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Theorem xpundir 4340
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 4294 . 2 ((AB) × 𝐶) = {⟨x, y⟩ ∣ (x (AB) y 𝐶)}
2 df-xp 4294 . . . 4 (A × 𝐶) = {⟨x, y⟩ ∣ (x A y 𝐶)}
3 df-xp 4294 . . . 4 (B × 𝐶) = {⟨x, y⟩ ∣ (x B y 𝐶)}
42, 3uneq12i 3089 . . 3 ((A × 𝐶) ∪ (B × 𝐶)) = ({⟨x, y⟩ ∣ (x A y 𝐶)} ∪ {⟨x, y⟩ ∣ (x B y 𝐶)})
5 elun 3078 . . . . . . 7 (x (AB) ↔ (x A x B))
65anbi1i 431 . . . . . 6 ((x (AB) y 𝐶) ↔ ((x A x B) y 𝐶))
7 andir 731 . . . . . 6 (((x A x B) y 𝐶) ↔ ((x A y 𝐶) (x B y 𝐶)))
86, 7bitri 173 . . . . 5 ((x (AB) y 𝐶) ↔ ((x A y 𝐶) (x B y 𝐶)))
98opabbii 3815 . . . 4 {⟨x, y⟩ ∣ (x (AB) y 𝐶)} = {⟨x, y⟩ ∣ ((x A y 𝐶) (x B y 𝐶))}
10 unopab 3827 . . . 4 ({⟨x, y⟩ ∣ (x A y 𝐶)} ∪ {⟨x, y⟩ ∣ (x B y 𝐶)}) = {⟨x, y⟩ ∣ ((x A y 𝐶) (x B y 𝐶))}
119, 10eqtr4i 2060 . . 3 {⟨x, y⟩ ∣ (x (AB) y 𝐶)} = ({⟨x, y⟩ ∣ (x A y 𝐶)} ∪ {⟨x, y⟩ ∣ (x B y 𝐶)})
124, 11eqtr4i 2060 . 2 ((A × 𝐶) ∪ (B × 𝐶)) = {⟨x, y⟩ ∣ (x (AB) y 𝐶)}
131, 12eqtr4i 2060 1 ((AB) × 𝐶) = ((A × 𝐶) ∪ (B × 𝐶))
Colors of variables: wff set class
Syntax hints:   wa 97   wo 628   = wceq 1242   wcel 1390  cun 2909  {copab 3808   × cxp 4286
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-opab 3810  df-xp 4294
This theorem is referenced by:  xpun  4344  resundi  4568
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