Theorem oat 927

Description: Transformation lemma for studying the orthoarguesian law.

Hypothesis

Ref	Expression
oat.1	(a⊥ ∩ (a ∪ b)) ≤ c

Assertion

Ref	Expression
oat	b ≤ (a⊥ →1 c)

Proof of Theorem oat

Step	Hyp	Ref	Expression
1		leor 159	. . 3 b ≤ (a ∪ b)
2		oml 445	. . . . 5 (a ∪ (a⊥ ∩ (a ∪ b))) = (a ∪ b)
3	2	ax-r1 35	. . . 4 (a ∪ b) = (a ∪ (a⊥ ∩ (a ∪ b)))
4		lea 160	. . . . . 6 (a⊥ ∩ (a ∪ b)) ≤ a⊥
5		oat.1	. . . . . 6 (a⊥ ∩ (a ∪ b)) ≤ c
6	4, 5	ler2an 173	. . . . 5 (a⊥ ∩ (a ∪ b)) ≤ (a⊥ ∩ c)
7	6	lelor 166	. . . 4 (a ∪ (a⊥ ∩ (a ∪ b))) ≤ (a ∪ (a⊥ ∩ c))
8	3, 7	bltr 138	. . 3 (a ∪ b) ≤ (a ∪ (a⊥ ∩ c))
9	1, 8	letr 137	. 2 b ≤ (a ∪ (a⊥ ∩ c))
10		ax-a1 30	. . . 4 a = a⊥ ⊥
11	10	ax-r5 38	. . 3 (a ∪ (a⊥ ∩ c)) = (a⊥ ⊥ ∪ (a⊥ ∩ c))
12		df-i1 44	. . . 4 (a⊥ →1 c) = (a⊥ ⊥ ∪ (a⊥ ∩ c))
13	12	ax-r1 35	. . 3 (a⊥ ⊥ ∪ (a⊥ ∩ c)) = (a⊥ →1 c)
14	11, 13	ax-r2 36	. 2 (a ∪ (a⊥ ∩ c)) = (a⊥ →1 c)
15	9, 14	lbtr 139	1 b ≤ (a⊥ →1 c)

Syntax hints:   ≤ wle 2  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₁ wi1 12

This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  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-i1 44  df-le1 130  df-le2 131

This theorem is referenced by:  oa4ctod  968