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Theorem xpindi 4726
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 4725 . 2  |-  ( ( A  X.  B )  i^i  ( A  X.  C ) )  =  ( ( A  i^i  A )  X.  ( B  i^i  C ) )
2 inidm 3285 . . 3  |-  ( A  i^i  A )  =  A
32xpeq1i 4616 . 2  |-  ( ( A  i^i  A )  X.  ( B  i^i  C ) )  =  ( A  X.  ( B  i^i  C ) )
41, 3eqtr2i 2274 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 3077    X. cxp 4578
This theorem is referenced by:  xpriindi  4729  xpcdaen  7693  fpwwe2lem13  8144  txhaus  17173  selsubf  25156
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 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108
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 1883  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-opab 3975  df-xp 4594  df-rel 4595
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