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Theorem xpundi 4832
 Description: Distributive law for cross product over union. Theorem 103 of [Suppes] p. 52. (Contributed by NM, 12-Aug-2004.)
Assertion
Ref Expression
xpundi (A × (BC)) = ((A × B) ∪ (A × C))

Proof of Theorem xpundi
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elun 3220 . . . . . 6 (y (BC) ↔ (y B y C))
21anbi2i 675 . . . . 5 ((x A y (BC)) ↔ (x A (y B y C)))
3 andi 837 . . . . 5 ((x A (y B y C)) ↔ ((x A y B) (x A y C)))
42, 3bitri 240 . . . 4 ((x A y (BC)) ↔ ((x A y B) (x A y C)))
54opabbii 4626 . . 3 {x, y (x A y (BC))} = {x, y ((x A y B) (x A y C))}
6 unopab 4638 . . 3 ({x, y (x A y B)} ∪ {x, y (x A y C)}) = {x, y ((x A y B) (x A y C))}
75, 6eqtr4i 2376 . 2 {x, y (x A y (BC))} = ({x, y (x A y B)} ∪ {x, y (x A y C)})
8 df-xp 4784 . 2 (A × (BC)) = {x, y (x A y (BC))}
9 df-xp 4784 . . 3 (A × B) = {x, y (x A y B)}
10 df-xp 4784 . . 3 (A × C) = {x, y (x A y C)}
119, 10uneq12i 3416 . 2 ((A × B) ∪ (A × C)) = ({x, y (x A y B)} ∪ {x, y (x A y C)})
127, 8, 113eqtr4i 2383 1 (A × (BC)) = ((A × B) ∪ (A × C))
 Colors of variables: wff setvar class Syntax hints:   ∨ wo 357   ∧ wa 358   = wceq 1642   ∈ wcel 1710   ∪ cun 3207  {copab 4622   × cxp 4770 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-un 3214  df-opab 4623  df-xp 4784 This theorem is referenced by:  xpun  4834  addcdi  6250
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