Theorem oaliv 983

Description: Orthoarguesian law. Godowski/Greechie, Eq. IV.

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 152	. . . 4 (b⊥ ∩ ((a →2 b) ∪ ((a →2 c) ∩ ((b ∪ c)⊥ ∪ ((a →2 b) ∩ (a →2 c)))))) ≤ b⊥
2		oalii 982	. . . 4 (b⊥ ∩ ((a →2 b) ∪ ((a →2 c) ∩ ((b ∪ c)⊥ ∪ ((a →2 b) ∩ (a →2 c)))))) ≤ a⊥
3	1, 2	ler2an 165	. . 3 (b⊥ ∩ ((a →2 b) ∪ ((a →2 c) ∩ ((b ∪ c)⊥ ∪ ((a →2 b) ∩ (a →2 c)))))) ≤ (b⊥ ∩ a⊥ )
4		df-i2 44	. . . . . . 7 (a →2 b) = (b ∪ (a⊥ ∩ b⊥ ))
5		ancom 68	. . . . . . . 8 (a⊥ ∩ b⊥ ) = (b⊥ ∩ a⊥ )
6	5	lor 66	. . . . . . 7 (b ∪ (a⊥ ∩ b⊥ )) = (b ∪ (b⊥ ∩ a⊥ ))
7	4, 6	ax-r2 35	. . . . . 6 (a →2 b) = (b ∪ (b⊥ ∩ a⊥ ))
8	7	lan 70	. . . . 5 (b⊥ ∩ (a →2 b)) = (b⊥ ∩ (b ∪ (b⊥ ∩ a⊥ )))
9		omlan 430	. . . . 5 (b⊥ ∩ (b ∪ (b⊥ ∩ a⊥ ))) = (b⊥ ∩ a⊥ )
10	8, 9	ax-r2 35	. . . 4 (b⊥ ∩ (a →2 b)) = (b⊥ ∩ a⊥ )
11	10	ax-r1 34	. . 3 (b⊥ ∩ a⊥ ) = (b⊥ ∩ (a →2 b))
12	3, 11	lbtr 131	. 2 (b⊥ ∩ ((a →2 b) ∪ ((a →2 c) ∩ ((b ∪ c)⊥ ∪ ((a →2 b) ∩ (a →2 c)))))) ≤ (b⊥ ∩ (a →2 b))
13		leo 150	. 2 (b⊥ ∩ (a →2 b)) ≤ ((b⊥ ∩ (a →2 b)) ∪ (c⊥ ∩ (a →2 c)))
14	12, 13	letr 129	1 (b⊥ ∩ ((a →2 b) ∪ ((a →2 c) ∩ ((b ∪ c)⊥ ∪ ((a →2 b) ∩ (a →2 c)))))) ≤ ((b⊥ ∩ (a →2 b)) ∪ (c⊥ ∩ (a →2 c)))

Syntax hints:   ≤ wle 2  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₂ wi2 14

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  ax-3oa 978

This theorem depends on definitions:  df-b 38  df-a 39  df-t 40  df-f 41  df-i1 43  df-i2 44  df-le1 122  df-le2 123

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