Theorem oal2 979

Description: Orthoarguesian law - →₂ version.

Assertion

Ref	Expression
oal2	((a →2 b) ∩ ((b ∪ c)⊥ ∪ ((a →2 b) ∩ (a →2 c)))) ≤ (a →2 c)

Proof of Theorem oal2

Step	Hyp	Ref	Expression
1		ax-3oa 978	. 2 ((b⊥ →1 a⊥ ) ∩ ((b⊥ ∩ c⊥ ) ∪ ((b⊥ →1 a⊥ ) ∩ (c⊥ →1 a⊥ )))) ≤ (c⊥ →1 a⊥ )
2		i2i1 259	. . 3 (a →2 b) = (b⊥ →1 a⊥ )
3		anor3 82	. . . . 5 (b⊥ ∩ c⊥ ) = (b ∪ c)⊥
4	3	ax-r1 34	. . . 4 (b ∪ c)⊥ = (b⊥ ∩ c⊥ )
5		i2i1 259	. . . . 5 (a →2 c) = (c⊥ →1 a⊥ )
6	2, 5	2an 72	. . . 4 ((a →2 b) ∩ (a →2 c)) = ((b⊥ →1 a⊥ ) ∩ (c⊥ →1 a⊥ ))
7	4, 6	2or 67	. . 3 ((b ∪ c)⊥ ∪ ((a →2 b) ∩ (a →2 c))) = ((b⊥ ∩ c⊥ ) ∪ ((b⊥ →1 a⊥ ) ∩ (c⊥ →1 a⊥ )))
8	2, 7	2an 72	. 2 ((a →2 b) ∩ ((b ∪ c)⊥ ∪ ((a →2 b) ∩ (a →2 c)))) = ((b⊥ →1 a⊥ ) ∩ ((b⊥ ∩ c⊥ ) ∪ ((b⊥ →1 a⊥ ) ∩ (c⊥ →1 a⊥ ))))
9	1, 8, 5	le3tr1 132	1 ((a →2 b) ∩ ((b ∪ c)⊥ ∪ ((a →2 b) ∩ (a →2 c)))) ≤ (a →2 c)

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

This theorem is referenced by:  oal1 980  oaliii 981  oagen2 996  mloa 998  oadistc0 1001

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a5 33  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37  ax-3oa 978

This theorem depends on definitions:  df-a 39  df-i1 43  df-i2 44  df-le1 122  df-le2 123

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