Theorem xpindi 4539

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 4538	. 2 |- ( ( A X. B ) i^i ( A X. C ) ) = ( ( A i^i A ) X. ( B i^i C ) )
2		inidm 3198	. . 3 |- ( A i^i A ) = A
3	2	xpeq1i 4431	. 2 |- ( ( A i^i A ) X. ( B i^i C ) ) = ( A X. ( B i^i C ) )
4	1, 3	eqtr2i 2106	1 |- ( A X. ( B i^i C ) ) = ( ( A X. B ) i^i ( A X. C ) )

Syntax hints:   = wceq 1287   i^i cin 2987   X. cxp 4409

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3932  ax-pow 3984  ax-pr 4010

This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2617  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-opab 3875  df-xp 4417  df-rel 4418

This theorem is referenced by:  xpriindim  4542