Theorem xpundir 5694

Description: Distributive law for Cartesian 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		df-xp 5630	. 2 ⊢ ((𝐴 ∪ 𝐵) × 𝐶) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴 ∪ 𝐵) ∧ 𝑦 ∈ 𝐶)}
2		df-xp 5630	. . . 4 ⊢ (𝐴 × 𝐶) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)}
3		df-xp 5630	. . . 4 ⊢ (𝐵 × 𝐶) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)}
4	2, 3	uneq12i 4118	. . 3 ⊢ ((𝐴 × 𝐶) ∪ (𝐵 × 𝐶)) = ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)} ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)})
5		elun 4105	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵))
6	5	anbi1i 624	. . . . . 6 ⊢ ((𝑥 ∈ (𝐴 ∪ 𝐵) ∧ 𝑦 ∈ 𝐶) ↔ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐶))
7		andir 1010	. . . . . 6 ⊢ (((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐶) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶) ∨ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)))
8	6, 7	bitri 275	. . . . 5 ⊢ ((𝑥 ∈ (𝐴 ∪ 𝐵) ∧ 𝑦 ∈ 𝐶) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶) ∨ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)))
9	8	opabbii 5165	. . . 4 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴 ∪ 𝐵) ∧ 𝑦 ∈ 𝐶)} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶) ∨ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))}
10		unopab 5178	. . . 4 ⊢ ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)} ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)}) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶) ∨ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))}
11	9, 10	eqtr4i 2762	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴 ∪ 𝐵) ∧ 𝑦 ∈ 𝐶)} = ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)} ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)})
12	4, 11	eqtr4i 2762	. 2 ⊢ ((𝐴 × 𝐶) ∪ (𝐵 × 𝐶)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴 ∪ 𝐵) ∧ 𝑦 ∈ 𝐶)}
13	1, 12	eqtr4i 2762	1 ⊢ ((𝐴 ∪ 𝐵) × 𝐶) = ((𝐴 × 𝐶) ∪ (𝐵 × 𝐶))

Syntax hints:   ∧ wa 395   ∨ wo 847   = wceq 1541   ∈ wcel 2113   ∪ cun 3899  {copab 5160   × cxp 5622

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-v 3442  df-un 3906  df-opab 5161  df-xp 5630

This theorem is referenced by:  xpun  5698  resundi  5952  xpprsng  7085  naddasslem1  8622  xp2dju  10087  alephadd  10488  hashxplem  14356  ustund  24166  cnmpopc  24878  poimirlem3  37824  poimirlem4  37825  poimirlem6  37827  poimirlem7  37828  poimirlem16  37837  poimirlem19  37840  fsuppssind  42836  pwssplit4  43331