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Theorem xpundir 4833
 Description: Distributive law for cross product over union. Similar to Theorem 103 of [Suppes] p. 52. (Contributed by NM, 30-Sep-2002.)
Assertion
Ref Expression
xpundir ((AB) × C) = ((A × C) ∪ (B × C))

Proof of Theorem xpundir
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elun 3220 . . . . . 6 (x (AB) ↔ (x A x B))
21anbi1i 676 . . . . 5 ((x (AB) y C) ↔ ((x A x B) y C))
3 andir 838 . . . . 5 (((x A x B) y C) ↔ ((x A y C) (x B y C)))
42, 3bitri 240 . . . 4 ((x (AB) y C) ↔ ((x A y C) (x B y C)))
54opabbii 4626 . . 3 {x, y (x (AB) y C)} = {x, y ((x A y C) (x B y C))}
6 unopab 4638 . . 3 ({x, y (x A y C)} ∪ {x, y (x B y C)}) = {x, y ((x A y C) (x B y C))}
75, 6eqtr4i 2376 . 2 {x, y (x (AB) y C)} = ({x, y (x A y C)} ∪ {x, y (x B y C)})
8 df-xp 4784 . 2 ((AB) × C) = {x, y (x (AB) y C)}
9 df-xp 4784 . . 3 (A × C) = {x, y (x A y C)}
10 df-xp 4784 . . 3 (B × C) = {x, y (x B y C)}
119, 10uneq12i 3416 . 2 ((A × C) ∪ (B × C)) = ({x, y (x A y C)} ∪ {x, y (x B y C)})
127, 8, 113eqtr4i 2383 1 ((AB) × C) = ((A × C) ∪ (B × 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  resundi  4981
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