Theorem anorabs2 216

Description: Absorption law for ortholattices.

Assertion

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

Proof of Theorem anorabs2

Step	Hyp	Ref	Expression
1		lea 152	. . 3 (a ∩ (b ∪ (a ∩ (b ∪ c)))) ≤ a
2		lear 153	. . . 4 (a ∩ (b ∪ (a ∩ (b ∪ c)))) ≤ (b ∪ (a ∩ (b ∪ c)))
3		leo 150	. . . . 5 b ≤ (b ∪ c)
4		lear 153	. . . . 5 (a ∩ (b ∪ c)) ≤ (b ∪ c)
5	3, 4	lel2or 162	. . . 4 (b ∪ (a ∩ (b ∪ c))) ≤ (b ∪ c)
6	2, 5	letr 129	. . 3 (a ∩ (b ∪ (a ∩ (b ∪ c)))) ≤ (b ∪ c)
7	1, 6	ler2an 165	. 2 (a ∩ (b ∪ (a ∩ (b ∪ c)))) ≤ (a ∩ (b ∪ c))
8		lea 152	. . 3 (a ∩ (b ∪ c)) ≤ a
9		leor 151	. . 3 (a ∩ (b ∪ c)) ≤ (b ∪ (a ∩ (b ∪ c)))
10	8, 9	ler2an 165	. 2 (a ∩ (b ∪ c)) ≤ (a ∩ (b ∪ (a ∩ (b ∪ c))))
11	7, 10	lebi 137	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 29  ax-a2 30  ax-a3 31  ax-a5 33  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37

This theorem depends on definitions:  df-a 39  df-t 40  df-f 41  df-le1 122  df-le2 123

metamath.org