Theorem xpundir 2462

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

Assertion

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

Proof of Theorem xpundir

Step	Hyp	Ref	Expression
1		elun 1601	. . . . . 6 |- (x e. (A u. B) <-> (x e. A \/ x e. B))
2	1	anbi1i 368	. . . . 5 |- ((x e. (A u. B) /\ y e. C) <-> ((x e. A \/ x e. B) /\ y e. C))
3		andir 457	. . . . 5 |- (((x e. A \/ x e. B) /\ y e. C) <-> ((x e. A /\ y e. C) \/ (x e. B /\ y e. C)))
4	2, 3	bitr 151	. . . 4 |- ((x e. (A u. B) /\ y e. C) <-> ((x e. A /\ y e. C) \/ (x e. B /\ y e. C)))
5	4	biopabi 2103	. . 3 |- {<.x, y>. | (x e. (A u. B) /\ y e. C)} = {<.x, y>. | ((x e. A /\ y e. C) \/ (x e. B /\ y e. C))}
6		unopab 2121	. . 3 |- ({<.x, y>. | (x e. A /\ y e. C)} u. {<.x, y>. | (x e. B /\ y e. C)}) = {<.x, y>. | ((x e. A /\ y e. C) \/ (x e. B /\ y e. C))}
7	5, 6	eqtr4 1122	. 2 |- {<.x, y>. | (x e. (A u. B) /\ y e. C)} = ({<.x, y>. | (x e. A /\ y e. C)} u. {<.x, y>. | (x e. B /\ y e. C)})
8		df-xp 2424	. 2 |- ((A u. B) X. C) = {<.x, y>. | (x e. (A u. B) /\ y e. C)}
9		df-xp 2424	. . 3 |- (A X. C) = {<.x, y>. | (x e. A /\ y e. C)}
10		df-xp 2424	. . 3 |- (B X. C) = {<.x, y>. | (x e. B /\ y e. C)}
11	9, 10	uneq12i 1609	. 2 |- ((A X. C) u. (B X. C)) = ({<.x, y>. | (x e. A /\ y e. C)} u. {<.x, y>. | (x e. B /\ y e. C)})
12	7, 8, 11	3eqtr4 1126	1 |- ((A u. B) X. C) = ((A X. C) u. (B X. C))

Syntax hints:   \/ wo 195   /\ wa 196   = wceq 1091   e. wcel 1092   u. cun 1485  {copab 2055   X. cxp 2408

This theorem is referenced by:  xpun 2463  resundi 2582  cdaassen 3725

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-un 1490  df-opab 2098  df-xp 2424

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