Theorem nomcon1 294

Description: Lemma for "Non-Orthomodular Models..." paper.

Assertion

Ref	Expression
nomcon1	(a ≡1 b) = (b⊥ ≡2 a⊥ )

Proof of Theorem nomcon1

Step	Hyp	Ref	Expression
1		ax-a2 30	. . . 4 (a ∪ b⊥ ) = (b⊥ ∪ a)
2		ax-a1 29	. . . . 5 a = a⊥ ⊥
3	2	lor 66	. . . 4 (b⊥ ∪ a) = (b⊥ ∪ a⊥ ⊥ )
4	1, 3	ax-r2 35	. . 3 (a ∪ b⊥ ) = (b⊥ ∪ a⊥ ⊥ )
5		ancom 68	. . . . 5 (a ∩ b) = (b ∩ a)
6		ax-a1 29	. . . . . 6 b = b⊥ ⊥
7	6, 2	2an 72	. . . . 5 (b ∩ a) = (b⊥ ⊥ ∩ a⊥ ⊥ )
8	5, 7	ax-r2 35	. . . 4 (a ∩ b) = (b⊥ ⊥ ∩ a⊥ ⊥ )
9	8	lor 66	. . 3 (a⊥ ∪ (a ∩ b)) = (a⊥ ∪ (b⊥ ⊥ ∩ a⊥ ⊥ ))
10	4, 9	2an 72	. 2 ((a ∪ b⊥ ) ∩ (a⊥ ∪ (a ∩ b))) = ((b⊥ ∪ a⊥ ⊥ ) ∩ (a⊥ ∪ (b⊥ ⊥ ∩ a⊥ ⊥ )))
11		df-id1 49	. 2 (a ≡1 b) = ((a ∪ b⊥ ) ∩ (a⊥ ∪ (a ∩ b)))
12		df-id2 50	. 2 (b⊥ ≡2 a⊥ ) = ((b⊥ ∪ a⊥ ⊥ ) ∩ (a⊥ ∪ (b⊥ ⊥ ∩ a⊥ ⊥ )))
13	10, 11, 12	3tr1 60	1 (a ≡1 b) = (b⊥ ≡2 a⊥ )

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   ≡₁ wid1 19   ≡₂ wid2 20

This theorem is referenced by:  nomcon4 297  nom51 324

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37

This theorem depends on definitions:  df-a 39  df-id1 49  df-id2 50

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