Theorem womaon 221

Description: Weak OM-like absorption law for ortholattices. (Contributed by NM, 8-Nov-1998.)

Assertion

Ref	Expression
womaon	(a ∩ (a⊥ ∪ (a ∩ (a⊥ ∪ b)))) = (a ∩ (a⊥ ∪ b))

Proof of Theorem womaon

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

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ 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:  kb10iii  893