Theorem oaidlem1 286

Description: Lemma for OA identity-like law.

Hypothesis

Ref	Expression
oaidlem1.1	(a ∩ b) ≤ c

Assertion

Ref	Expression
oaidlem1	(a⊥ ∪ (b →1 c)) = 1

Proof of Theorem oaidlem1

Step	Hyp	Ref	Expression
1		df-i1 43	. . 3 (b →1 c) = (b⊥ ∪ (b ∩ c))
2	1	lor 66	. 2 (a⊥ ∪ (b →1 c)) = (a⊥ ∪ (b⊥ ∪ (b ∩ c)))
3		oran3 85	. . . 4 (a⊥ ∪ b⊥ ) = (a ∩ b)⊥
4	3	ax-r5 37	. . 3 ((a⊥ ∪ b⊥ ) ∪ (b ∩ c)) = ((a ∩ b)⊥ ∪ (b ∩ c))
5		ax-a3 31	. . 3 ((a⊥ ∪ b⊥ ) ∪ (b ∩ c)) = (a⊥ ∪ (b⊥ ∪ (b ∩ c)))
6		lear 153	. . . . 5 (a ∩ b) ≤ b
7		oaidlem1.1	. . . . 5 (a ∩ b) ≤ c
8	6, 7	ler2an 165	. . . 4 (a ∩ b) ≤ (b ∩ c)
9	8	sklem 222	. . 3 ((a ∩ b)⊥ ∪ (b ∩ c)) = 1
10	4, 5, 9	3tr2 61	. 2 (a⊥ ∪ (b⊥ ∪ (b ∩ c))) = 1
11	2, 10	ax-r2 35	1 (a⊥ ∪ (b →1 c)) = 1

Syntax hints:   = wb 1   ≤ wle 2  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 9   →₁ wi1 13

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

This theorem depends on definitions:  df-a 39  df-t 40  df-f 41  df-i1 43  df-le1 122  df-le2 123

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