Theorem xpindi 4780

Description: Distributive law for cross product over intersection. Theorem 102 of [Suppes] p. 52. (Contributed by NM, 26-Sep-2004.)

Assertion

Ref	Expression
xpindi	|- ( A X. ( B i^i C ) ) = ( ( A X. B ) i^i ( A X. C ) )

Proof of Theorem xpindi

Step	Hyp	Ref	Expression
1		inxp 4779	. 2 |- ( ( A X. B ) i^i ( A X. C ) ) = ( ( A i^i A ) X. ( B i^i C ) )
2		inidm 3359	. . 3 |- ( A i^i A ) = A
3	2	xpeq1i 4664	. 2 |- ( ( A i^i A ) X. ( B i^i C ) ) = ( A X. ( B i^i C ) )
4	1, 3	eqtr2i 2211	1 |- ( A X. ( B i^i C ) ) = ( ( A X. B ) i^i ( A X. C ) )

Syntax hints:   = wceq 1364   i^i cin 3143   X. cxp 4642

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-opab 4080  df-xp 4650  df-rel 4651

This theorem is referenced by:  xpriindim  4783  djuassen  7245  xpdjuen  7246