Theorem oal1 1000

Description: Orthoarguesian law - →₁ version derived from →₁ version.

Assertion

Ref	Expression
oal1	((a →1 c) ∩ ((a ∩ b) ∪ ((a →1 c) ∩ (b →1 c)))) ≤ (b →1 c)

Proof of Theorem oal1

Step	Hyp	Ref	Expression
1		oal2 999	. 2 ((c⊥ →2 a⊥ ) ∩ ((a⊥ ∪ b⊥ )⊥ ∪ ((c⊥ →2 a⊥ ) ∩ (c⊥ →2 b⊥ )))) ≤ (c⊥ →2 b⊥ )
2		i1i2 266	. . 3 (a →1 c) = (c⊥ →2 a⊥ )
3		df-a 40	. . . 4 (a ∩ b) = (a⊥ ∪ b⊥ )⊥
4		i1i2 266	. . . . 5 (b →1 c) = (c⊥ →2 b⊥ )
5	2, 4	2an 79	. . . 4 ((a →1 c) ∩ (b →1 c)) = ((c⊥ →2 a⊥ ) ∩ (c⊥ →2 b⊥ ))
6	3, 5	2or 72	. . 3 ((a ∩ b) ∪ ((a →1 c) ∩ (b →1 c))) = ((a⊥ ∪ b⊥ )⊥ ∪ ((c⊥ →2 a⊥ ) ∩ (c⊥ →2 b⊥ )))
7	2, 6	2an 79	. 2 ((a →1 c) ∩ ((a ∩ b) ∪ ((a →1 c) ∩ (b →1 c)))) = ((c⊥ →2 a⊥ ) ∩ ((a⊥ ∪ b⊥ )⊥ ∪ ((c⊥ →2 a⊥ ) ∩ (c⊥ →2 b⊥ ))))
8	1, 7, 4	le3tr1 140	1 ((a →1 c) ∩ ((a ∩ b) ∪ ((a →1 c) ∩ (b →1 c)))) ≤ (b →1 c)

Syntax hints:   ≤ wle 2  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₁ wi1 12   →₂ wi2 13

This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38  ax-3oa 998

This theorem depends on definitions:  df-a 40  df-i1 44  df-i2 45  df-le1 130  df-le2 131

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