Theorem u3lemnana 647

Description: Lemma for Kalmbach implication study. (Contributed by NM, 16-Dec-1997.)

Assertion

Ref	Expression
u3lemnana	((a →3 b)⊥ ∩ a⊥ ) = (a⊥ ∩ ((a ∪ b) ∩ (a ∪ b⊥ )))

Proof of Theorem u3lemnana

Step	Hyp	Ref	Expression
1		u3lemoa 622	. . . 4 ((a →3 b) ∪ a) = (a ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )))
2		ax-a2 31	. . . . . 6 ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) = ((a⊥ ∩ b⊥ ) ∪ (a⊥ ∩ b))
3		anor3 90	. . . . . . . 8 (a⊥ ∩ b⊥ ) = (a ∪ b)⊥
4		anor2 89	. . . . . . . 8 (a⊥ ∩ b) = (a ∪ b⊥ )⊥
5	3, 4	2or 72	. . . . . . 7 ((a⊥ ∩ b⊥ ) ∪ (a⊥ ∩ b)) = ((a ∪ b)⊥ ∪ (a ∪ b⊥ )⊥ )
6		oran3 93	. . . . . . 7 ((a ∪ b)⊥ ∪ (a ∪ b⊥ )⊥ ) = ((a ∪ b) ∩ (a ∪ b⊥ ))⊥
7	5, 6	ax-r2 36	. . . . . 6 ((a⊥ ∩ b⊥ ) ∪ (a⊥ ∩ b)) = ((a ∪ b) ∩ (a ∪ b⊥ ))⊥
8	2, 7	ax-r2 36	. . . . 5 ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) = ((a ∪ b) ∩ (a ∪ b⊥ ))⊥
9	8	lor 70	. . . 4 (a ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))) = (a ∪ ((a ∪ b) ∩ (a ∪ b⊥ ))⊥ )
10	1, 9	ax-r2 36	. . 3 ((a →3 b) ∪ a) = (a ∪ ((a ∪ b) ∩ (a ∪ b⊥ ))⊥ )
11		oran 87	. . 3 ((a →3 b) ∪ a) = ((a →3 b)⊥ ∩ a⊥ )⊥
12		oran1 91	. . 3 (a ∪ ((a ∪ b) ∩ (a ∪ b⊥ ))⊥ ) = (a⊥ ∩ ((a ∪ b) ∩ (a ∪ b⊥ )))⊥
13	10, 11, 12	3tr2 64	. 2 ((a →3 b)⊥ ∩ a⊥ )⊥ = (a⊥ ∩ ((a ∪ b) ∩ (a ∪ b⊥ )))⊥
14	13	con1 66	1 ((a →3 b)⊥ ∩ a⊥ ) = (a⊥ ∩ ((a ∪ b) ∩ (a ∪ b⊥ )))

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₃ wi3 14

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-i3 46  df-le1 130  df-le2 131

This theorem is referenced by:  u3lem13a  789  u3lem13b  790