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Theorem xpindir 4976
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 4974 . 2  |-  ( ( A  X.  C )  i^i  ( B  X.  C ) )  =  ( ( A  i^i  B )  X.  ( C  i^i  C ) )
2 inidm 3518 . . 3  |-  ( C  i^i  C )  =  C
32xpeq2i 4866 . 2  |-  ( ( A  i^i  B )  X.  ( C  i^i  C ) )  =  ( ( A  i^i  B
)  X.  C )
41, 3eqtr2i 2433 1  |-  ( ( A  i^i  B )  X.  C )  =  ( ( A  X.  C )  i^i  ( B  X.  C ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1649    i^i cin 3287    X. cxp 4843
This theorem is referenced by:  resres  5126  resindi  5129  imainrect  5279  resdmres  5328  cdaassen  8026  txhaus  17640  ustund  18212
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pr 4371
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-opab 4235  df-xp 4851  df-rel 4852
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