Theorem womaa 222

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

Assertion

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

Proof of Theorem womaa

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

This proof depends on syntax axioms:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7

This proof depends on 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 proof depends on definitions:  df-a 40  df-t 41  df-f 42  df-le1 130  df-le2 131

This theorem is used by:  nom13  310  lem3.3.7i3e1  1066