Theorem 1oaii 806

Description: OML analog to orthoarguesian law of Godowski/Greechie, Eq. II with →₁ instead of →₀ .

Assertion

Ref	Expression
1oaii	(b⊥ ∩ ((a →2 b) ∪ ((a →2 c) ∩ ((b ∪ c) →1 ((a →2 b) ∩ (a →2 c)))))) ≤ a⊥

Proof of Theorem 1oaii

Step	Hyp	Ref	Expression
1		a5b 112	. . . . 5 ((a →2 b) ∪ ((a →2 b) ∩ ((b ∪ c) →1 ((a →2 b) ∩ (a →2 c))))) = (a →2 b)
2		1oaiii 805	. . . . . 6 ((a →2 b) ∩ ((b ∪ c) →1 ((a →2 b) ∩ (a →2 c)))) = ((a →2 c) ∩ ((b ∪ c) →1 ((a →2 b) ∩ (a →2 c))))
3	2	lor 66	. . . . 5 ((a →2 b) ∪ ((a →2 b) ∩ ((b ∪ c) →1 ((a →2 b) ∩ (a →2 c))))) = ((a →2 b) ∪ ((a →2 c) ∩ ((b ∪ c) →1 ((a →2 b) ∩ (a →2 c)))))
4		df-i2 44	. . . . . 6 (a →2 b) = (b ∪ (a⊥ ∩ b⊥ ))
5		ancom 68	. . . . . . 7 (a⊥ ∩ b⊥ ) = (b⊥ ∩ a⊥ )
6	5	lor 66	. . . . . 6 (b ∪ (a⊥ ∩ b⊥ )) = (b ∪ (b⊥ ∩ a⊥ ))
7	4, 6	ax-r2 35	. . . . 5 (a →2 b) = (b ∪ (b⊥ ∩ a⊥ ))
8	1, 3, 7	3tr2 61	. . . 4 ((a →2 b) ∪ ((a →2 c) ∩ ((b ∪ c) →1 ((a →2 b) ∩ (a →2 c))))) = (b ∪ (b⊥ ∩ a⊥ ))
9	8	lan 70	. . 3 (b⊥ ∩ ((a →2 b) ∪ ((a →2 c) ∩ ((b ∪ c) →1 ((a →2 b) ∩ (a →2 c)))))) = (b⊥ ∩ (b ∪ (b⊥ ∩ a⊥ )))
10		omlan 430	. . 3 (b⊥ ∩ (b ∪ (b⊥ ∩ a⊥ ))) = (b⊥ ∩ a⊥ )
11	9, 10	ax-r2 35	. 2 (b⊥ ∩ ((a →2 b) ∪ ((a →2 c) ∩ ((b ∪ c) →1 ((a →2 b) ∩ (a →2 c)))))) = (b⊥ ∩ a⊥ )
12		lear 153	. 2 (b⊥ ∩ a⊥ ) ≤ a⊥
13	11, 12	bltr 130	1 (b⊥ ∩ ((a →2 b) ∪ ((a →2 c) ∩ ((b ∪ c) →1 ((a →2 b) ∩ (a →2 c)))))) ≤ a⊥

Syntax hints:   ≤ wle 2  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₁ wi1 13   →₂ wi2 14

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-i2 44  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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