Theorem u3lemoa 622

Description: Lemma for Kalmbach implication study. (Contributed by NM, 15-Dec-1997.)

Assertion

Ref	Expression
u3lemoa	((a →3 b) ∪ a) = (a ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )))

Proof of Theorem u3lemoa

Step	Hyp	Ref	Expression
1		df-i3 46	. . 3 (a →3 b) = (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b)))
2	1	ax-r5 38	. 2 ((a →3 b) ∪ a) = ((((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b))) ∪ a)
3		ax-a3 32	. . 3 ((((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b))) ∪ a) = (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ ((a ∩ (a⊥ ∪ b)) ∪ a))
4		lea 160	. . . . . 6 (a ∩ (a⊥ ∪ b)) ≤ a
5	4	df-le2 131	. . . . 5 ((a ∩ (a⊥ ∪ b)) ∪ a) = a
6	5	lor 70	. . . 4 (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ ((a ∩ (a⊥ ∪ b)) ∪ a)) = (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ a)
7		ax-a2 31	. . . 4 (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ a) = (a ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )))
8	6, 7	ax-r2 36	. . 3 (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ ((a ∩ (a⊥ ∪ b)) ∪ a)) = (a ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )))
9	3, 8	ax-r2 36	. 2 ((((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b))) ∪ a) = (a ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )))
10	2, 9	ax-r2 36	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-a2 31  ax-a3 32  ax-a5 34  ax-r1 35  ax-r2 36  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:  u3lemnana  647