Theorem 1oai1 821

Description: Orthoarguesian-like OM law. (Contributed by NM, 30-Dec-1998.)

Assertion

Ref	Expression
1oai1	((a →1 c) ∩ ((a ∩ b)⊥ →1 ((a →1 c) ∩ (b →1 c)))) ≤ (b →1 c)

Proof of Theorem 1oai1

Step	Hyp	Ref	Expression
1		1oa 820	. 2 ((c⊥ →2 a⊥ ) ∩ ((a⊥ ∪ b⊥ ) →1 ((c⊥ →2 a⊥ ) ∩ (c⊥ →2 b⊥ )))) ≤ (c⊥ →2 b⊥ )
2		i1i2 266	. . 3 (a →1 c) = (c⊥ →2 a⊥ )
3		oran3 93	. . . . 5 (a⊥ ∪ b⊥ ) = (a ∩ b)⊥
4	3	ax-r1 35	. . . 4 (a ∩ b)⊥ = (a⊥ ∪ b⊥ )
5		i1i2 266	. . . . 5 (b →1 c) = (c⊥ →2 b⊥ )
6	2, 5	2an 79	. . . 4 ((a →1 c) ∩ (b →1 c)) = ((c⊥ →2 a⊥ ) ∩ (c⊥ →2 b⊥ ))
7	4, 6	ud1lem0ab 257	. . 3 ((a ∩ b)⊥ →1 ((a →1 c) ∩ (b →1 c))) = ((a⊥ ∪ b⊥ ) →1 ((c⊥ →2 a⊥ ) ∩ (c⊥ →2 b⊥ )))
8	2, 7	2an 79	. 2 ((a →1 c) ∩ ((a ∩ b)⊥ →1 ((a →1 c) ∩ (b →1 c)))) = ((c⊥ →2 a⊥ ) ∩ ((a⊥ ∪ b⊥ ) →1 ((c⊥ →2 a⊥ ) ∩ (c⊥ →2 b⊥ ))))
9	1, 8, 5	le3tr1 140	1 ((a →1 c) ∩ ((a ∩ b)⊥ →1 ((a →1 c) ∩ (b →1 c)))) ≤ (b →1 c)

This proof depends on syntax axioms:   ≤ wle 2  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₁ wi1 12   →₂ wi2 13

This proof depends on axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a4 33  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38  ax-r3 439

This proof 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  df-c1 132  df-c2 133

This theorem is used by:  2oai1u  822  d3oa  995