Theorem nom20 313

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

Assertion

Ref	Expression
nom20	(a ≡0 (a ∩ b)) = (a →1 b)

Proof of Theorem nom20

Step	Hyp	Ref	Expression
1		lea 160	. . . . . 6 (a ∩ b) ≤ a
2		leor 159	. . . . . 6 a ≤ (b⊥ ∪ a)
3	1, 2	letr 137	. . . . 5 (a ∩ b) ≤ (b⊥ ∪ a)
4	3	lelor 166	. . . 4 (a⊥ ∪ (a ∩ b)) ≤ (a⊥ ∪ (b⊥ ∪ a))
5		ax-a3 32	. . . . . 6 ((a⊥ ∪ b⊥ ) ∪ a) = (a⊥ ∪ (b⊥ ∪ a))
6	5	ax-r1 35	. . . . 5 (a⊥ ∪ (b⊥ ∪ a)) = ((a⊥ ∪ b⊥ ) ∪ a)
7		oran3 93	. . . . . 6 (a⊥ ∪ b⊥ ) = (a ∩ b)⊥
8	7	ax-r5 38	. . . . 5 ((a⊥ ∪ b⊥ ) ∪ a) = ((a ∩ b)⊥ ∪ a)
9	6, 8	ax-r2 36	. . . 4 (a⊥ ∪ (b⊥ ∪ a)) = ((a ∩ b)⊥ ∪ a)
10	4, 9	lbtr 139	. . 3 (a⊥ ∪ (a ∩ b)) ≤ ((a ∩ b)⊥ ∪ a)
11	10	df2le2 136	. 2 ((a⊥ ∪ (a ∩ b)) ∩ ((a ∩ b)⊥ ∪ a)) = (a⊥ ∪ (a ∩ b))
12		df-id0 49	. 2 (a ≡0 (a ∩ b)) = ((a⊥ ∪ (a ∩ b)) ∩ ((a ∩ b)⊥ ∪ a))
13		df-i1 44	. 2 (a →1 b) = (a⊥ ∪ (a ∩ b))
14	11, 12, 13	3tr1 63	1 (a ≡0 (a ∩ b)) = (a →1 b)

This proof depends on syntax axioms:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₁ wi1 12   ≡₀ wid0 17

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-id0 49  df-le1 130  df-le2 131

This theorem is used by:  nom30  319  nom50  331