ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  xpundir GIF version

Theorem xpundir 4443
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 ((𝐴𝐵) × 𝐶) = ((𝐴 × 𝐶) ∪ (𝐵 × 𝐶))

Proof of Theorem xpundir
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-xp 4397 . 2 ((𝐴𝐵) × 𝐶) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴𝐵) ∧ 𝑦𝐶)}
2 df-xp 4397 . . . 4 (𝐴 × 𝐶) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐶)}
3 df-xp 4397 . . . 4 (𝐵 × 𝐶) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦𝐶)}
42, 3uneq12i 3134 . . 3 ((𝐴 × 𝐶) ∪ (𝐵 × 𝐶)) = ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐶)} ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦𝐶)})
5 elun 3123 . . . . . . 7 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
65anbi1i 446 . . . . . 6 ((𝑥 ∈ (𝐴𝐵) ∧ 𝑦𝐶) ↔ ((𝑥𝐴𝑥𝐵) ∧ 𝑦𝐶))
7 andir 766 . . . . . 6 (((𝑥𝐴𝑥𝐵) ∧ 𝑦𝐶) ↔ ((𝑥𝐴𝑦𝐶) ∨ (𝑥𝐵𝑦𝐶)))
86, 7bitri 182 . . . . 5 ((𝑥 ∈ (𝐴𝐵) ∧ 𝑦𝐶) ↔ ((𝑥𝐴𝑦𝐶) ∨ (𝑥𝐵𝑦𝐶)))
98opabbii 3865 . . . 4 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴𝐵) ∧ 𝑦𝐶)} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐶) ∨ (𝑥𝐵𝑦𝐶))}
10 unopab 3877 . . . 4 ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐶)} ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦𝐶)}) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐶) ∨ (𝑥𝐵𝑦𝐶))}
119, 10eqtr4i 2106 . . 3 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴𝐵) ∧ 𝑦𝐶)} = ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐶)} ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦𝐶)})
124, 11eqtr4i 2106 . 2 ((𝐴 × 𝐶) ∪ (𝐵 × 𝐶)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴𝐵) ∧ 𝑦𝐶)}
131, 12eqtr4i 2106 1 ((𝐴𝐵) × 𝐶) = ((𝐴 × 𝐶) ∪ (𝐵 × 𝐶))
Colors of variables: wff set class
Syntax hints:  wa 102  wo 662   = wceq 1285  wcel 1434  cun 2980  {copab 3858   × cxp 4389
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-un 2986  df-opab 3860  df-xp 4397
This theorem is referenced by:  xpun  4447  resundi  4673  xpfi  6472  hashxp  9902
  Copyright terms: Public domain W3C validator