Theorem oatr 928

Description: Reverse transformation lemma for studying the orthoarguesian law. (Contributed by NM, 26-Dec-1998.)

Hypothesis

Ref	Expression
oatr.1	b ≤ (a⊥ →1 c)

Assertion

Ref	Expression
oatr	(a⊥ ∩ (a ∪ b)) ≤ c

Proof of Theorem oatr

Step	Hyp	Ref	Expression
1		leo 158	. . . . 5 a ≤ (a ∪ (a⊥ ∩ c))
2		oatr.1	. . . . . 6 b ≤ (a⊥ →1 c)
3		df-i1 44	. . . . . . 7 (a⊥ →1 c) = (a⊥ ⊥ ∪ (a⊥ ∩ c))
4		ax-a1 30	. . . . . . . . 9 a = a⊥ ⊥
5	4	ax-r5 38	. . . . . . . 8 (a ∪ (a⊥ ∩ c)) = (a⊥ ⊥ ∪ (a⊥ ∩ c))
6	5	ax-r1 35	. . . . . . 7 (a⊥ ⊥ ∪ (a⊥ ∩ c)) = (a ∪ (a⊥ ∩ c))
7	3, 6	ax-r2 36	. . . . . 6 (a⊥ →1 c) = (a ∪ (a⊥ ∩ c))
8	2, 7	lbtr 139	. . . . 5 b ≤ (a ∪ (a⊥ ∩ c))
9	1, 8	lel2or 170	. . . 4 (a ∪ b) ≤ (a ∪ (a⊥ ∩ c))
10	9	lelan 167	. . 3 (a⊥ ∩ (a ∪ b)) ≤ (a⊥ ∩ (a ∪ (a⊥ ∩ c)))
11		omlan 448	. . 3 (a⊥ ∩ (a ∪ (a⊥ ∩ c))) = (a⊥ ∩ c)
12	10, 11	lbtr 139	. 2 (a⊥ ∩ (a ∪ b)) ≤ (a⊥ ∩ c)
13		lear 161	. 2 (a⊥ ∩ c) ≤ c
14	12, 13	letr 137	1 (a⊥ ∩ (a ∪ b)) ≤ 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:  oa4dtoc  969