Theorem nom24 309

Description: Part of Lemma 3.3(14) from "Non-Orthomodular Models..." paper.

Assertion

Ref	Expression
nom24	(a ≡4 (a ∩ b)) = (a →1 b)

Proof of Theorem nom24

Step	Hyp	Ref	Expression
1		leo 150	. . . . 5 a⊥ ≤ (a⊥ ∪ b⊥ )
2	1	leror 144	. . . 4 (a⊥ ∪ (a ∩ b)) ≤ ((a⊥ ∪ b⊥ ) ∪ (a ∩ b))
3		oran3 85	. . . . 5 (a⊥ ∪ b⊥ ) = (a ∩ b)⊥
4		anidm 103	. . . . . . . 8 (a ∩ a) = a
5	4	ran 71	. . . . . . 7 ((a ∩ a) ∩ b) = (a ∩ b)
6	5	ax-r1 34	. . . . . 6 (a ∩ b) = ((a ∩ a) ∩ b)
7		anass 69	. . . . . 6 ((a ∩ a) ∩ b) = (a ∩ (a ∩ b))
8	6, 7	ax-r2 35	. . . . 5 (a ∩ b) = (a ∩ (a ∩ b))
9	3, 8	2or 67	. . . 4 ((a⊥ ∪ b⊥ ) ∪ (a ∩ b)) = ((a ∩ b)⊥ ∪ (a ∩ (a ∩ b)))
10	2, 9	lbtr 131	. . 3 (a⊥ ∪ (a ∩ b)) ≤ ((a ∩ b)⊥ ∪ (a ∩ (a ∩ b)))
11	10	df2le2 128	. 2 ((a⊥ ∪ (a ∩ b)) ∩ ((a ∩ b)⊥ ∪ (a ∩ (a ∩ b)))) = (a⊥ ∪ (a ∩ b))
12		df-id4 52	. 2 (a ≡4 (a ∩ b)) = ((a⊥ ∪ (a ∩ b)) ∩ ((a ∩ b)⊥ ∪ (a ∩ (a ∩ b))))
13		df-i1 43	. 2 (a →1 b) = (a⊥ ∪ (a ∩ b))
14	11, 12, 13	3tr1 60	1 (a ≡4 (a ∩ b)) = (a →1 b)

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₁ wi1 13   ≡₄ wid4 22

This theorem is referenced by:  nom31 312  nom53 326

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-i1 43  df-id4 52  df-le1 122  df-le2 123

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