Theorem womao 212

Description: Weak OM-like absorption law for ortholattices.

Assertion

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

Proof of Theorem womao

Step	Hyp	Ref	Expression
1		lea 152	. . 3 (a⊥ ∩ (a ∪ (a⊥ ∩ (a ∪ b)))) ≤ a⊥
2		lear 153	. . . 4 (a⊥ ∩ (a ∪ (a⊥ ∩ (a ∪ b)))) ≤ (a ∪ (a⊥ ∩ (a ∪ b)))
3		leo 150	. . . . 5 a ≤ (a ∪ b)
4		lear 153	. . . . 5 (a⊥ ∩ (a ∪ b)) ≤ (a ∪ b)
5	3, 4	lel2or 162	. . . 4 (a ∪ (a⊥ ∩ (a ∪ b))) ≤ (a ∪ b)
6	2, 5	letr 129	. . 3 (a⊥ ∩ (a ∪ (a⊥ ∩ (a ∪ b)))) ≤ (a ∪ b)
7	1, 6	ler2an 165	. 2 (a⊥ ∩ (a ∪ (a⊥ ∩ (a ∪ b)))) ≤ (a⊥ ∩ (a ∪ b))
8		lea 152	. . 3 (a⊥ ∩ (a ∪ b)) ≤ a⊥
9		leor 151	. . 3 (a⊥ ∩ (a ∪ b)) ≤ (a ∪ (a⊥ ∩ (a ∪ b)))
10	8, 9	ler2an 165	. 2 (a⊥ ∩ (a ∪ b)) ≤ (a⊥ ∩ (a ∪ (a⊥ ∩ (a ∪ b))))
11	7, 10	lebi 137	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 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

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