Theorem xpundi 3307

Description: Distributive law for cross product over union. Theorem 103 of [Suppes] p. 52.

Assertion

Ref	Expression
xpundi	|- (A X. (B u. C)) = ((A X. B) u. (A X. C))

Proof of Theorem xpundi

Step	Hyp	Ref	Expression
1		elun 2223	. . . . . 6 |- (y e. (B u. C) <-> (y e. B \/ y e. C))
2	1	anbi2i 482	. . . . 5 |- ((x e. A /\ y e. (B u. C)) <-> (x e. A /\ (y e. B \/ y e. C)))
3		andi 606	. . . . 5 |- ((x e. A /\ (y e. B \/ y e. C)) <-> ((x e. A /\ y e. B) \/ (x e. A /\ y e. C)))
4	2, 3	bitri 171	. . . 4 |- ((x e. A /\ y e. (B u. C)) <-> ((x e. A /\ y e. B) \/ (x e. A /\ y e. C)))
5	4	opabbii 2740	. . 3 |- {<.x, y>. | (x e. A /\ y e. (B u. C))} = {<.x, y>. | ((x e. A /\ y e. B) \/ (x e. A /\ y e. C))}
6		unopab 2748	. . 3 |- ({<.x, y>. | (x e. A /\ y e. B)} u. {<.x, y>. | (x e. A /\ y e. C)}) = {<.x, y>. | ((x e. A /\ y e. B) \/ (x e. A /\ y e. C))}
7	5, 6	eqtr4i 1539	. 2 |- {<.x, y>. | (x e. A /\ y e. (B u. C))} = ({<.x, y>. | (x e. A /\ y e. B)} u. {<.x, y>. | (x e. A /\ y e. C)})
8		df-xp 3262	. 2 |- (A X. (B u. C)) = {<.x, y>. | (x e. A /\ y e. (B u. C))}
9		df-xp 3262	. . 3 |- (A X. B) = {<.x, y>. | (x e. A /\ y e. B)}
10		df-xp 3262	. . 3 |- (A X. C) = {<.x, y>. | (x e. A /\ y e. C)}
11	9, 10	uneq12i 2232	. 2 |- ((A X. B) u. (A X. C)) = ({<.x, y>. | (x e. A /\ y e. B)} u. {<.x, y>. | (x e. A /\ y e. C)})
12	7, 8, 11	3eqtr4i 1546	1 |- (A X. (B u. C)) = ((A X. B) u. (A X. C))

Syntax hints:   \/ wo 220   /\ wa 221   = wceq 989   e. wcel 991   u. cun 2095  {copab 2735   X. cxp 3246

This theorem is referenced by:  xpun 3309  xp2cda 5071  xpcdaen 5074  alephadd 7781

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 995  ax-gen 996  ax-8 997  ax-10 999  ax-12 1001  ax-17 1004  ax-4 1006  ax-5o 1008  ax-6o 1011  ax-9o 1156  ax-10o 1174  ax-16 1244  ax-11o 1252  ax-ext 1498

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-ex 1014  df-sb 1206  df-clab 1504  df-cleq 1509  df-clel 1512  df-v 1856  df-un 2100  df-opab 2736  df-xp 3262

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