Theorem i3lem4 507

Description: Lemma for Kalmbach implication. (Contributed by NM, 7-Nov-1997.)

Hypothesis

Ref	Expression
i3lem.1	(a →3 b) = 1

Assertion

Ref	Expression
i3lem4	(a⊥ ∪ b) = 1

Proof of Theorem i3lem4

Step	Hyp	Ref	Expression
1		i3lem.1	. . . . 5 (a →3 b) = 1
2	1	i3lem1 504	. . . 4 ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) = a⊥
3	2	ax-r5 38	. . 3 (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b))) = (a⊥ ∪ (a ∩ (a⊥ ∪ b)))
4	3	ax-r1 35	. 2 (a⊥ ∪ (a ∩ (a⊥ ∪ b))) = (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b)))
5		omln 446	. 2 (a⊥ ∪ (a ∩ (a⊥ ∪ b))) = (a⊥ ∪ b)
6		df-i3 46	. . . 4 (a →3 b) = (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b)))
7	6	ax-r1 35	. . 3 (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b))) = (a →3 b)
8	7, 1	ax-r2 36	. 2 (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b))) = 1
9	4, 5, 8	3tr2 64	1 (a⊥ ∪ b) = 1

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8   →₃ wi3 14

This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a4 33  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38  ax-r3 439

This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i3 46  df-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is referenced by:  i3le  515