womaan - Quantum Logic Explorer

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Theorem womaan 223

Description: Weak OM-like absorption law for ortholattices.

**Assertion**
Ref	Expression
womaan	(a ∪ (a^⊥ ∩ (a ∪ (a^⊥ ∩ b)))) = (a ∪ (a^⊥ ∩ b))

**Proof of Theorem womaan**
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))

Colors of variables: term

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)