Theorem u3lemoa 604

Description: Lemma for Kalmbach implication study.

Assertion

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

Proof of Theorem u3lemoa

Step	Hyp	Ref	Expression
1		df-i3 45	. . 3 (a →3 b) = (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b)))
2	1	ax-r5 37	. 2 ((a →3 b) ∪ a) = ((((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b))) ∪ a)
3		ax-a3 31	. . 3 ((((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b))) ∪ a) = (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ ((a ∩ (a⊥ ∪ b)) ∪ a))
4		lea 152	. . . . . 6 (a ∩ (a⊥ ∪ b)) ≤ a
5	4	df-le2 123	. . . . 5 ((a ∩ (a⊥ ∪ b)) ∪ a) = a
6	5	lor 66	. . . 4 (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ ((a ∩ (a⊥ ∪ b)) ∪ a)) = (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ a)
7		ax-a2 30	. . . 4 (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ a) = (a ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )))
8	6, 7	ax-r2 35	. . 3 (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ ((a ∩ (a⊥ ∪ b)) ∪ a)) = (a ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )))
9	3, 8	ax-r2 35	. 2 ((((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b))) ∪ a) = (a ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )))
10	2, 9	ax-r2 35	1 ((a →3 b) ∪ a) = (a ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )))

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₃ wi3 15

This theorem is referenced by:  u3lemnana 629

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

This theorem depends on definitions:  df-a 39  df-i3 45  df-le1 122  df-le2 123

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