Theorem nom23 316

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

Assertion

Ref	Expression
nom23	(a ≡3 (a ∩ b)) = (a →1 b)

Proof of Theorem nom23

Step	Hyp	Ref	Expression
1		le1 146	. . . 4 (a⊥ ∪ (a ∩ b)) ≤ 1
2		df-t 41	. . . . 5 1 = (a ∪ a⊥ )
3		anabs 121	. . . . . . . 8 (a⊥ ∩ (a⊥ ∪ b⊥ )) = a⊥
4	3	ax-r1 35	. . . . . . 7 a⊥ = (a⊥ ∩ (a⊥ ∪ b⊥ ))
5		oran3 93	. . . . . . . 8 (a⊥ ∪ b⊥ ) = (a ∩ b)⊥
6	5	lan 77	. . . . . . 7 (a⊥ ∩ (a⊥ ∪ b⊥ )) = (a⊥ ∩ (a ∩ b)⊥ )
7	4, 6	ax-r2 36	. . . . . 6 a⊥ = (a⊥ ∩ (a ∩ b)⊥ )
8	7	lor 70	. . . . 5 (a ∪ a⊥ ) = (a ∪ (a⊥ ∩ (a ∩ b)⊥ ))
9	2, 8	ax-r2 36	. . . 4 1 = (a ∪ (a⊥ ∩ (a ∩ b)⊥ ))
10	1, 9	lbtr 139	. . 3 (a⊥ ∪ (a ∩ b)) ≤ (a ∪ (a⊥ ∩ (a ∩ b)⊥ ))
11	10	df2le2 136	. 2 ((a⊥ ∪ (a ∩ b)) ∩ (a ∪ (a⊥ ∩ (a ∩ b)⊥ ))) = (a⊥ ∪ (a ∩ b))
12		df-id3 52	. 2 (a ≡3 (a ∩ b)) = ((a⊥ ∪ (a ∩ b)) ∩ (a ∪ (a⊥ ∩ (a ∩ b)⊥ )))
13		df-i1 44	. 2 (a →1 b) = (a⊥ ∪ (a ∩ b))
14	11, 12, 13	3tr1 63	1 (a ≡3 (a ∩ b)) = (a →1 b)

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8   →₁ wi1 12   ≡₃ wid3 20

This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a4 33  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38

This theorem depends on definitions:  df-a 40  df-t 41  df-i1 44  df-id3 52  df-le1 130  df-le2 131

This theorem is referenced by:  nom32  321  nom54  335