Theorem nom54 335

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

Assertion

Ref	Expression
nom54	((a ∪ b) ≡4 b) = (a →2 b)

Proof of Theorem nom54

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	lor 70	. . . . . 6 (b⊥ ⊥ ∪ (b⊥ ∩ a⊥ )) = (b⊥ ⊥ ∪ (a ∪ b)⊥ )
5	3	ax-r4 37	. . . . . . . 8 (b⊥ ∩ a⊥ )⊥ = (a ∪ b)⊥ ⊥
6	5	lan 77	. . . . . . 7 (b⊥ ⊥ ∩ (b⊥ ∩ a⊥ )⊥ ) = (b⊥ ⊥ ∩ (a ∪ b)⊥ ⊥ )
7	6	lor 70	. . . . . 6 (b⊥ ∪ (b⊥ ⊥ ∩ (b⊥ ∩ a⊥ )⊥ )) = (b⊥ ∪ (b⊥ ⊥ ∩ (a ∪ b)⊥ ⊥ ))
8	4, 7	2an 79	. . . . 5 ((b⊥ ⊥ ∪ (b⊥ ∩ a⊥ )) ∩ (b⊥ ∪ (b⊥ ⊥ ∩ (b⊥ ∩ a⊥ )⊥ ))) = ((b⊥ ⊥ ∪ (a ∪ b)⊥ ) ∩ (b⊥ ∪ (b⊥ ⊥ ∩ (a ∪ b)⊥ ⊥ )))
9		df-id3 52	. . . . 5 (b⊥ ≡3 (b⊥ ∩ a⊥ )) = ((b⊥ ⊥ ∪ (b⊥ ∩ a⊥ )) ∩ (b⊥ ∪ (b⊥ ⊥ ∩ (b⊥ ∩ a⊥ )⊥ )))
10		df-id3 52	. . . . 5 (b⊥ ≡3 (a ∪ b)⊥ ) = ((b⊥ ⊥ ∪ (a ∪ b)⊥ ) ∩ (b⊥ ∪ (b⊥ ⊥ ∩ (a ∪ b)⊥ ⊥ )))
11	8, 9, 10	3tr1 63	. . . 4 (b⊥ ≡3 (b⊥ ∩ a⊥ )) = (b⊥ ≡3 (a ∪ b)⊥ )
12	11	ax-r1 35	. . 3 (b⊥ ≡3 (a ∪ b)⊥ ) = (b⊥ ≡3 (b⊥ ∩ a⊥ ))
13		nom23 316	. . 3 (b⊥ ≡3 (b⊥ ∩ a⊥ )) = (b⊥ →1 a⊥ )
14	12, 13	ax-r2 36	. 2 (b⊥ ≡3 (a ∪ b)⊥ ) = (b⊥ →1 a⊥ )
15		nomcon4 305	. 2 ((a ∪ b) ≡4 b) = (b⊥ ≡3 (a ∪ b)⊥ )
16		i2i1 267	. 2 (a →2 b) = (b⊥ →1 a⊥ )
17	14, 15, 16	3tr1 63	1 ((a ∪ b) ≡4 b) = (a →2 b)

This proof depends on syntax axioms:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₁ wi1 12   →₂ wi2 13   ≡₃ wid3 20   ≡₄ wid4 21

This proof depends on 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 proof depends on definitions:  df-a 40  df-t 41  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 used by:  nom61  338