Theorem distlem 188

Description: Distributive law inference (uses OL only). (Contributed by NM, 17-Nov-1998.)

Hypothesis

Ref	Expression
distlem.1	(a ∩ (b ∪ c)) ≤ b

Assertion

Ref	Expression
distlem	(a ∩ (b ∪ c)) = ((a ∩ b) ∪ (a ∩ c))

Proof of Theorem distlem

Step	Hyp	Ref	Expression
1		lea 160	. . . 4 (a ∩ (b ∪ c)) ≤ a
2		distlem.1	. . . 4 (a ∩ (b ∪ c)) ≤ b
3	1, 2	ler2an 173	. . 3 (a ∩ (b ∪ c)) ≤ (a ∩ b)
4		leo 158	. . 3 (a ∩ b) ≤ ((a ∩ b) ∪ (a ∩ c))
5	3, 4	letr 137	. 2 (a ∩ (b ∪ c)) ≤ ((a ∩ b) ∪ (a ∩ c))
6		ledi 174	. 2 ((a ∩ b) ∪ (a ∩ c)) ≤ (a ∩ (b ∪ c))
7	5, 6	lebi 145	1 (a ∩ (b ∪ c)) = ((a ∩ b) ∪ (a ∩ c))

Syntax hints:   = wb 1   ≤ 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:  oadist2a  1007