Theorem xpindi 3270

Description: Distributive law for cross product over intersection. Theorem 102 of [Suppes] p. 52.

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 3269	. 2 |- ((A X. B) i^i (A X. C)) = ((A i^i A) X. (B i^i C))
2		inidm 2222	. . 3 |- (A i^i A) = A
3		xpeq1 3200	. . 3 |- ((A i^i A) = A -> ((A i^i A) X. (B i^i C)) = (A X. (B i^i C)))
4	2, 3	ax-mp 7	. 2 |- ((A i^i A) X. (B i^i C)) = (A X. (B i^i C))
5	1, 4	eqtr2 1496	1 |- (A X. (B i^i C)) = ((A X. B) i^i (A X. C))

Syntax hints:   = wceq 956   i^i cin 2046   X. cxp 3168

This theorem is referenced by:  xpcdaen 4931

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-opab 2667  df-xp 3184  df-rel 3185

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