Theorem xpindi 4714

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 4713	. 2 |- ( ( A X. B ) i^i ( A X. C ) ) = ( ( A i^i A ) X. ( B i^i C ) )
2		inidm 3312	. . 3 |- ( A i^i A ) = A
3	2	xpeq1i 4599	. 2 |- ( ( A i^i A ) X. ( B i^i C ) ) = ( A X. ( B i^i C ) )
4	1, 3	eqtr2i 2176	1 |- ( A X. ( B i^i C ) ) = ( ( A X. B ) i^i ( A X. C ) )

Syntax hints:   = wceq 1332   i^i cin 3097   X. cxp 4577

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-14 2128  ax-ext 2136  ax-sep 4078  ax-pow 4130  ax-pr 4164

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1740  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ral 2437  df-rex 2438  df-v 2711  df-un 3102  df-in 3104  df-ss 3111  df-pw 3541  df-sn 3562  df-pr 3563  df-op 3565  df-opab 4022  df-xp 4585  df-rel 4586

This theorem is referenced by:  xpriindim  4717  djuassen  7131  xpdjuen  7132