Theorem ledir 175

Description: Half of distributive law. (Contributed by NM, 30-Nov-1998.)

Assertion

Ref	Expression
ledir	((b ∩ a) ∪ (c ∩ a)) ≤ ((b ∪ c) ∩ a)

Proof of Theorem ledir

Step	Hyp	Ref	Expression
1		ledi 174	. 2 ((a ∩ b) ∪ (a ∩ c)) ≤ (a ∩ (b ∪ c))
2		ancom 74	. . 3 (b ∩ a) = (a ∩ b)
3		ancom 74	. . 3 (c ∩ a) = (a ∩ c)
4	2, 3	2or 72	. 2 ((b ∩ a) ∪ (c ∩ a)) = ((a ∩ b) ∪ (a ∩ c))
5		ancom 74	. 2 ((b ∪ c) ∩ a) = (a ∩ (b ∪ c))
6	1, 4, 5	le3tr1 140	1 ((b ∩ a) ∪ (c ∩ a)) ≤ ((b ∪ c) ∩ a)

Syntax hints:   ≤ wle 2   ∪ wo 6   ∩ wa 7

This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38

This theorem depends on definitions:  df-a 40  df-t 41  df-f 42  df-le1 130  df-le2 131

This theorem is referenced by: (None)