Theorem i3lem4 489

Description: Lemma for Kalmbach implication.

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 486	. . . 4 ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) = a⊥
3	2	ax-r5 37	. . 3 (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b))) = (a⊥ ∪ (a ∩ (a⊥ ∪ b)))
4	3	ax-r1 34	. 2 (a⊥ ∪ (a ∩ (a⊥ ∪ b))) = (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b)))
5		omln 428	. 2 (a⊥ ∪ (a ∩ (a⊥ ∪ b))) = (a⊥ ∪ b)
6		df-i3 45	. . . 4 (a →3 b) = (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b)))
7	6	ax-r1 34	. . 3 (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b))) = (a →3 b)
8	7, 1	ax-r2 35	. 2 (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b))) = 1
9	4, 5, 8	3tr2 61	1 (a⊥ ∪ b) = 1

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 9   →₃ wi3 15

This theorem is referenced by:  i3le 497

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a3 31  ax-a4 32  ax-a5 33  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37  ax-r3 421

This theorem depends on definitions:  df-b 38  df-a 39  df-t 40  df-f 41  df-i3 45  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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