Theorem 1oai1 803

Description: Orthoarguesian-like OM law.

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 802	. 2 ((c⊥ →2 a⊥ ) ∩ ((a⊥ ∪ b⊥ ) →1 ((c⊥ →2 a⊥ ) ∩ (c⊥ →2 b⊥ )))) ≤ (c⊥ →2 b⊥ )
2		i1i2 258	. . 3 (a →1 c) = (c⊥ →2 a⊥ )
3		oran3 85	. . . . 5 (a⊥ ∪ b⊥ ) = (a ∩ b)⊥
4	3	ax-r1 34	. . . 4 (a ∩ b)⊥ = (a⊥ ∪ b⊥ )
5		i1i2 258	. . . . 5 (b →1 c) = (c⊥ →2 b⊥ )
6	2, 5	2an 72	. . . 4 ((a →1 c) ∩ (b →1 c)) = ((c⊥ →2 a⊥ ) ∩ (c⊥ →2 b⊥ ))
7	4, 6	ud1lem0ab 249	. . 3 ((a ∩ b)⊥ →1 ((a →1 c) ∩ (b →1 c))) = ((a⊥ ∪ b⊥ ) →1 ((c⊥ →2 a⊥ ) ∩ (c⊥ →2 b⊥ )))
8	2, 7	2an 72	. 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 132	1 ((a →1 c) ∩ ((a ∩ b)⊥ →1 ((a →1 c) ∩ (b →1 c)))) ≤ (b →1 c)

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

This theorem is referenced by:  2oai1u 804  d3oa 975

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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