Theorem nom20 305

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

Assertion

Ref	Expression
nom20	(a ≡0 (a ∩ b)) = (a →1 b)

Proof of Theorem nom20

Step	Hyp	Ref	Expression
1		lea 152	. . . . . 6 (a ∩ b) ≤ a
2		leor 151	. . . . . 6 a ≤ (b⊥ ∪ a)
3	1, 2	letr 129	. . . . 5 (a ∩ b) ≤ (b⊥ ∪ a)
4	3	lelor 158	. . . 4 (a⊥ ∪ (a ∩ b)) ≤ (a⊥ ∪ (b⊥ ∪ a))
5		ax-a3 31	. . . . . 6 ((a⊥ ∪ b⊥ ) ∪ a) = (a⊥ ∪ (b⊥ ∪ a))
6	5	ax-r1 34	. . . . 5 (a⊥ ∪ (b⊥ ∪ a)) = ((a⊥ ∪ b⊥ ) ∪ a)
7		oran3 85	. . . . . 6 (a⊥ ∪ b⊥ ) = (a ∩ b)⊥
8	7	ax-r5 37	. . . . 5 ((a⊥ ∪ b⊥ ) ∪ a) = ((a ∩ b)⊥ ∪ a)
9	6, 8	ax-r2 35	. . . 4 (a⊥ ∪ (b⊥ ∪ a)) = ((a ∩ b)⊥ ∪ a)
10	4, 9	lbtr 131	. . 3 (a⊥ ∪ (a ∩ b)) ≤ ((a ∩ b)⊥ ∪ a)
11	10	df2le2 128	. 2 ((a⊥ ∪ (a ∩ b)) ∩ ((a ∩ b)⊥ ∪ a)) = (a⊥ ∪ (a ∩ b))
12		df-id0 48	. 2 (a ≡0 (a ∩ b)) = ((a⊥ ∪ (a ∩ b)) ∩ ((a ∩ b)⊥ ∪ a))
13		df-i1 43	. 2 (a →1 b) = (a⊥ ∪ (a ∩ b))
14	11, 12, 13	3tr1 60	1 (a ≡0 (a ∩ b)) = (a →1 b)

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₁ wi1 13   ≡₀ wid0 18

This theorem is referenced by:  nom30 311  nom50 323

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-id0 48  df-le1 122  df-le2 123

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