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Theorem xpun 4499
Description: The cross product of two unions. (Contributed by NM, 12-Aug-2004.)
Assertion
Ref Expression
xpun  |-  ( ( A  u.  B )  X.  ( C  u.  D ) )  =  ( ( ( A  X.  C )  u.  ( A  X.  D
) )  u.  (
( B  X.  C
)  u.  ( B  X.  D ) ) )

Proof of Theorem xpun
StepHypRef Expression
1 xpundi 4494 . 2  |-  ( ( A  u.  B )  X.  ( C  u.  D ) )  =  ( ( ( A  u.  B )  X.  C )  u.  (
( A  u.  B
)  X.  D ) )
2 xpundir 4495 . . 3  |-  ( ( A  u.  B )  X.  C )  =  ( ( A  X.  C )  u.  ( B  X.  C ) )
3 xpundir 4495 . . 3  |-  ( ( A  u.  B )  X.  D )  =  ( ( A  X.  D )  u.  ( B  X.  D ) )
42, 3uneq12i 3152 . 2  |-  ( ( ( A  u.  B
)  X.  C )  u.  ( ( A  u.  B )  X.  D ) )  =  ( ( ( A  X.  C )  u.  ( B  X.  C
) )  u.  (
( A  X.  D
)  u.  ( B  X.  D ) ) )
5 un4 3160 . 2  |-  ( ( ( A  X.  C
)  u.  ( B  X.  C ) )  u.  ( ( A  X.  D )  u.  ( B  X.  D
) ) )  =  ( ( ( A  X.  C )  u.  ( A  X.  D
) )  u.  (
( B  X.  C
)  u.  ( B  X.  D ) ) )
61, 4, 53eqtri 2112 1  |-  ( ( A  u.  B )  X.  ( C  u.  D ) )  =  ( ( ( A  X.  C )  u.  ( A  X.  D
) )  u.  (
( B  X.  C
)  u.  ( B  X.  D ) ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1289    u. cun 2997    X. cxp 4436
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 3003  df-opab 3900  df-xp 4444
This theorem is referenced by: (None)
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