Theorem nom13 310

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

Assertion

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

Proof of Theorem nom13

Step	Hyp	Ref	Expression
1		oran 87	. . . . . . . . 9 (a ∪ (a ∩ b)) = (a⊥ ∩ (a ∩ b)⊥ )⊥
2	1	ax-r1 35	. . . . . . . 8 (a⊥ ∩ (a ∩ b)⊥ )⊥ = (a ∪ (a ∩ b))
3		orabs 120	. . . . . . . 8 (a ∪ (a ∩ b)) = a
4	2, 3	ax-r2 36	. . . . . . 7 (a⊥ ∩ (a ∩ b)⊥ )⊥ = a
5	4	con3 68	. . . . . 6 (a⊥ ∩ (a ∩ b)⊥ ) = a⊥
6	5	lor 70	. . . . 5 ((a⊥ ∩ (a ∩ b)) ∪ (a⊥ ∩ (a ∩ b)⊥ )) = ((a⊥ ∩ (a ∩ b)) ∪ a⊥ )
7		lea 160	. . . . . 6 (a⊥ ∩ (a ∩ b)) ≤ a⊥
8	7	df-le2 131	. . . . 5 ((a⊥ ∩ (a ∩ b)) ∪ a⊥ ) = a⊥
9	6, 8	ax-r2 36	. . . 4 ((a⊥ ∩ (a ∩ b)) ∪ (a⊥ ∩ (a ∩ b)⊥ )) = a⊥
10	9	ax-r5 38	. . 3 (((a⊥ ∩ (a ∩ b)) ∪ (a⊥ ∩ (a ∩ b)⊥ )) ∪ (a ∩ (a⊥ ∪ (a ∩ b)))) = (a⊥ ∪ (a ∩ (a⊥ ∪ (a ∩ b))))
11		womaa 222	. . 3 (a⊥ ∪ (a ∩ (a⊥ ∪ (a ∩ b)))) = (a⊥ ∪ (a ∩ b))
12	10, 11	ax-r2 36	. 2 (((a⊥ ∩ (a ∩ b)) ∪ (a⊥ ∩ (a ∩ b)⊥ )) ∪ (a ∩ (a⊥ ∪ (a ∩ b)))) = (a⊥ ∪ (a ∩ b))
13		df-i3 46	. 2 (a →3 (a ∩ b)) = (((a⊥ ∩ (a ∩ b)) ∪ (a⊥ ∩ (a ∩ b)⊥ )) ∪ (a ∩ (a⊥ ∪ (a ∩ b))))
14		df-i1 44	. 2 (a →1 b) = (a⊥ ∪ (a ∩ b))
15	12, 13, 14	3tr1 63	1 (a →3 (a ∩ b)) = (a →1 b)

This proof depends on syntax axioms:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₁ wi1 12   →₃ wi3 14

This proof depends on axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38

This proof depends on definitions:  df-a 40  df-t 41  df-f 42  df-i1 44  df-i3 46  df-le1 130  df-le2 131

This theorem is used by:  nom44  329  lem3.3.7i3e3  1068