Theorem xpundir 2462

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

Assertion

Ref	Expression
xpundir	⊢ ((A ∪ B) × C) = ((A × C) ∪ (B × C))

Proof of Theorem xpundir

Step	Hyp	Ref	Expression
1		elun 1601	. . . . . 6 ⊢ (x ∈ (A ∪ B) ↔ (x ∈ A ∨ x ∈ B))
2	1	anbi1i 368	. . . . 5 ⊢ ((x ∈ (A ∪ B) ∧ y ∈ C) ↔ ((x ∈ A ∨ x ∈ B) ∧ y ∈ C))
3		andir 457	. . . . 5 ⊢ (((x ∈ A ∨ x ∈ B) ∧ y ∈ C) ↔ ((x ∈ A ∧ y ∈ C) ∨ (x ∈ B ∧ y ∈ C)))
4	2, 3	bitr 151	. . . 4 ⊢ ((x ∈ (A ∪ B) ∧ y ∈ C) ↔ ((x ∈ A ∧ y ∈ C) ∨ (x ∈ B ∧ y ∈ C)))
5	4	biopabi 2103	. . 3 ⊢ {⟨x, y⟩∣(x ∈ (A ∪ B) ∧ y ∈ C)} = {⟨x, y⟩∣((x ∈ A ∧ y ∈ C) ∨ (x ∈ B ∧ y ∈ C))}
6		unopab 2121	. . 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	eqtr4 1122	. 2 ⊢ {⟨x, y⟩∣(x ∈ (A ∪ B) ∧ y ∈ C)} = ({⟨x, y⟩∣(x ∈ A ∧ y ∈ C)} ∪ {⟨x, y⟩∣(x ∈ B ∧ y ∈ C)})
8		df-xp 2424	. 2 ⊢ ((A ∪ B) × C) = {⟨x, y⟩∣(x ∈ (A ∪ B) ∧ y ∈ C)}
9		df-xp 2424	. . 3 ⊢ (A × C) = {⟨x, y⟩∣(x ∈ A ∧ y ∈ C)}
10		df-xp 2424	. . 3 ⊢ (B × C) = {⟨x, y⟩∣(x ∈ B ∧ y ∈ C)}
11	9, 10	uneq12i 1609	. 2 ⊢ ((A × C) ∪ (B × C)) = ({⟨x, y⟩∣(x ∈ A ∧ y ∈ C)} ∪ {⟨x, y⟩∣(x ∈ B ∧ y ∈ C)})
12	7, 8, 11	3eqtr4 1126	1 ⊢ ((A ∪ B) × C) = ((A × C) ∪ (B × C))

Syntax hints:   ∨ wo 195   ∧ wa 196   = wceq 1091   ∈ wcel 1092   ∪ cun 1485  {copab 2055   × 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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