Theorem anorabs 225

Description: Absorption law for ortholattices. (Contributed by NM, 8-Nov-1998.)

Assertion

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

Proof of Theorem anorabs

Step	Hyp	Ref	Expression
1		anorabs2 224	. 2 (a⊥ ∩ (b ∪ (a⊥ ∩ (b ∪ a)))) = (a⊥ ∩ (b ∪ a))
2		ax-a2 31	. . . . 5 (a ∪ b) = (b ∪ a)
3	2	lan 77	. . . 4 (a⊥ ∩ (a ∪ b)) = (a⊥ ∩ (b ∪ a))
4	3	lor 70	. . 3 (b ∪ (a⊥ ∩ (a ∪ b))) = (b ∪ (a⊥ ∩ (b ∪ a)))
5	4	lan 77	. 2 (a⊥ ∩ (b ∪ (a⊥ ∩ (a ∪ b)))) = (a⊥ ∩ (b ∪ (a⊥ ∩ (b ∪ a))))
6	1, 5, 3	3tr1 63	1 (a⊥ ∩ (b ∪ (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: (None)