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Theorem xpindi 4819
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 4818 . 2  |-  ( ( A  X.  B )  i^i  ( A  X.  C ) )  =  ( ( A  i^i  A )  X.  ( B  i^i  C ) )
2 inidm 3378 . . 3  |-  ( A  i^i  A )  =  A
32xpeq1i 4709 . 2  |-  ( ( A  i^i  A )  X.  ( B  i^i  C ) )  =  ( A  X.  ( B  i^i  C ) )
41, 3eqtr2i 2304 1  |-  ( A  X.  ( B  i^i  C ) )  =  ( ( A  X.  B
)  i^i  ( A  X.  C ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1623    i^i cin 3151    X. cxp 4687
This theorem is referenced by:  xpriindi  4822  xpcdaen  7809  fpwwe2lem13  8264  txhaus  17341  selsubf  25990
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-opab 4078  df-xp 4695  df-rel 4696
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