Theorem xpundir 4397

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 4351	. 2 ⊢ ((𝐴 ∪ 𝐵) × 𝐶) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴 ∪ 𝐵) ∧ 𝑦 ∈ 𝐶)}
2		df-xp 4351	. . . 4 ⊢ (𝐴 × 𝐶) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)}
3		df-xp 4351	. . . 4 ⊢ (𝐵 × 𝐶) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)}
4	2, 3	uneq12i 3095	. . 3 ⊢ ((𝐴 × 𝐶) ∪ (𝐵 × 𝐶)) = ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)} ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)})
5		elun 3084	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵))
6	5	anbi1i 431	. . . . . 6 ⊢ ((𝑥 ∈ (𝐴 ∪ 𝐵) ∧ 𝑦 ∈ 𝐶) ↔ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐶))
7		andir 732	. . . . . 6 ⊢ (((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐶) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶) ∨ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)))
8	6, 7	bitri 173	. . . . 5 ⊢ ((𝑥 ∈ (𝐴 ∪ 𝐵) ∧ 𝑦 ∈ 𝐶) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶) ∨ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)))
9	8	opabbii 3824	. . . 4 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴 ∪ 𝐵) ∧ 𝑦 ∈ 𝐶)} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶) ∨ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))}
10		unopab 3836	. . . 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 2063	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴 ∪ 𝐵) ∧ 𝑦 ∈ 𝐶)} = ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)} ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)})
12	4, 11	eqtr4i 2063	. 2 ⊢ ((𝐴 × 𝐶) ∪ (𝐵 × 𝐶)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴 ∪ 𝐵) ∧ 𝑦 ∈ 𝐶)}
13	1, 12	eqtr4i 2063	1 ⊢ ((𝐴 ∪ 𝐵) × 𝐶) = ((𝐴 × 𝐶) ∪ (𝐵 × 𝐶))

Syntax hints:   ∧ wa 97   ∨ wo 629   = wceq 1243   ∈ wcel 1393   ∪ cun 2915  {copab 3817   × cxp 4343

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-opab 3819  df-xp 4351

This theorem is referenced by:  xpun  4401  resundi  4625