Theorem 2oath1i1 827

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

Assertion

Ref	Expression
2oath1i1	((a →1 c) ∩ ((a ∩ b)⊥ →2 ((a →1 c) ∩ (b →1 c)))) = ((a →1 c) ∩ (b →1 c))

Proof of Theorem 2oath1i1

Step	Hyp	Ref	Expression
1		2oath1 826	. 2 ((c⊥ →2 a⊥ ) ∩ ((a⊥ ∪ b⊥ ) →2 ((c⊥ →2 a⊥ ) ∩ (c⊥ →2 b⊥ )))) = ((c⊥ →2 a⊥ ) ∩ (c⊥ →2 b⊥ ))
2		i1i2 266	. . 3 (a →1 c) = (c⊥ →2 a⊥ )
3		i1i2 266	. . . . . 6 (b →1 c) = (c⊥ →2 b⊥ )
4	2, 3	2an 79	. . . . 5 ((a →1 c) ∩ (b →1 c)) = ((c⊥ →2 a⊥ ) ∩ (c⊥ →2 b⊥ ))
5	4	ud2lem0a 258	. . . 4 ((a ∩ b)⊥ →2 ((a →1 c) ∩ (b →1 c))) = ((a ∩ b)⊥ →2 ((c⊥ →2 a⊥ ) ∩ (c⊥ →2 b⊥ )))
6		oran3 93	. . . . . 6 (a⊥ ∪ b⊥ ) = (a ∩ b)⊥
7	6	ax-r1 35	. . . . 5 (a ∩ b)⊥ = (a⊥ ∪ b⊥ )
8	7	ud2lem0b 259	. . . 4 ((a ∩ b)⊥ →2 ((c⊥ →2 a⊥ ) ∩ (c⊥ →2 b⊥ ))) = ((a⊥ ∪ b⊥ ) →2 ((c⊥ →2 a⊥ ) ∩ (c⊥ →2 b⊥ )))
9	5, 8	ax-r2 36	. . 3 ((a ∩ b)⊥ →2 ((a →1 c) ∩ (b →1 c))) = ((a⊥ ∪ b⊥ ) →2 ((c⊥ →2 a⊥ ) ∩ (c⊥ →2 b⊥ )))
10	2, 9	2an 79	. 2 ((a →1 c) ∩ ((a ∩ b)⊥ →2 ((a →1 c) ∩ (b →1 c)))) = ((c⊥ →2 a⊥ ) ∩ ((a⊥ ∪ b⊥ ) →2 ((c⊥ →2 a⊥ ) ∩ (c⊥ →2 b⊥ ))))
11	1, 10, 4	3tr1 63	1 ((a →1 c) ∩ ((a ∩ b)⊥ →2 ((a →1 c) ∩ (b →1 c)))) = ((a →1 c) ∩ (b →1 c))

Syntax hints:   = wb 1  ^⊥ 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-a3 32  ax-a4 33  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38  ax-r3 439

This theorem 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 referenced by:  1oath1i1u  828  d3oa  995