Theorem xpindir 4675

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 4673	. 2 |- ( ( A X. C ) i^i ( B X. C ) ) = ( ( A i^i B ) X. ( C i^i C ) )
2		inidm 3285	. . 3 |- ( C i^i C ) = C
3	2	xpeq2i 4560	. 2 |- ( ( A i^i B ) X. ( C i^i C ) ) = ( ( A i^i B ) X. C )
4	1, 3	eqtr2i 2161	1 |- ( ( A i^i B ) X. C ) = ( ( A X. C ) i^i ( B X. C ) )

Syntax hints:   = wceq 1331   i^i cin 3070   X. cxp 4537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-opab 3990  df-xp 4545  df-rel 4546

This theorem is referenced by:  resres  4831  resindi  4834  imainrect  4984  resdmres  5030