Theorem anorabs2 224

Description: Absorption law for ortholattices. (Contributed by NM, 13-Nov-1998.)

Assertion

Ref	Expression
anorabs2	(a ∩ (b ∪ (a ∩ (b ∪ c)))) = (a ∩ (b ∪ c))

Proof of Theorem anorabs2

Step	Hyp	Ref	Expression
1		lea 160	. . 3 (a ∩ (b ∪ (a ∩ (b ∪ c)))) ≤ a
2		lear 161	. . . 4 (a ∩ (b ∪ (a ∩ (b ∪ c)))) ≤ (b ∪ (a ∩ (b ∪ c)))
3		leo 158	. . . . 5 b ≤ (b ∪ c)
4		lear 161	. . . . 5 (a ∩ (b ∪ c)) ≤ (b ∪ c)
5	3, 4	lel2or 170	. . . 4 (b ∪ (a ∩ (b ∪ c))) ≤ (b ∪ c)
6	2, 5	letr 137	. . 3 (a ∩ (b ∪ (a ∩ (b ∪ c)))) ≤ (b ∪ c)
7	1, 6	ler2an 173	. 2 (a ∩ (b ∪ (a ∩ (b ∪ c)))) ≤ (a ∩ (b ∪ c))
8		lea 160	. . 3 (a ∩ (b ∪ c)) ≤ a
9		leor 159	. . 3 (a ∩ (b ∪ c)) ≤ (b ∪ (a ∩ (b ∪ c)))
10	8, 9	ler2an 173	. 2 (a ∩ (b ∪ c)) ≤ (a ∩ (b ∪ (a ∩ (b ∪ c))))
11	7, 10	lebi 145	1 (a ∩ (b ∪ (a ∩ (b ∪ c)))) = (a ∩ (b ∪ c))

Syntax hints:   = wb 1   ∪ 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:  anorabs  225