Theorem oalii 1002

Description: Orthoarguesian law. Godowski/Greechie, Eq. II. This proof references oaliii 1001 only. (Contributed by NM, 22-Sep-1998.)

Assertion

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

Proof of Theorem oalii

Step	Hyp	Ref	Expression
1		orabs 120	. . . . 5 ((a →2 b) ∪ ((a →2 b) ∩ ((b ∪ c)⊥ ∪ ((a →2 b) ∩ (a →2 c))))) = (a →2 b)
2		oaliii 1001	. . . . . 6 ((a →2 b) ∩ ((b ∪ c)⊥ ∪ ((a →2 b) ∩ (a →2 c)))) = ((a →2 c) ∩ ((b ∪ c)⊥ ∪ ((a →2 b) ∩ (a →2 c))))
3	2	lor 70	. . . . 5 ((a →2 b) ∪ ((a →2 b) ∩ ((b ∪ c)⊥ ∪ ((a →2 b) ∩ (a →2 c))))) = ((a →2 b) ∪ ((a →2 c) ∩ ((b ∪ c)⊥ ∪ ((a →2 b) ∩ (a →2 c)))))
4		df-i2 45	. . . . . 6 (a →2 b) = (b ∪ (a⊥ ∩ b⊥ ))
5		ancom 74	. . . . . . 7 (a⊥ ∩ b⊥ ) = (b⊥ ∩ a⊥ )
6	5	lor 70	. . . . . 6 (b ∪ (a⊥ ∩ b⊥ )) = (b ∪ (b⊥ ∩ a⊥ ))
7	4, 6	ax-r2 36	. . . . 5 (a →2 b) = (b ∪ (b⊥ ∩ a⊥ ))
8	1, 3, 7	3tr2 64	. . . 4 ((a →2 b) ∪ ((a →2 c) ∩ ((b ∪ c)⊥ ∪ ((a →2 b) ∩ (a →2 c))))) = (b ∪ (b⊥ ∩ a⊥ ))
9	8	lan 77	. . 3 (b⊥ ∩ ((a →2 b) ∪ ((a →2 c) ∩ ((b ∪ c)⊥ ∪ ((a →2 b) ∩ (a →2 c)))))) = (b⊥ ∩ (b ∪ (b⊥ ∩ a⊥ )))
10		omlan 448	. . 3 (b⊥ ∩ (b ∪ (b⊥ ∩ a⊥ ))) = (b⊥ ∩ a⊥ )
11	9, 10	ax-r2 36	. 2 (b⊥ ∩ ((a →2 b) ∪ ((a →2 c) ∩ ((b ∪ c)⊥ ∪ ((a →2 b) ∩ (a →2 c)))))) = (b⊥ ∩ a⊥ )
12		lear 161	. 2 (b⊥ ∩ a⊥ ) ≤ a⊥
13	11, 12	bltr 138	1 (b⊥ ∩ ((a →2 b) ∪ ((a →2 c) ∩ ((b ∪ c)⊥ ∪ ((a →2 b) ∩ (a →2 c)))))) ≤ a⊥

Syntax hints:   ≤ wle 2  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₂ wi2 13

This theorem was proved from 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 theorem 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 referenced by:  oaliv  1003  oalem1  1005