Theorem ledior 177

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

Assertion

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

Proof of Theorem ledior

Step	Hyp	Ref	Expression
1		ledio 176	. 2 (a ∪ (b ∩ c)) ≤ ((a ∪ b) ∩ (a ∪ c))
2		ax-a2 31	. 2 ((b ∩ c) ∪ a) = (a ∪ (b ∩ c))
3		ax-a2 31	. . 3 (b ∪ a) = (a ∪ b)
4		ax-a2 31	. . 3 (c ∪ a) = (a ∪ c)
5	3, 4	2an 79	. 2 ((b ∪ a) ∩ (c ∪ a)) = ((a ∪ b) ∩ (a ∪ c))
6	1, 2, 5	le3tr1 140	1 ((b ∩ c) ∪ a) ≤ ((b ∪ a) ∩ (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:  oadistc0  1021