Theorem xpundi 5754

Description: Distributive law for Cartesian 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		df-xp 5691	. 2 ⊢ (𝐴 × (𝐵 ∪ 𝐶)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐵 ∪ 𝐶))}
2		df-xp 5691	. . . 4 ⊢ (𝐴 × 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)}
3		df-xp 5691	. . . 4 ⊢ (𝐴 × 𝐶) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)}
4	2, 3	uneq12i 4166	. . 3 ⊢ ((𝐴 × 𝐵) ∪ (𝐴 × 𝐶)) = ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)} ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)})
5		elun 4153	. . . . . . 7 ⊢ (𝑦 ∈ (𝐵 ∪ 𝐶) ↔ (𝑦 ∈ 𝐵 ∨ 𝑦 ∈ 𝐶))
6	5	anbi2i 623	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐵 ∪ 𝐶)) ↔ (𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝐵 ∨ 𝑦 ∈ 𝐶)))
7		andi 1010	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝐵 ∨ 𝑦 ∈ 𝐶)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∨ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)))
8	6, 7	bitri 275	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐵 ∪ 𝐶)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∨ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)))
9	8	opabbii 5210	. . . 4 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐵 ∪ 𝐶))} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∨ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶))}
10		unopab 5224	. . . 4 ⊢ ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)} ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)}) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∨ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶))}
11	9, 10	eqtr4i 2768	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐵 ∪ 𝐶))} = ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)} ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)})
12	4, 11	eqtr4i 2768	. 2 ⊢ ((𝐴 × 𝐵) ∪ (𝐴 × 𝐶)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐵 ∪ 𝐶))}
13	1, 12	eqtr4i 2768	1 ⊢ (𝐴 × (𝐵 ∪ 𝐶)) = ((𝐴 × 𝐵) ∪ (𝐴 × 𝐶))

Syntax hints:   ∧ wa 395   ∨ wo 848   = wceq 1540   ∈ wcel 2108   ∪ cun 3949  {copab 5205   × cxp 5683

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-un 3956  df-opab 5206  df-xp 5691

This theorem is referenced by:  xpun  5759  naddasslem2  8733  djuassen  10219  xpdjuen  10220  ustund  24230  bj-2upln1upl  37025