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Theorem 1oai1 821
 Description: Orthoarguesian-like OM law.
Assertion
Ref Expression
1oai1 ((a1 c) ∩ ((ab)1 ((a1 c) ∩ (b1 c)))) ≤ (b1 c)

Proof of Theorem 1oai1
StepHypRef Expression
1 1oa 820 . 2 ((c2 a ) ∩ ((ab ) →1 ((c2 a ) ∩ (c2 b )))) ≤ (c2 b )
2 i1i2 266 . . 3 (a1 c) = (c2 a )
3 oran3 93 . . . . 5 (ab ) = (ab)
43ax-r1 35 . . . 4 (ab) = (ab )
5 i1i2 266 . . . . 5 (b1 c) = (c2 b )
62, 52an 79 . . . 4 ((a1 c) ∩ (b1 c)) = ((c2 a ) ∩ (c2 b ))
74, 6ud1lem0ab 257 . . 3 ((ab)1 ((a1 c) ∩ (b1 c))) = ((ab ) →1 ((c2 a ) ∩ (c2 b )))
82, 72an 79 . 2 ((a1 c) ∩ ((ab)1 ((a1 c) ∩ (b1 c)))) = ((c2 a ) ∩ ((ab ) →1 ((c2 a ) ∩ (c2 b ))))
91, 8, 5le3tr1 140 1 ((a1 c) ∩ ((ab)1 ((a1 c) ∩ (b1 c)))) ≤ (b1 c)
 Colors of variables: term 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-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:  2oai1u  822  d3oa  995
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