  10540
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