Theorem nom21 306

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

Assertion

Ref	Expression
nom21	(a ≡1 (a ∩ b)) = (a →1 b)

Proof of Theorem nom21

Step	Hyp	Ref	Expression
1		ancom 68	. . 3 ((a⊥ ∪ (a ∪ b⊥ )) ∩ (a⊥ ∪ (a ∩ b))) = ((a⊥ ∪ (a ∩ b)) ∩ (a⊥ ∪ (a ∪ b⊥ )))
2		or12 73	. . . . 5 (a⊥ ∪ (a ∪ b⊥ )) = (a ∪ (a⊥ ∪ b⊥ ))
3		oran3 85	. . . . . 6 (a⊥ ∪ b⊥ ) = (a ∩ b)⊥
4	3	lor 66	. . . . 5 (a ∪ (a⊥ ∪ b⊥ )) = (a ∪ (a ∩ b)⊥ )
5	2, 4	ax-r2 35	. . . 4 (a⊥ ∪ (a ∪ b⊥ )) = (a ∪ (a ∩ b)⊥ )
6		anidm 103	. . . . . . . 8 (a ∩ a) = a
7	6	ran 71	. . . . . . 7 ((a ∩ a) ∩ b) = (a ∩ b)
8	7	ax-r1 34	. . . . . 6 (a ∩ b) = ((a ∩ a) ∩ b)
9		anass 69	. . . . . 6 ((a ∩ a) ∩ b) = (a ∩ (a ∩ b))
10	8, 9	ax-r2 35	. . . . 5 (a ∩ b) = (a ∩ (a ∩ b))
11	10	lor 66	. . . 4 (a⊥ ∪ (a ∩ b)) = (a⊥ ∪ (a ∩ (a ∩ b)))
12	5, 11	2an 72	. . 3 ((a⊥ ∪ (a ∪ b⊥ )) ∩ (a⊥ ∪ (a ∩ b))) = ((a ∪ (a ∩ b)⊥ ) ∩ (a⊥ ∪ (a ∩ (a ∩ b))))
13		lea 152	. . . . . 6 (a ∩ b) ≤ a
14		leo 150	. . . . . 6 a ≤ (a ∪ b⊥ )
15	13, 14	letr 129	. . . . 5 (a ∩ b) ≤ (a ∪ b⊥ )
16	15	lelor 158	. . . 4 (a⊥ ∪ (a ∩ b)) ≤ (a⊥ ∪ (a ∪ b⊥ ))
17	16	df2le2 128	. . 3 ((a⊥ ∪ (a ∩ b)) ∩ (a⊥ ∪ (a ∪ b⊥ ))) = (a⊥ ∪ (a ∩ b))
18	1, 12, 17	3tr2 61	. 2 ((a ∪ (a ∩ b)⊥ ) ∩ (a⊥ ∪ (a ∩ (a ∩ b)))) = (a⊥ ∪ (a ∩ b))
19		df-id1 49	. 2 (a ≡1 (a ∩ b)) = ((a ∪ (a ∩ b)⊥ ) ∩ (a⊥ ∪ (a ∩ (a ∩ b))))
20		df-i1 43	. 2 (a →1 b) = (a⊥ ∪ (a ∩ b))
21	18, 19, 20	3tr1 60	1 (a ≡1 (a ∩ b)) = (a →1 b)

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₁ wi1 13   ≡₁ wid1 19

This theorem is referenced by:  nom34 315  nom52 325

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-id1 49  df-le1 122  df-le2 123

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