Theorem nom23 308

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

Assertion

Ref	Expression
nom23	(a ≡3 (a ∩ b)) = (a →1 b)

Proof of Theorem nom23

Step	Hyp	Ref	Expression
1		le1 138	. . . 4 (a⊥ ∪ (a ∩ b)) ≤ 1
2		df-t 40	. . . . 5 1 = (a ∪ a⊥ )
3		a5c 113	. . . . . . . 8 (a⊥ ∩ (a⊥ ∪ b⊥ )) = a⊥
4	3	ax-r1 34	. . . . . . 7 a⊥ = (a⊥ ∩ (a⊥ ∪ b⊥ ))
5		oran3 85	. . . . . . . 8 (a⊥ ∪ b⊥ ) = (a ∩ b)⊥
6	5	lan 70	. . . . . . 7 (a⊥ ∩ (a⊥ ∪ b⊥ )) = (a⊥ ∩ (a ∩ b)⊥ )
7	4, 6	ax-r2 35	. . . . . 6 a⊥ = (a⊥ ∩ (a ∩ b)⊥ )
8	7	lor 66	. . . . 5 (a ∪ a⊥ ) = (a ∪ (a⊥ ∩ (a ∩ b)⊥ ))
9	2, 8	ax-r2 35	. . . 4 1 = (a ∪ (a⊥ ∩ (a ∩ b)⊥ ))
10	1, 9	lbtr 131	. . 3 (a⊥ ∪ (a ∩ b)) ≤ (a ∪ (a⊥ ∩ (a ∩ b)⊥ ))
11	10	df2le2 128	. 2 ((a⊥ ∪ (a ∩ b)) ∩ (a ∪ (a⊥ ∩ (a ∩ b)⊥ ))) = (a⊥ ∪ (a ∩ b))
12		df-id3 51	. 2 (a ≡3 (a ∩ b)) = ((a⊥ ∪ (a ∩ b)) ∩ (a ∪ (a⊥ ∩ (a ∩ b)⊥ )))
13		df-i1 43	. 2 (a →1 b) = (a⊥ ∪ (a ∩ b))
14	11, 12, 13	3tr1 60	1 (a ≡3 (a ∩ b)) = (a →1 b)

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 9   →₁ wi1 13   ≡₃ wid3 21

This theorem is referenced by:  nom32 313  nom54 327

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a4 32  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-i1 43  df-id3 51  df-le1 122  df-le2 123

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