Theorem oaliv 1003

Description: Orthoarguesian law. Godowski/Greechie, Eq. IV. (Contributed by NM, 25-Nov-1998.)

Assertion

Ref	Expression
oaliv	(b⊥ ∩ ((a →2 b) ∪ ((a →2 c) ∩ ((b ∪ c)⊥ ∪ ((a →2 b) ∩ (a →2 c)))))) ≤ ((b⊥ ∩ (a →2 b)) ∪ (c⊥ ∩ (a →2 c)))

Proof of Theorem oaliv

Step	Hyp	Ref	Expression
1		lea 160	. . . 4 (b⊥ ∩ ((a →2 b) ∪ ((a →2 c) ∩ ((b ∪ c)⊥ ∪ ((a →2 b) ∩ (a →2 c)))))) ≤ b⊥
2		oalii 1002	. . . 4 (b⊥ ∩ ((a →2 b) ∪ ((a →2 c) ∩ ((b ∪ c)⊥ ∪ ((a →2 b) ∩ (a →2 c)))))) ≤ a⊥
3	1, 2	ler2an 173	. . 3 (b⊥ ∩ ((a →2 b) ∪ ((a →2 c) ∩ ((b ∪ c)⊥ ∪ ((a →2 b) ∩ (a →2 c)))))) ≤ (b⊥ ∩ a⊥ )
4		df-i2 45	. . . . . . 7 (a →2 b) = (b ∪ (a⊥ ∩ b⊥ ))
5		ancom 74	. . . . . . . 8 (a⊥ ∩ b⊥ ) = (b⊥ ∩ a⊥ )
6	5	lor 70	. . . . . . 7 (b ∪ (a⊥ ∩ b⊥ )) = (b ∪ (b⊥ ∩ a⊥ ))
7	4, 6	ax-r2 36	. . . . . 6 (a →2 b) = (b ∪ (b⊥ ∩ a⊥ ))
8	7	lan 77	. . . . 5 (b⊥ ∩ (a →2 b)) = (b⊥ ∩ (b ∪ (b⊥ ∩ a⊥ )))
9		omlan 448	. . . . 5 (b⊥ ∩ (b ∪ (b⊥ ∩ a⊥ ))) = (b⊥ ∩ a⊥ )
10	8, 9	ax-r2 36	. . . 4 (b⊥ ∩ (a →2 b)) = (b⊥ ∩ a⊥ )
11	10	ax-r1 35	. . 3 (b⊥ ∩ a⊥ ) = (b⊥ ∩ (a →2 b))
12	3, 11	lbtr 139	. 2 (b⊥ ∩ ((a →2 b) ∪ ((a →2 c) ∩ ((b ∪ c)⊥ ∪ ((a →2 b) ∩ (a →2 c)))))) ≤ (b⊥ ∩ (a →2 b))
13		leo 158	. 2 (b⊥ ∩ (a →2 b)) ≤ ((b⊥ ∩ (a →2 b)) ∪ (c⊥ ∩ (a →2 c)))
14	12, 13	letr 137	1 (b⊥ ∩ ((a →2 b) ∪ ((a →2 c) ∩ ((b ∪ c)⊥ ∪ ((a →2 b) ∩ (a →2 c)))))) ≤ ((b⊥ ∩ (a →2 b)) ∪ (c⊥ ∩ (a →2 c)))

This proof depends on syntax axioms:   ≤ wle 2  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₂ wi2 13

This proof depends on 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  ax-3oa 998

This proof depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i1 44  df-i2 45  df-le1 130  df-le2 131

This theorem is used by: (None)