Theorem xpindir 4765

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

Step	Hyp	Ref	Expression
1		inxp 4763	. 2 |- ( ( A X. C ) i^i ( B X. C ) ) = ( ( A i^i B ) X. ( C i^i C ) )
2		inidm 3346	. . 3 |- ( C i^i C ) = C
3	2	xpeq2i 4649	. 2 |- ( ( A i^i B ) X. ( C i^i C ) ) = ( ( A i^i B ) X. C )
4	1, 3	eqtr2i 2199	1 |- ( ( A i^i B ) X. C ) = ( ( A X. C ) i^i ( B X. C ) )

Syntax hints:   = wceq 1353   i^i cin 3130   X. cxp 4626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-opab 4067  df-xp 4634  df-rel 4635

This theorem is referenced by:  resres  4921  resindi  4924  imainrect  5076  resdmres  5122