Theorem xpundir 4834

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	⊢ ((A ∪ B) × C) = ((A × C) ∪ (B × C))

Proof of Theorem xpundir

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elun 3221	. . . . . 6 ⊢ (x ∈ (A ∪ B) ↔ (x ∈ A ∨ x ∈ B))
2	1	anbi1i 676	. . . . 5 ⊢ ((x ∈ (A ∪ B) ∧ 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)))
4	2, 3	bitri 240	. . . 4 ⊢ ((x ∈ (A ∪ B) ∧ y ∈ C) ↔ ((x ∈ A ∧ y ∈ C) ∨ (x ∈ B ∧ y ∈ C)))
5	4	opabbii 4627	. . 3 ⊢ {⟨x, y⟩ ∣ (x ∈ (A ∪ B) ∧ y ∈ C)} = {⟨x, y⟩ ∣ ((x ∈ A ∧ y ∈ C) ∨ (x ∈ B ∧ y ∈ C))}
6		unopab 4639	. . 3 ⊢ ({⟨x, y⟩ ∣ (x ∈ A ∧ y ∈ C)} ∪ {⟨x, y⟩ ∣ (x ∈ B ∧ y ∈ C)}) = {⟨x, y⟩ ∣ ((x ∈ A ∧ y ∈ C) ∨ (x ∈ B ∧ y ∈ C))}
7	5, 6	eqtr4i 2376	. 2 ⊢ {⟨x, y⟩ ∣ (x ∈ (A ∪ B) ∧ y ∈ C)} = ({⟨x, y⟩ ∣ (x ∈ A ∧ y ∈ C)} ∪ {⟨x, y⟩ ∣ (x ∈ B ∧ y ∈ C)})
8		df-xp 4785	. 2 ⊢ ((A ∪ B) × C) = {⟨x, y⟩ ∣ (x ∈ (A ∪ B) ∧ y ∈ C)}
9		df-xp 4785	. . 3 ⊢ (A × C) = {⟨x, y⟩ ∣ (x ∈ A ∧ y ∈ C)}
10		df-xp 4785	. . 3 ⊢ (B × C) = {⟨x, y⟩ ∣ (x ∈ B ∧ y ∈ C)}
11	9, 10	uneq12i 3417	. 2 ⊢ ((A × C) ∪ (B × C)) = ({⟨x, y⟩ ∣ (x ∈ A ∧ y ∈ C)} ∪ {⟨x, y⟩ ∣ (x ∈ B ∧ y ∈ C)})
12	7, 8, 11	3eqtr4i 2383	1 ⊢ ((A ∪ B) × C) = ((A × C) ∪ (B × C))

Syntax hints:   ∨ wo 357   ∧ wa 358   = wceq 1642   ∈ wcel 1710   ∪ cun 3208  {copab 4623   × cxp 4771

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 2479  df-v 2862  df-nin 3212  df-compl 3213  df-un 3215  df-opab 4624  df-xp 4785

This theorem is referenced by:  xpun  4835  resundi  4982