Theorem nom53 334

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

Assertion

Ref	Expression
nom53	((a ∪ b) ≡3 b) = (a →2 b)

Proof of Theorem nom53

Step	Hyp	Ref	Expression
1		ancom 74	. . . . . . . 8 (b⊥ ∩ a⊥ ) = (a⊥ ∩ b⊥ )
2		anor3 90	. . . . . . . 8 (a⊥ ∩ b⊥ ) = (a ∪ b)⊥
3	1, 2	ax-r2 36	. . . . . . 7 (b⊥ ∩ a⊥ ) = (a ∪ b)⊥
4	3	ax-r1 35	. . . . . 6 (a ∪ b)⊥ = (b⊥ ∩ a⊥ )
5	4	lor 70	. . . . 5 (b⊥ ⊥ ∪ (a ∪ b)⊥ ) = (b⊥ ⊥ ∪ (b⊥ ∩ a⊥ ))
6	4	ax-r4 37	. . . . . 6 (a ∪ b)⊥ ⊥ = (b⊥ ∩ a⊥ )⊥
7	4	lan 77	. . . . . 6 (b⊥ ∩ (a ∪ b)⊥ ) = (b⊥ ∩ (b⊥ ∩ a⊥ ))
8	6, 7	2or 72	. . . . 5 ((a ∪ b)⊥ ⊥ ∪ (b⊥ ∩ (a ∪ b)⊥ )) = ((b⊥ ∩ a⊥ )⊥ ∪ (b⊥ ∩ (b⊥ ∩ a⊥ )))
9	5, 8	2an 79	. . . 4 ((b⊥ ⊥ ∪ (a ∪ b)⊥ ) ∩ ((a ∪ b)⊥ ⊥ ∪ (b⊥ ∩ (a ∪ b)⊥ ))) = ((b⊥ ⊥ ∪ (b⊥ ∩ a⊥ )) ∩ ((b⊥ ∩ a⊥ )⊥ ∪ (b⊥ ∩ (b⊥ ∩ a⊥ ))))
10		df-id4 53	. . . 4 (b⊥ ≡4 (a ∪ b)⊥ ) = ((b⊥ ⊥ ∪ (a ∪ b)⊥ ) ∩ ((a ∪ b)⊥ ⊥ ∪ (b⊥ ∩ (a ∪ b)⊥ )))
11		df-id4 53	. . . 4 (b⊥ ≡4 (b⊥ ∩ a⊥ )) = ((b⊥ ⊥ ∪ (b⊥ ∩ a⊥ )) ∩ ((b⊥ ∩ a⊥ )⊥ ∪ (b⊥ ∩ (b⊥ ∩ a⊥ ))))
12	9, 10, 11	3tr1 63	. . 3 (b⊥ ≡4 (a ∪ b)⊥ ) = (b⊥ ≡4 (b⊥ ∩ a⊥ ))
13		nom24 317	. . 3 (b⊥ ≡4 (b⊥ ∩ a⊥ )) = (b⊥ →1 a⊥ )
14	12, 13	ax-r2 36	. 2 (b⊥ ≡4 (a ∪ b)⊥ ) = (b⊥ →1 a⊥ )
15		nomcon3 304	. 2 ((a ∪ b) ≡3 b) = (b⊥ ≡4 (a ∪ b)⊥ )
16		i2i1 267	. 2 (a →2 b) = (b⊥ →1 a⊥ )
17	14, 15, 16	3tr1 63	1 ((a ∪ b) ≡3 b) = (a →2 b)

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₁ wi1 12   →₂ wi2 13   ≡₃ wid3 20   ≡₄ wid4 21

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-id3 52  df-id4 53  df-le1 130  df-le2 131

This theorem is referenced by:  nom62  339