Theorem xpundir 4655

Description: Distributive law for cross 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 4604	. 2 ⊢ ((𝐴 ∪ 𝐵) × 𝐶) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴 ∪ 𝐵) ∧ 𝑦 ∈ 𝐶)}
2		df-xp 4604	. . . 4 ⊢ (𝐴 × 𝐶) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)}
3		df-xp 4604	. . . 4 ⊢ (𝐵 × 𝐶) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)}
4	2, 3	uneq12i 3269	. . 3 ⊢ ((𝐴 × 𝐶) ∪ (𝐵 × 𝐶)) = ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)} ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)})
5		elun 3258	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵))
6	5	anbi1i 454	. . . . . 6 ⊢ ((𝑥 ∈ (𝐴 ∪ 𝐵) ∧ 𝑦 ∈ 𝐶) ↔ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐶))
7		andir 809	. . . . . 6 ⊢ (((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐶) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶) ∨ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)))
8	6, 7	bitri 183	. . . . 5 ⊢ ((𝑥 ∈ (𝐴 ∪ 𝐵) ∧ 𝑦 ∈ 𝐶) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶) ∨ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)))
9	8	opabbii 4043	. . . 4 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴 ∪ 𝐵) ∧ 𝑦 ∈ 𝐶)} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶) ∨ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))}
10		unopab 4055	. . . 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 2188	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴 ∪ 𝐵) ∧ 𝑦 ∈ 𝐶)} = ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)} ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)})
12	4, 11	eqtr4i 2188	. 2 ⊢ ((𝐴 × 𝐶) ∪ (𝐵 × 𝐶)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴 ∪ 𝐵) ∧ 𝑦 ∈ 𝐶)}
13	1, 12	eqtr4i 2188	1 ⊢ ((𝐴 ∪ 𝐵) × 𝐶) = ((𝐴 × 𝐶) ∪ (𝐵 × 𝐶))

Syntax hints:   ∧ wa 103   ∨ wo 698   = wceq 1342   ∈ wcel 2135   ∪ cun 3109  {copab 4036   × cxp 4596

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146

This theorem depends on definitions:  df-bi 116  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-v 2723  df-un 3115  df-opab 4038  df-xp 4604

This theorem is referenced by:  xpun  4659  resundi  4891  xpfi  6886  xp2dju  7162  hashxp  10728