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Theorem xpindir 4819
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 4817 . 2  |-  ( ( A  X.  C )  i^i  ( B  X.  C ) )  =  ( ( A  i^i  B )  X.  ( C  i^i  C ) )
2 inidm 3379 . . 3  |-  ( C  i^i  C )  =  C
32xpeq2i 4709 . 2  |-  ( ( A  i^i  B )  X.  ( C  i^i  C ) )  =  ( ( A  i^i  B
)  X.  C )
41, 3eqtr2i 2305 1  |-  ( ( A  i^i  B )  X.  C )  =  ( ( A  X.  C )  i^i  ( B  X.  C ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1624    i^i cin 3152    X. cxp 4686
This theorem is referenced by:  resres  4967  resindi  4970  imainrect  5118  resdmres  5162  cdaassen  7803  txhaus  17335
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-opab 4079  df-xp 4694  df-rel 4695
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