Theorem nom12 301

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

Assertion

Ref	Expression
nom12	(a →2 (a ∩ b)) = (a →1 b)

Proof of Theorem nom12

Step	Hyp	Ref	Expression
1		oran 79	. . . . . . 7 (a ∪ (a ∩ b)) = (a⊥ ∩ (a ∩ b)⊥ )⊥
2	1	ax-r1 34	. . . . . 6 (a⊥ ∩ (a ∩ b)⊥ )⊥ = (a ∪ (a ∩ b))
3		a5b 112	. . . . . 6 (a ∪ (a ∩ b)) = a
4	2, 3	ax-r2 35	. . . . 5 (a⊥ ∩ (a ∩ b)⊥ )⊥ = a
5	4	con3 65	. . . 4 (a⊥ ∩ (a ∩ b)⊥ ) = a⊥
6	5	lor 66	. . 3 ((a ∩ b) ∪ (a⊥ ∩ (a ∩ b)⊥ )) = ((a ∩ b) ∪ a⊥ )
7		ax-a2 30	. . 3 ((a ∩ b) ∪ a⊥ ) = (a⊥ ∪ (a ∩ b))
8	6, 7	ax-r2 35	. 2 ((a ∩ b) ∪ (a⊥ ∩ (a ∩ b)⊥ )) = (a⊥ ∪ (a ∩ b))
9		df-i2 44	. 2 (a →2 (a ∩ b)) = ((a ∩ b) ∪ (a⊥ ∩ (a ∩ b)⊥ ))
10		df-i1 43	. 2 (a →1 b) = (a⊥ ∪ (a ∩ b))
11	8, 9, 10	3tr1 60	1 (a →2 (a ∩ b)) = (a →1 b)

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₁ wi1 13   →₂ wi2 14

This theorem is referenced by:  nom41 318

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a5 33  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37

This theorem depends on definitions:  df-a 39  df-i1 43  df-i2 44

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