anorabs2 - Quantum Logic Explorer

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Theorem anorabs2 224

Description: Absorption law for ortholattices.

**Assertion**
Ref	Expression
anorabs2	(a ∩ (b ∪ (a ∩ (b ∪ c)))) = (a ∩ (b ∪ c))

**Proof of Theorem anorabs2**
Step	Hyp	Ref	Expression
1		lea 160	. . 3 (a ∩ (b ∪ (a ∩ (b ∪ c)))) ≤ a
2		lear 161	. . . 4 (a ∩ (b ∪ (a ∩ (b ∪ c)))) ≤ (b ∪ (a ∩ (b ∪ c)))
3		leo 158	. . . . 5 b ≤ (b ∪ c)
4		lear 161	. . . . 5 (a ∩ (b ∪ c)) ≤ (b ∪ c)
5	3, 4	lel2or 170	. . . 4 (b ∪ (a ∩ (b ∪ c))) ≤ (b ∪ c)
6	2, 5	letr 137	. . 3 (a ∩ (b ∪ (a ∩ (b ∪ c)))) ≤ (b ∪ c)
7	1, 6	ler2an 173	. 2 (a ∩ (b ∪ (a ∩ (b ∪ c)))) ≤ (a ∩ (b ∪ c))
8		lea 160	. . 3 (a ∩ (b ∪ c)) ≤ a
9		leor 159	. . 3 (a ∩ (b ∪ c)) ≤ (b ∪ (a ∩ (b ∪ c)))
10	8, 9	ler2an 173	. 2 (a ∩ (b ∪ c)) ≤ (a ∩ (b ∪ (a ∩ (b ∪ c))))
11	7, 10	lebi 145	1 (a ∩ (b ∪ (a ∩ (b ∪ c)))) = (a ∩ (b ∪ c))

Colors of variables: term

Syntax hints: = wb 1 ∪ 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: anorabs 225