Theorem indir

Description: Distributive law for intersection over union. Theorem 28 of Suppes p. 27. (Contributed by NM, 30-Sep-2002)

	Ref	Expression
Assertion	indir	⊢ ( ( 𝐴 ∪ 𝐵 ) ∩ 𝐶 ) = ( ( 𝐴 ∩ 𝐶 ) ∪ ( 𝐵 ∩ 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		indi	⊢ ( 𝐶 ∩ ( 𝐴 ∪ 𝐵 ) ) = ( ( 𝐶 ∩ 𝐴 ) ∪ ( 𝐶 ∩ 𝐵 ) )
2		incom	⊢ ( ( 𝐴 ∪ 𝐵 ) ∩ 𝐶 ) = ( 𝐶 ∩ ( 𝐴 ∪ 𝐵 ) )
3		incom	⊢ ( 𝐴 ∩ 𝐶 ) = ( 𝐶 ∩ 𝐴 )
4		incom	⊢ ( 𝐵 ∩ 𝐶 ) = ( 𝐶 ∩ 𝐵 )
5	3 4	uneq12i	⊢ ( ( 𝐴 ∩ 𝐶 ) ∪ ( 𝐵 ∩ 𝐶 ) ) = ( ( 𝐶 ∩ 𝐴 ) ∪ ( 𝐶 ∩ 𝐵 ) )
6	1 2 5	3eqtr4i	⊢ ( ( 𝐴 ∪ 𝐵 ) ∩ 𝐶 ) = ( ( 𝐴 ∩ 𝐶 ) ∪ ( 𝐵 ∩ 𝐶 ) )