Theorem xpundi 4833

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 × (B ∪ C)) = ((A × B) ∪ (A × C))

Proof of Theorem xpundi

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

Step	Hyp	Ref	Expression
1		elun 3221	. . . . . 6 ⊢ (y ∈ (B ∪ C) ↔ (y ∈ B ∨ y ∈ C))
2	1	anbi2i 675	. . . . 5 ⊢ ((x ∈ A ∧ y ∈ (B ∪ C)) ↔ (x ∈ A ∧ (y ∈ B ∨ y ∈ C)))
3		andi 837	. . . . 5 ⊢ ((x ∈ A ∧ (y ∈ B ∨ y ∈ C)) ↔ ((x ∈ A ∧ y ∈ B) ∨ (x ∈ A ∧ y ∈ C)))
4	2, 3	bitri 240	. . . 4 ⊢ ((x ∈ A ∧ y ∈ (B ∪ C)) ↔ ((x ∈ A ∧ y ∈ B) ∨ (x ∈ A ∧ y ∈ C)))
5	4	opabbii 4627	. . 3 ⊢ {⟨x, y⟩ ∣ (x ∈ A ∧ y ∈ (B ∪ C))} = {⟨x, y⟩ ∣ ((x ∈ A ∧ y ∈ B) ∨ (x ∈ A ∧ y ∈ C))}
6		unopab 4639	. . 3 ⊢ ({⟨x, y⟩ ∣ (x ∈ A ∧ y ∈ B)} ∪ {⟨x, y⟩ ∣ (x ∈ A ∧ y ∈ C)}) = {⟨x, y⟩ ∣ ((x ∈ A ∧ y ∈ B) ∨ (x ∈ A ∧ y ∈ C))}
7	5, 6	eqtr4i 2376	. 2 ⊢ {⟨x, y⟩ ∣ (x ∈ A ∧ y ∈ (B ∪ C))} = ({⟨x, y⟩ ∣ (x ∈ A ∧ y ∈ B)} ∪ {⟨x, y⟩ ∣ (x ∈ A ∧ y ∈ C)})
8		df-xp 4785	. 2 ⊢ (A × (B ∪ C)) = {⟨x, y⟩ ∣ (x ∈ A ∧ y ∈ (B ∪ C))}
9		df-xp 4785	. . 3 ⊢ (A × B) = {⟨x, y⟩ ∣ (x ∈ A ∧ y ∈ B)}
10		df-xp 4785	. . 3 ⊢ (A × C) = {⟨x, y⟩ ∣ (x ∈ A ∧ y ∈ C)}
11	9, 10	uneq12i 3417	. 2 ⊢ ((A × B) ∪ (A × C)) = ({⟨x, y⟩ ∣ (x ∈ A ∧ y ∈ B)} ∪ {⟨x, y⟩ ∣ (x ∈ A ∧ y ∈ C)})
12	7, 8, 11	3eqtr4i 2383	1 ⊢ (A × (B ∪ C)) = ((A × B) ∪ (A × C))

This proof depends on syntax axioms:   ∨ wo 357   ∧ wa 358   = wceq 1642   ∈ wcel 1710   ∪ cun 3208  {copab 4623   × cxp 4771

This proof depends on 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 proof 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 used by:  xpun  4835  addcdi  6251