Theorem nom51 332

Description: Part of Lemma 3.3(15) from "Non-Orthomodular Models..." paper. (Contributed by NM, 7-Feb-1999.)

Assertion

Ref	Expression
nom51	((a ∪ b) ≡1 b) = (a →2 b)

Proof of Theorem nom51

Step	Hyp	Ref	Expression
1		ancom 74	. . . . . . . . 9 (b⊥ ∩ a⊥ ) = (a⊥ ∩ b⊥ )
2		anor3 90	. . . . . . . . 9 (a⊥ ∩ b⊥ ) = (a ∪ b)⊥
3	1, 2	ax-r2 36	. . . . . . . 8 (b⊥ ∩ a⊥ ) = (a ∪ b)⊥
4	3	ax-r1 35	. . . . . . 7 (a ∪ b)⊥ = (b⊥ ∩ a⊥ )
5	4	ax-r4 37	. . . . . 6 (a ∪ b)⊥ ⊥ = (b⊥ ∩ a⊥ )⊥
6	5	lor 70	. . . . 5 (b⊥ ∪ (a ∪ b)⊥ ⊥ ) = (b⊥ ∪ (b⊥ ∩ a⊥ )⊥ )
7	2	ax-r1 35	. . . . . . . . 9 (a ∪ b)⊥ = (a⊥ ∩ b⊥ )
8		ancom 74	. . . . . . . . 9 (a⊥ ∩ b⊥ ) = (b⊥ ∩ a⊥ )
9	7, 8	ax-r2 36	. . . . . . . 8 (a ∪ b)⊥ = (b⊥ ∩ a⊥ )
10	9	ax-r4 37	. . . . . . 7 (a ∪ b)⊥ ⊥ = (b⊥ ∩ a⊥ )⊥
11	10	lan 77	. . . . . 6 (b⊥ ⊥ ∩ (a ∪ b)⊥ ⊥ ) = (b⊥ ⊥ ∩ (b⊥ ∩ a⊥ )⊥ )
12	4, 11	2or 72	. . . . 5 ((a ∪ b)⊥ ∪ (b⊥ ⊥ ∩ (a ∪ b)⊥ ⊥ )) = ((b⊥ ∩ a⊥ ) ∪ (b⊥ ⊥ ∩ (b⊥ ∩ a⊥ )⊥ ))
13	6, 12	2an 79	. . . 4 ((b⊥ ∪ (a ∪ b)⊥ ⊥ ) ∩ ((a ∪ b)⊥ ∪ (b⊥ ⊥ ∩ (a ∪ b)⊥ ⊥ ))) = ((b⊥ ∪ (b⊥ ∩ a⊥ )⊥ ) ∩ ((b⊥ ∩ a⊥ ) ∪ (b⊥ ⊥ ∩ (b⊥ ∩ a⊥ )⊥ )))
14		df-id2 51	. . . 4 (b⊥ ≡2 (a ∪ b)⊥ ) = ((b⊥ ∪ (a ∪ b)⊥ ⊥ ) ∩ ((a ∪ b)⊥ ∪ (b⊥ ⊥ ∩ (a ∪ b)⊥ ⊥ )))
15		df-id2 51	. . . 4 (b⊥ ≡2 (b⊥ ∩ a⊥ )) = ((b⊥ ∪ (b⊥ ∩ a⊥ )⊥ ) ∩ ((b⊥ ∩ a⊥ ) ∪ (b⊥ ⊥ ∩ (b⊥ ∩ a⊥ )⊥ )))
16	13, 14, 15	3tr1 63	. . 3 (b⊥ ≡2 (a ∪ b)⊥ ) = (b⊥ ≡2 (b⊥ ∩ a⊥ ))
17		nom22 315	. . 3 (b⊥ ≡2 (b⊥ ∩ a⊥ )) = (b⊥ →1 a⊥ )
18	16, 17	ax-r2 36	. 2 (b⊥ ≡2 (a ∪ b)⊥ ) = (b⊥ →1 a⊥ )
19		nomcon1 302	. 2 ((a ∪ b) ≡1 b) = (b⊥ ≡2 (a ∪ b)⊥ )
20		i2i1 267	. 2 (a →2 b) = (b⊥ →1 a⊥ )
21	18, 19, 20	3tr1 63	1 ((a ∪ b) ≡1 b) = (a →2 b)

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₁ wi1 12   →₂ wi2 13   ≡₁ wid1 18   ≡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-i2 45  df-id1 50  df-id2 51  df-le1 130  df-le2 131

This theorem is referenced by:  nom64  341