Theorem nom43 320

Description: Part of Lemma 3.3(15) from "Non-Orthomodular Models..." paper.

Assertion

Ref	Expression
nom43	((a ∪ b) →3 b) = (a →2 b)

Proof of Theorem nom43

Step	Hyp	Ref	Expression
1		ancom 68	. . . . . 6 (b⊥ ∩ a⊥ ) = (a⊥ ∩ b⊥ )
2		anor3 82	. . . . . 6 (a⊥ ∩ b⊥ ) = (a ∪ b)⊥
3	1, 2	ax-r2 35	. . . . 5 (b⊥ ∩ a⊥ ) = (a ∪ b)⊥
4	3	ud4lem0a 254	. . . 4 (b⊥ →4 (b⊥ ∩ a⊥ )) = (b⊥ →4 (a ∪ b)⊥ )
5	4	ax-r1 34	. . 3 (b⊥ →4 (a ∪ b)⊥ ) = (b⊥ →4 (b⊥ ∩ a⊥ ))
6		nom14 303	. . 3 (b⊥ →4 (b⊥ ∩ a⊥ )) = (b⊥ →1 a⊥ )
7	5, 6	ax-r2 35	. 2 (b⊥ →4 (a ∪ b)⊥ ) = (b⊥ →1 a⊥ )
8		i3i4 262	. 2 ((a ∪ b) →3 b) = (b⊥ →4 (a ∪ b)⊥ )
9		i2i1 259	. 2 (a →2 b) = (b⊥ →1 a⊥ )
10	7, 8, 9	3tr1 60	1 ((a ∪ b) →3 b) = (a →2 b)

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₁ wi1 13   →₂ wi2 14   →₃ wi3 15   →₄ wi4 16

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a3 31  ax-a5 33  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37

This theorem depends on definitions:  df-a 39  df-t 40  df-f 41  df-i1 43  df-i2 44  df-i3 45  df-i4 46  df-le1 122  df-le2 123

metamath.org