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Metamath Proof Explorer

Theorem xpundi

Description: Distributive law for Cartesian product over union. Theorem 103 of Suppes p. 52. (Contributed by NM, 12-Aug-2004)

Ref Expression
Assertion xpundi ( 𝐴 × ( 𝐵 ∪ 𝐶 ) ) = ( ( 𝐴 × 𝐵 ) ∪ ( 𝐴 × 𝐶 ) )

Proof

Step Hyp Ref Expression
1 df-xp ⊢ ( 𝐴 × ( 𝐵 ∪ 𝐶 ) ) = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ( 𝐵 ∪ 𝐶 ) ) }
2 df-xp ⊢ ( 𝐴 × 𝐵 ) = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) }
3 df-xp ⊢ ( 𝐴 × 𝐶 ) = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶 ) }
4 2 3 uneq12i ⊢ ( ( 𝐴 × 𝐵 ) ∪ ( 𝐴 × 𝐶 ) ) = ( { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) } ∪ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶 ) } )
5 elun ⊢ ( 𝑦 ∈ ( 𝐵 ∪ 𝐶 ) ↔ ( 𝑦 ∈ 𝐵 ∨ 𝑦 ∈ 𝐶 ) )
6 5 anbi2i ⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ( 𝐵 ∪ 𝐶 ) ) ↔ ( 𝑥 ∈ 𝐴 ∧ ( 𝑦 ∈ 𝐵 ∨ 𝑦 ∈ 𝐶 ) ) )
7 andi ⊢ ( ( 𝑥 ∈ 𝐴 ∧ ( 𝑦 ∈ 𝐵 ∨ 𝑦 ∈ 𝐶 ) ) ↔ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ∨ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶 ) ) )
8 6 7 bitri ⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ( 𝐵 ∪ 𝐶 ) ) ↔ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ∨ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶 ) ) )
9 8 opabbii ⊢ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ( 𝐵 ∪ 𝐶 ) ) } = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ∨ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶 ) ) }
10 unopab ⊢ ( { ⟨ 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 ⊢ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ( 𝐵 ∪ 𝐶 ) ) } = ( { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) } ∪ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶 ) } )
12 4 11 eqtr4i ⊢ ( ( 𝐴 × 𝐵 ) ∪ ( 𝐴 × 𝐶 ) ) = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ( 𝐵 ∪ 𝐶 ) ) }
13 1 12 eqtr4i ⊢ ( 𝐴 × ( 𝐵 ∪ 𝐶 ) ) = ( ( 𝐴 × 𝐵 ) ∪ ( 𝐴 × 𝐶 ) )