Theorem nom13 302

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

Assertion

Ref	Expression
nom13	(a →3 (a ∩ b)) = (a →1 b)

Proof of Theorem nom13

Step	Hyp	Ref	Expression
1		oran 79	. . . . . . . . 9 (a ∪ (a ∩ b)) = (a⊥ ∩ (a ∩ b)⊥ )⊥
2	1	ax-r1 34	. . . . . . . 8 (a⊥ ∩ (a ∩ b)⊥ )⊥ = (a ∪ (a ∩ b))
3		a5b 112	. . . . . . . 8 (a ∪ (a ∩ b)) = a
4	2, 3	ax-r2 35	. . . . . . 7 (a⊥ ∩ (a ∩ b)⊥ )⊥ = a
5	4	con3 65	. . . . . 6 (a⊥ ∩ (a ∩ b)⊥ ) = a⊥
6	5	lor 66	. . . . 5 ((a⊥ ∩ (a ∩ b)) ∪ (a⊥ ∩ (a ∩ b)⊥ )) = ((a⊥ ∩ (a ∩ b)) ∪ a⊥ )
7		lea 152	. . . . . 6 (a⊥ ∩ (a ∩ b)) ≤ a⊥
8	7	df-le2 123	. . . . 5 ((a⊥ ∩ (a ∩ b)) ∪ a⊥ ) = a⊥
9	6, 8	ax-r2 35	. . . 4 ((a⊥ ∩ (a ∩ b)) ∪ (a⊥ ∩ (a ∩ b)⊥ )) = a⊥
10	9	ax-r5 37	. . 3 (((a⊥ ∩ (a ∩ b)) ∪ (a⊥ ∩ (a ∩ b)⊥ )) ∪ (a ∩ (a⊥ ∪ (a ∩ b)))) = (a⊥ ∪ (a ∩ (a⊥ ∪ (a ∩ b))))
11		womaa 214	. . 3 (a⊥ ∪ (a ∩ (a⊥ ∪ (a ∩ b)))) = (a⊥ ∪ (a ∩ b))
12	10, 11	ax-r2 35	. 2 (((a⊥ ∩ (a ∩ b)) ∪ (a⊥ ∩ (a ∩ b)⊥ )) ∪ (a ∩ (a⊥ ∪ (a ∩ b)))) = (a⊥ ∪ (a ∩ b))
13		df-i3 45	. 2 (a →3 (a ∩ b)) = (((a⊥ ∩ (a ∩ b)) ∪ (a⊥ ∩ (a ∩ b)⊥ )) ∪ (a ∩ (a⊥ ∪ (a ∩ b))))
14		df-i1 43	. 2 (a →1 b) = (a⊥ ∪ (a ∩ b))
15	12, 13, 14	3tr1 60	1 (a →3 (a ∩ b)) = (a →1 b)

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₁ wi1 13   →₃ wi3 15

This theorem is referenced by:  nom44 321

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-i3 45  df-le1 122  df-le2 123

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