Theorem oat 907

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 151	. . 3 b ≤ (a ∪ b)
2		oml 427	. . . . 5 (a ∪ (a⊥ ∩ (a ∪ b))) = (a ∪ b)
3	2	ax-r1 34	. . . 4 (a ∪ b) = (a ∪ (a⊥ ∩ (a ∪ b)))
4		lea 152	. . . . . 6 (a⊥ ∩ (a ∪ b)) ≤ a⊥
5		oat.1	. . . . . 6 (a⊥ ∩ (a ∪ b)) ≤ c
6	4, 5	ler2an 165	. . . . 5 (a⊥ ∩ (a ∪ b)) ≤ (a⊥ ∩ c)
7	6	lelor 158	. . . 4 (a ∪ (a⊥ ∩ (a ∪ b))) ≤ (a ∪ (a⊥ ∩ c))
8	3, 7	bltr 130	. . 3 (a ∪ b) ≤ (a ∪ (a⊥ ∩ c))
9	1, 8	letr 129	. 2 b ≤ (a ∪ (a⊥ ∩ c))
10		ax-a1 29	. . . 4 a = a⊥ ⊥
11	10	ax-r5 37	. . 3 (a ∪ (a⊥ ∩ c)) = (a⊥ ⊥ ∪ (a⊥ ∩ c))
12		df-i1 43	. . . 4 (a⊥ →1 c) = (a⊥ ⊥ ∪ (a⊥ ∩ c))
13	12	ax-r1 34	. . 3 (a⊥ ⊥ ∪ (a⊥ ∩ c)) = (a⊥ →1 c)
14	11, 13	ax-r2 35	. 2 (a ∪ (a⊥ ∩ c)) = (a⊥ →1 c)
15	9, 14	lbtr 131	1 b ≤ (a⊥ →1 c)

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

This theorem is referenced by:  oa4ctod 948

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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