Theorem nom22 315

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

Assertion

Ref	Expression
nom22	(a ≡2 (a ∩ b)) = (a →1 b)

Proof of Theorem nom22

Step	Hyp	Ref	Expression
1		oran3 93	. . . . . . 7 (a⊥ ∪ b⊥ ) = (a ∩ b)⊥
2	1	lor 70	. . . . . 6 (a ∪ (a⊥ ∪ b⊥ )) = (a ∪ (a ∩ b)⊥ )
3	2	ax-r1 35	. . . . 5 (a ∪ (a ∩ b)⊥ ) = (a ∪ (a⊥ ∪ b⊥ ))
4		or12 80	. . . . 5 (a ∪ (a⊥ ∪ b⊥ )) = (a⊥ ∪ (a ∪ b⊥ ))
5	3, 4	ax-r2 36	. . . 4 (a ∪ (a ∩ b)⊥ ) = (a⊥ ∪ (a ∪ b⊥ ))
6		ax-a2 31	. . . . 5 ((a ∩ b) ∪ (a⊥ ∩ (a ∩ b)⊥ )) = ((a⊥ ∩ (a ∩ b)⊥ ) ∪ (a ∩ b))
7	1	lan 77	. . . . . . . 8 (a⊥ ∩ (a⊥ ∪ b⊥ )) = (a⊥ ∩ (a ∩ b)⊥ )
8	7	ax-r1 35	. . . . . . 7 (a⊥ ∩ (a ∩ b)⊥ ) = (a⊥ ∩ (a⊥ ∪ b⊥ ))
9		anabs 121	. . . . . . 7 (a⊥ ∩ (a⊥ ∪ b⊥ )) = a⊥
10	8, 9	ax-r2 36	. . . . . 6 (a⊥ ∩ (a ∩ b)⊥ ) = a⊥
11	10	ax-r5 38	. . . . 5 ((a⊥ ∩ (a ∩ b)⊥ ) ∪ (a ∩ b)) = (a⊥ ∪ (a ∩ b))
12	6, 11	ax-r2 36	. . . 4 ((a ∩ b) ∪ (a⊥ ∩ (a ∩ b)⊥ )) = (a⊥ ∪ (a ∩ b))
13	5, 12	2an 79	. . 3 ((a ∪ (a ∩ b)⊥ ) ∩ ((a ∩ b) ∪ (a⊥ ∩ (a ∩ b)⊥ ))) = ((a⊥ ∪ (a ∪ b⊥ )) ∩ (a⊥ ∪ (a ∩ b)))
14		ancom 74	. . 3 ((a⊥ ∪ (a ∪ b⊥ )) ∩ (a⊥ ∪ (a ∩ b))) = ((a⊥ ∪ (a ∩ b)) ∩ (a⊥ ∪ (a ∪ b⊥ )))
15		lea 160	. . . . . 6 (a ∩ b) ≤ a
16		leo 158	. . . . . 6 a ≤ (a ∪ b⊥ )
17	15, 16	letr 137	. . . . 5 (a ∩ b) ≤ (a ∪ b⊥ )
18	17	lelor 166	. . . 4 (a⊥ ∪ (a ∩ b)) ≤ (a⊥ ∪ (a ∪ b⊥ ))
19	18	df2le2 136	. . 3 ((a⊥ ∪ (a ∩ b)) ∩ (a⊥ ∪ (a ∪ b⊥ ))) = (a⊥ ∪ (a ∩ b))
20	13, 14, 19	3tr 65	. 2 ((a ∪ (a ∩ b)⊥ ) ∩ ((a ∩ b) ∪ (a⊥ ∩ (a ∩ b)⊥ ))) = (a⊥ ∪ (a ∩ b))
21		df-id2 51	. 2 (a ≡2 (a ∩ b)) = ((a ∪ (a ∩ b)⊥ ) ∩ ((a ∩ b) ∪ (a⊥ ∩ (a ∩ b)⊥ )))
22		df-i1 44	. 2 (a →1 b) = (a⊥ ∪ (a ∩ b))
23	20, 21, 22	3tr1 63	1 (a ≡2 (a ∩ b)) = (a →1 b)

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₁ wi1 12   ≡₂ wid2 19

This theorem was proved from 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 theorem depends on definitions:  df-a 40  df-t 41  df-f 42  df-i1 44  df-id2 51  df-le1 130  df-le2 131

This theorem is referenced by:  nom33  322  nom51  332