Theorem xpundi 4719

Description: Distributive law for cross 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 4669	. 2 ⊢ (𝐴 × (𝐵 ∪ 𝐶)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐵 ∪ 𝐶))}
2		df-xp 4669	. . . 4 ⊢ (𝐴 × 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)}
3		df-xp 4669	. . . 4 ⊢ (𝐴 × 𝐶) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)}
4	2, 3	uneq12i 3315	. . 3 ⊢ ((𝐴 × 𝐵) ∪ (𝐴 × 𝐶)) = ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)} ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)})
5		elun 3304	. . . . . . 7 ⊢ (𝑦 ∈ (𝐵 ∪ 𝐶) ↔ (𝑦 ∈ 𝐵 ∨ 𝑦 ∈ 𝐶))
6	5	anbi2i 457	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐵 ∪ 𝐶)) ↔ (𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝐵 ∨ 𝑦 ∈ 𝐶)))
7		andi 819	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝐵 ∨ 𝑦 ∈ 𝐶)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∨ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)))
8	6, 7	bitri 184	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐵 ∪ 𝐶)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∨ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)))
9	8	opabbii 4100	. . . 4 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐵 ∪ 𝐶))} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∨ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶))}
10		unopab 4112	. . . 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 2220	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐵 ∪ 𝐶))} = ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)} ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)})
12	4, 11	eqtr4i 2220	. 2 ⊢ ((𝐴 × 𝐵) ∪ (𝐴 × 𝐶)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐵 ∪ 𝐶))}
13	1, 12	eqtr4i 2220	1 ⊢ (𝐴 × (𝐵 ∪ 𝐶)) = ((𝐴 × 𝐵) ∪ (𝐴 × 𝐶))

Syntax hints:   ∧ wa 104   ∨ wo 709   = wceq 1364   ∈ wcel 2167   ∪ cun 3155  {copab 4093   × cxp 4661

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-opab 4095  df-xp 4669

This theorem is referenced by:  xpun  4724  djuassen  7284  xpdjuen  7285