Theorem oaur 910

Description: Transformation lemma for studying the orthoarguesian law.

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 140	. . . . 5 (a →1 c) ≤ (a →1 c)
2		oaur.1	. . . . 5 b ≤ (a →1 c)
3	1, 2	lel2or 162	. . . 4 ((a →1 c) ∪ b) ≤ (a →1 c)
4	3	lelan 159	. . 3 (a ∩ ((a →1 c) ∪ b)) ≤ (a ∩ (a →1 c))
5		ancom 68	. . . 4 (a ∩ (a →1 c)) = ((a →1 c) ∩ a)
6		u1lemaa 582	. . . 4 ((a →1 c) ∩ a) = (a ∩ c)
7	5, 6	ax-r2 35	. . 3 (a ∩ (a →1 c)) = (a ∩ c)
8	4, 7	lbtr 131	. 2 (a ∩ ((a →1 c) ∪ b)) ≤ (a ∩ c)
9		lear 153	. 2 (a ∩ c) ≤ c
10	8, 9	letr 129	1 (a ∩ ((a →1 c) ∪ b)) ≤ c

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

This theorem is referenced by:  oa4gto4u 956

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a3 31  ax-a4 32  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  df-c1 124  df-c2 125

metamath.org