Theorem xpindir 4585

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 4583	. 2 |- ( ( A X. C ) i^i ( B X. C ) ) = ( ( A i^i B ) X. ( C i^i C ) )
2		inidm 3210	. . 3 |- ( C i^i C ) = C
3	2	xpeq2i 4473	. 2 |- ( ( A i^i B ) X. ( C i^i C ) ) = ( ( A i^i B ) X. C )
4	1, 3	eqtr2i 2110	1 |- ( ( A i^i B ) X. C ) = ( ( A X. C ) i^i ( B X. C ) )

Syntax hints:   = wceq 1290   i^i cin 2999   X. cxp 4450

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-opab 3906  df-xp 4458  df-rel 4459

This theorem is referenced by:  resres  4738  resindi  4741  imainrect  4889  resdmres  4935