Theorem nom24 317

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

Assertion

Ref	Expression
nom24	(a ≡4 (a ∩ b)) = (a →1 b)

Proof of Theorem nom24

Step	Hyp	Ref	Expression
1		leo 158	. . . . 5 a⊥ ≤ (a⊥ ∪ b⊥ )
2	1	leror 152	. . . 4 (a⊥ ∪ (a ∩ b)) ≤ ((a⊥ ∪ b⊥ ) ∪ (a ∩ b))
3		oran3 93	. . . . 5 (a⊥ ∪ b⊥ ) = (a ∩ b)⊥
4		anidm 111	. . . . . . . 8 (a ∩ a) = a
5	4	ran 78	. . . . . . 7 ((a ∩ a) ∩ b) = (a ∩ b)
6	5	ax-r1 35	. . . . . 6 (a ∩ b) = ((a ∩ a) ∩ b)
7		anass 76	. . . . . 6 ((a ∩ a) ∩ b) = (a ∩ (a ∩ b))
8	6, 7	ax-r2 36	. . . . 5 (a ∩ b) = (a ∩ (a ∩ b))
9	3, 8	2or 72	. . . 4 ((a⊥ ∪ b⊥ ) ∪ (a ∩ b)) = ((a ∩ b)⊥ ∪ (a ∩ (a ∩ b)))
10	2, 9	lbtr 139	. . 3 (a⊥ ∪ (a ∩ b)) ≤ ((a ∩ b)⊥ ∪ (a ∩ (a ∩ b)))
11	10	df2le2 136	. 2 ((a⊥ ∪ (a ∩ b)) ∩ ((a ∩ b)⊥ ∪ (a ∩ (a ∩ b)))) = (a⊥ ∪ (a ∩ b))
12		df-id4 53	. 2 (a ≡4 (a ∩ b)) = ((a⊥ ∪ (a ∩ b)) ∩ ((a ∩ b)⊥ ∪ (a ∩ (a ∩ b))))
13		df-i1 44	. 2 (a →1 b) = (a⊥ ∪ (a ∩ b))
14	11, 12, 13	3tr1 63	1 (a ≡4 (a ∩ b)) = (a →1 b)

This proof depends on syntax axioms:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₁ wi1 12   ≡₄ wid4 21

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-id4 53  df-le1 130  df-le2 131

This theorem is used by:  nom31  320  nom53  334