Theorem oatr 908

Description: Reverse transformation lemma for studying the orthoarguesian law.

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 150	. . . . 5 a ≤ (a ∪ (a⊥ ∩ c))
2		oatr.1	. . . . . 6 b ≤ (a⊥ →1 c)
3		df-i1 43	. . . . . . 7 (a⊥ →1 c) = (a⊥ ⊥ ∪ (a⊥ ∩ c))
4		ax-a1 29	. . . . . . . . 9 a = a⊥ ⊥
5	4	ax-r5 37	. . . . . . . 8 (a ∪ (a⊥ ∩ c)) = (a⊥ ⊥ ∪ (a⊥ ∩ c))
6	5	ax-r1 34	. . . . . . 7 (a⊥ ⊥ ∪ (a⊥ ∩ c)) = (a ∪ (a⊥ ∩ c))
7	3, 6	ax-r2 35	. . . . . 6 (a⊥ →1 c) = (a ∪ (a⊥ ∩ c))
8	2, 7	lbtr 131	. . . . 5 b ≤ (a ∪ (a⊥ ∩ c))
9	1, 8	lel2or 162	. . . 4 (a ∪ b) ≤ (a ∪ (a⊥ ∩ c))
10	9	lelan 159	. . 3 (a⊥ ∩ (a ∪ b)) ≤ (a⊥ ∩ (a ∪ (a⊥ ∩ c)))
11		omlan 430	. . 3 (a⊥ ∩ (a ∪ (a⊥ ∩ c))) = (a⊥ ∩ c)
12	10, 11	lbtr 131	. 2 (a⊥ ∩ (a ∪ b)) ≤ (a⊥ ∩ c)
13		lear 153	. 2 (a⊥ ∩ c) ≤ c
14	12, 13	letr 129	1 (a⊥ ∩ (a ∪ b)) ≤ c

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

This theorem is referenced by:  oa4dtoc 949

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a3 31  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-i1 43  df-le1 122  df-le2 123

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