Theorem nom12 309

Description: Part of Lemma 3.3(14) from "Non-Orthomodular Models..." paper. (Contributed by NM, 7-Feb-1999.)

Assertion

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

Proof of Theorem nom12

Step	Hyp	Ref	Expression
1		oran 87	. . . . . . 7 (a ∪ (a ∩ b)) = (a⊥ ∩ (a ∩ b)⊥ )⊥
2	1	ax-r1 35	. . . . . 6 (a⊥ ∩ (a ∩ b)⊥ )⊥ = (a ∪ (a ∩ b))
3		orabs 120	. . . . . 6 (a ∪ (a ∩ b)) = a
4	2, 3	ax-r2 36	. . . . 5 (a⊥ ∩ (a ∩ b)⊥ )⊥ = a
5	4	con3 68	. . . 4 (a⊥ ∩ (a ∩ b)⊥ ) = a⊥
6	5	lor 70	. . 3 ((a ∩ b) ∪ (a⊥ ∩ (a ∩ b)⊥ )) = ((a ∩ b) ∪ a⊥ )
7		ax-a2 31	. . 3 ((a ∩ b) ∪ a⊥ ) = (a⊥ ∪ (a ∩ b))
8	6, 7	ax-r2 36	. 2 ((a ∩ b) ∪ (a⊥ ∩ (a ∩ b)⊥ )) = (a⊥ ∪ (a ∩ b))
9		df-i2 45	. 2 (a →2 (a ∩ b)) = ((a ∩ b) ∪ (a⊥ ∩ (a ∩ b)⊥ ))
10		df-i1 44	. 2 (a →1 b) = (a⊥ ∪ (a ∩ b))
11	8, 9, 10	3tr1 63	1 (a →2 (a ∩ b)) = (a →1 b)

This proof depends on syntax axioms:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₁ wi1 12   →₂ wi2 13

This proof depends on axioms:  ax-a1 30  ax-a2 31  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38

This proof depends on definitions:  df-a 40  df-i1 44  df-i2 45

This theorem is used by:  nom41  326  lem3.3.7i2e3  1065