Theorem indir 2253

Description: Distributive law for intersection over union. Theorem 28 of [Suppes] p. 27.

Assertion

Ref	Expression
indir	|- ((A u. B) i^i C) = ((A i^i C) u. (B i^i C))

Proof of Theorem indir

Step	Hyp	Ref	Expression
1		indi 2251	. 2 |- (C i^i (A u. B)) = ((C i^i A) u. (C i^i B))
2		incom 2208	. 2 |- ((A u. B) i^i C) = (C i^i (A u. B))
3		incom 2208	. . 3 |- (A i^i C) = (C i^i A)
4		incom 2208	. . 3 |- (B i^i C) = (C i^i B)
5	3, 4	uneq12i 2182	. 2 |- ((A i^i C) u. (B i^i C)) = ((C i^i A) u. (C i^i B))
6	1, 2, 5	3eqtr4 1505	1 |- ((A u. B) i^i C) = ((A i^i C) u. (B i^i C))

Syntax hints:   = wceq 956   u. cun 2045   i^i cin 2046

This theorem is referenced by:  difundir 2258  undisj1 2320  resundir 3379

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-12 968  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

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-un 2050  df-in 2051

Copyright terms: Public domain