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Theorem xpindi 4772
Description: Distributive law for cross product over intersection. Theorem 102 of [Suppes] p. 52. (Contributed by NM, 26-Sep-2004.)
Assertion
Ref Expression
xpindi  |-  ( A  X.  ( B  i^i  C ) )  =  ( ( A  X.  B
)  i^i  ( A  X.  C ) )

Proof of Theorem xpindi
StepHypRef Expression
1 inxp 4771 . 2  |-  ( ( A  X.  B )  i^i  ( A  X.  C ) )  =  ( ( A  i^i  A )  X.  ( B  i^i  C ) )
2 inidm 3320 . . 3  |-  ( A  i^i  A )  =  A
32xpeq1i 4662 . 2  |-  ( ( A  i^i  A )  X.  ( B  i^i  C ) )  =  ( A  X.  ( B  i^i  C ) )
41, 3eqtr2i 2277 1  |-  ( A  X.  ( B  i^i  C ) )  =  ( ( A  X.  B
)  i^i  ( A  X.  C ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1619    i^i cin 3093    X. cxp 4624
This theorem is referenced by:  xpriindi  4775  xpcdaen  7742  fpwwe2lem13  8197  txhaus  17268  selsubf  25322
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 4081  ax-nul 4089  ax-pr 4152
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 2520  df-rex 2521  df-rab 2523  df-v 2742  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-sn 3587  df-pr 3588  df-op 3590  df-opab 4018  df-xp 4640  df-rel 4641
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