Theorem oaur 930

Description: Transformation lemma for studying the orthoarguesian law. (Contributed by NM, 28-Dec-1998.)

Hypothesis

Ref	Expression
oaur.1	b ≤ (a →1 c)

Assertion

Ref	Expression
oaur	(a ∩ ((a →1 c) ∪ b)) ≤ c

Proof of Theorem oaur

Step	Hyp	Ref	Expression
1		leid 148	. . . . 5 (a →1 c) ≤ (a →1 c)
2		oaur.1	. . . . 5 b ≤ (a →1 c)
3	1, 2	lel2or 170	. . . 4 ((a →1 c) ∪ b) ≤ (a →1 c)
4	3	lelan 167	. . 3 (a ∩ ((a →1 c) ∪ b)) ≤ (a ∩ (a →1 c))
5		ancom 74	. . . 4 (a ∩ (a →1 c)) = ((a →1 c) ∩ a)
6		u1lemaa 600	. . . 4 ((a →1 c) ∩ a) = (a ∩ c)
7	5, 6	ax-r2 36	. . 3 (a ∩ (a →1 c)) = (a ∩ c)
8	4, 7	lbtr 139	. 2 (a ∩ ((a →1 c) ∪ b)) ≤ (a ∩ c)
9		lear 161	. 2 (a ∩ c) ≤ c
10	8, 9	letr 137	1 (a ∩ ((a →1 c) ∪ b)) ≤ c

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

This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a4 33  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  df-c1 132  df-c2 133

This theorem is referenced by:  oa4gto4u  976