Theorem nom14 311

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

Assertion

Ref	Expression
nom14	(a →4 (a ∩ b)) = (a →1 b)

Proof of Theorem nom14

Step	Hyp	Ref	Expression
1		ax-a2 31	. . . . 5 ((a ∩ (a ∩ b)) ∪ (a⊥ ∩ (a ∩ b))) = ((a⊥ ∩ (a ∩ b)) ∪ (a ∩ (a ∩ b)))
2		anass 76	. . . . . . . 8 ((a ∩ a) ∩ b) = (a ∩ (a ∩ b))
3	2	ax-r1 35	. . . . . . 7 (a ∩ (a ∩ b)) = ((a ∩ a) ∩ b)
4		anidm 111	. . . . . . . 8 (a ∩ a) = a
5	4	ran 78	. . . . . . 7 ((a ∩ a) ∩ b) = (a ∩ b)
6	3, 5	ax-r2 36	. . . . . 6 (a ∩ (a ∩ b)) = (a ∩ b)
7	6	lor 70	. . . . 5 ((a⊥ ∩ (a ∩ b)) ∪ (a ∩ (a ∩ b))) = ((a⊥ ∩ (a ∩ b)) ∪ (a ∩ b))
8		lear 161	. . . . . 6 (a⊥ ∩ (a ∩ b)) ≤ (a ∩ b)
9	8	df-le2 131	. . . . 5 ((a⊥ ∩ (a ∩ b)) ∪ (a ∩ b)) = (a ∩ b)
10	1, 7, 9	3tr 65	. . . 4 ((a ∩ (a ∩ b)) ∪ (a⊥ ∩ (a ∩ b))) = (a ∩ b)
11	10	ax-r5 38	. . 3 (((a ∩ (a ∩ b)) ∪ (a⊥ ∩ (a ∩ b))) ∪ ((a⊥ ∪ (a ∩ b)) ∩ (a ∩ b)⊥ )) = ((a ∩ b) ∪ ((a⊥ ∪ (a ∩ b)) ∩ (a ∩ b)⊥ ))
12		leo 158	. . . . 5 (a ∩ b) ≤ ((a ∩ b) ∪ a⊥ )
13		lea 160	. . . . . 6 ((a⊥ ∪ (a ∩ b)) ∩ (a ∩ b)⊥ ) ≤ (a⊥ ∪ (a ∩ b))
14		ax-a2 31	. . . . . 6 (a⊥ ∪ (a ∩ b)) = ((a ∩ b) ∪ a⊥ )
15	13, 14	lbtr 139	. . . . 5 ((a⊥ ∪ (a ∩ b)) ∩ (a ∩ b)⊥ ) ≤ ((a ∩ b) ∪ a⊥ )
16	12, 15	lel2or 170	. . . 4 ((a ∩ b) ∪ ((a⊥ ∪ (a ∩ b)) ∩ (a ∩ b)⊥ )) ≤ ((a ∩ b) ∪ a⊥ )
17		leo 158	. . . . . 6 a⊥ ≤ (a⊥ ∪ (a ∩ b))
18		lea 160	. . . . . . 7 (a ∩ b) ≤ a
19	18	lecon 154	. . . . . 6 a⊥ ≤ (a ∩ b)⊥
20	17, 19	ler2an 173	. . . . 5 a⊥ ≤ ((a⊥ ∪ (a ∩ b)) ∩ (a ∩ b)⊥ )
21	20	lelor 166	. . . 4 ((a ∩ b) ∪ a⊥ ) ≤ ((a ∩ b) ∪ ((a⊥ ∪ (a ∩ b)) ∩ (a ∩ b)⊥ ))
22	16, 21	lebi 145	. . 3 ((a ∩ b) ∪ ((a⊥ ∪ (a ∩ b)) ∩ (a ∩ b)⊥ )) = ((a ∩ b) ∪ a⊥ )
23		ax-a2 31	. . 3 ((a ∩ b) ∪ a⊥ ) = (a⊥ ∪ (a ∩ b))
24	11, 22, 23	3tr 65	. 2 (((a ∩ (a ∩ b)) ∪ (a⊥ ∩ (a ∩ b))) ∪ ((a⊥ ∪ (a ∩ b)) ∩ (a ∩ b)⊥ )) = (a⊥ ∪ (a ∩ b))
25		df-i4 47	. 2 (a →4 (a ∩ b)) = (((a ∩ (a ∩ b)) ∪ (a⊥ ∩ (a ∩ b))) ∪ ((a⊥ ∪ (a ∩ b)) ∩ (a ∩ b)⊥ ))
26		df-i1 44	. 2 (a →1 b) = (a⊥ ∪ (a ∩ b))
27	24, 25, 26	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 wi4 15

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-i4 47  df-le1 130  df-le2 131

This theorem is used by:  nom43  328  lem3.3.7i4e3  1071