Theorem xpundir 3232

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 2176	. . . . . 6 ⊢ (x ∈ (A ∪ B) ↔ (x ∈ A ⋁ x ∈ B))
2	1	anbi1i 483	. . . . 5 ⊢ ((x ∈ (A ∪ B) ⋀ y ∈ C) ↔ ((x ∈ A ⋁ x ∈ B) ⋀ y ∈ C))
3		andir 607	. . . . 5 ⊢ (((x ∈ A ⋁ x ∈ B) ⋀ y ∈ C) ↔ ((x ∈ A ⋀ y ∈ C) ⋁ (x ∈ B ⋀ y ∈ C)))
4	2, 3	bitr 173	. . . 4 ⊢ ((x ∈ (A ∪ B) ⋀ y ∈ C) ↔ ((x ∈ A ⋀ y ∈ C) ⋁ (x ∈ B ⋀ y ∈ C)))
5	4	opabbii 2676	. . 3 ⊢ {⟨x, y⟩∣(x ∈ (A ∪ B) ⋀ y ∈ C)} = {⟨x, y⟩∣((x ∈ A ⋀ y ∈ C) ⋁ (x ∈ B ⋀ y ∈ C))}
6		unopab 2684	. . 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 1501	. 2 ⊢ {⟨x, y⟩∣(x ∈ (A ∪ B) ⋀ y ∈ C)} = ({⟨x, y⟩∣(x ∈ A ⋀ y ∈ C)} ∪ {⟨x, y⟩∣(x ∈ B ⋀ y ∈ C)})
8		df-xp 3190	. 2 ⊢ ((A ∪ B) × C) = {⟨x, y⟩∣(x ∈ (A ∪ B) ⋀ y ∈ C)}
9		df-xp 3190	. . 3 ⊢ (A × C) = {⟨x, y⟩∣(x ∈ A ⋀ y ∈ C)}
10		df-xp 3190	. . 3 ⊢ (B × C) = {⟨x, y⟩∣(x ∈ B ⋀ y ∈ C)}
11	9, 10	uneq12i 2185	. 2 ⊢ ((A × C) ∪ (B × C)) = ({⟨x, y⟩∣(x ∈ A ⋀ y ∈ C)} ∪ {⟨x, y⟩∣(x ∈ B ⋀ y ∈ C)})
12	7, 8, 11	3eqtr4 1508	1 ⊢ ((A ∪ B) × C) = ((A × C) ∪ (B × C))

Syntax hints:   ⋁ wo 222   ⋀ wa 223   = wceq 958   ∈ wcel 960   ∪ cun 2048  {copab 2671   × cxp 3174

This theorem is referenced by:  xpun 3233  resundi 3384  cdaassen 4942

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-v 1815  df-un 2053  df-opab 2672  df-xp 3190

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