Theorem xpundi 4215

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 2992	. . . . . 6 |- (y e. (B u. C) <-> (y e. B \/ y e. C))
2	1	anbi2i 804	. . . . 5 |- ((x e. A /\ y e. (B u. C)) <-> (x e. A /\ (y e. B \/ y e. C)))
3		andi 989	. . . . 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 306	. . . 4 |- ((x e. A /\ y e. (B u. C)) <-> ((x e. A /\ y e. B) \/ (x e. A /\ y e. C)))
5	4	opabbii 3602	. . 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 3610	. . 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 2193	. 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 4165	. 2 |- (A X. (B u. C)) = {<.x, y>. | (x e. A /\ y e. (B u. C))}
9		df-xp 4165	. . 3 |- (A X. B) = {<.x, y>. | (x e. A /\ y e. B)}
10		df-xp 4165	. . 3 |- (A X. C) = {<.x, y>. | (x e. A /\ y e. C)}
11	9, 10	uneq12i 3003	. 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 2200	1 |- (A X. (B u. C)) = ((A X. B) u. (A X. C))

Syntax hints:   \/ wo 432   /\ wa 433   = wceq 1615   e. wcel 1617   u. cun 2857  {copab 3597   X. cxp 4149

This theorem is referenced by:  xpun 4217  xp2cda 6315  xpcdaen 6318  alephadd 9381  phtpycolem5 17140

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1621  ax-gen 1622  ax-8 1623  ax-9 1624  ax-10 1625  ax-11 1626  ax-12 1627  ax-17 1634  ax-4 1637  ax-5o 1639  ax-6o 1642  ax-9o 1792  ax-10o 1810  ax-16 1883  ax-11o 1893  ax-ext 2152

This theorem depends on definitions:  df-bi 232  df-or 434  df-an 435  df-ex 1645  df-sb 1845  df-clab 2158  df-cleq 2163  df-clel 2166  df-v 2571  df-un 2864  df-opab 3598  df-xp 4165

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