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 ) ) |
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 ) ) |