Theorem nom45 330

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

Assertion

Ref	Expression
nom45	((a ∪ b) →5 b) = (a →2 b)

Proof of Theorem nom45

Step	Hyp	Ref	Expression
1		ancom 74	. . . . . 6 (b⊥ ∩ a⊥ ) = (a⊥ ∩ b⊥ )
2		anor3 90	. . . . . 6 (a⊥ ∩ b⊥ ) = (a ∪ b)⊥
3	1, 2	ax-r2 36	. . . . 5 (b⊥ ∩ a⊥ ) = (a ∪ b)⊥
4	3	ud5lem0a 264	. . . 4 (b⊥ →5 (b⊥ ∩ a⊥ )) = (b⊥ →5 (a ∪ b)⊥ )
5	4	ax-r1 35	. . 3 (b⊥ →5 (a ∪ b)⊥ ) = (b⊥ →5 (b⊥ ∩ a⊥ ))
6		nom15 312	. . 3 (b⊥ →5 (b⊥ ∩ a⊥ )) = (b⊥ →1 a⊥ )
7	5, 6	ax-r2 36	. 2 (b⊥ →5 (a ∪ b)⊥ ) = (b⊥ →1 a⊥ )
8		i5con 272	. 2 ((a ∪ b) →5 b) = (b⊥ →5 (a ∪ b)⊥ )
9		i2i1 267	. 2 (a →2 b) = (b⊥ →1 a⊥ )
10	7, 8, 9	3tr1 63	1 ((a ∪ b) →5 b) = (a →2 b)

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₁ wi1 12   →₂ wi2 13   →₅ wi5 16

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-i5 48  df-le1 130  df-le2 131

This theorem is referenced by: (None)