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Theorem xpindir 4794
Description: Distributive law for cross product over intersection. Similar to Theorem 102 of [Suppes] p. 52. (Contributed by NM, 26-Sep-2004.)
Assertion
Ref Expression
xpindir  |-  ( ( A  i^i  B )  X.  C )  =  ( ( A  X.  C )  i^i  ( B  X.  C ) )

Proof of Theorem xpindir
StepHypRef Expression
1 inxp 4792 . 2  |-  ( ( A  X.  C )  i^i  ( B  X.  C ) )  =  ( ( A  i^i  B )  X.  ( C  i^i  C ) )
2 inidm 3339 . . 3  |-  ( C  i^i  C )  =  C
32xpeq2i 4684 . 2  |-  ( ( A  i^i  B )  X.  ( C  i^i  C ) )  =  ( ( A  i^i  B
)  X.  C )
41, 3eqtr2i 2277 1  |-  ( ( A  i^i  B )  X.  C )  =  ( ( A  X.  C )  i^i  ( B  X.  C ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1619    i^i cin 3112    X. cxp 4645
This theorem is referenced by:  resres  4942  resindi  4945  imainrect  5093  resdmres  5137  cdaassen  7762  txhaus  17289
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4101  ax-nul 4109  ax-pr 4172
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2521  df-rex 2522  df-rab 2525  df-v 2759  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3417  df-if 3526  df-sn 3606  df-pr 3607  df-op 3609  df-opab 4038  df-xp 4661  df-rel 4662
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