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Theorem 1oaii 824
Description: OML analog to orthoarguesian law of Godowski/Greechie, Eq. II with 1 instead of 0 . (Contributed by NM, 1-Nov-1998.)
Assertion
Ref Expression
1oaii (b ∩ ((a2 b) ∪ ((a2 c) ∩ ((bc) →1 ((a2 b) ∩ (a2 c)))))) ≤ a

Proof of Theorem 1oaii
StepHypRef Expression
1 orabs 120 . . . . 5 ((a2 b) ∪ ((a2 b) ∩ ((bc) →1 ((a2 b) ∩ (a2 c))))) = (a2 b)
2 1oaiii 823 . . . . . 6 ((a2 b) ∩ ((bc) →1 ((a2 b) ∩ (a2 c)))) = ((a2 c) ∩ ((bc) →1 ((a2 b) ∩ (a2 c))))
32lor 70 . . . . 5 ((a2 b) ∪ ((a2 b) ∩ ((bc) →1 ((a2 b) ∩ (a2 c))))) = ((a2 b) ∪ ((a2 c) ∩ ((bc) →1 ((a2 b) ∩ (a2 c)))))
4 df-i2 45 . . . . . 6 (a2 b) = (b ∪ (ab ))
5 ancom 74 . . . . . . 7 (ab ) = (ba )
65lor 70 . . . . . 6 (b ∪ (ab )) = (b ∪ (ba ))
74, 6ax-r2 36 . . . . 5 (a2 b) = (b ∪ (ba ))
81, 3, 73tr2 64 . . . 4 ((a2 b) ∪ ((a2 c) ∩ ((bc) →1 ((a2 b) ∩ (a2 c))))) = (b ∪ (ba ))
98lan 77 . . 3 (b ∩ ((a2 b) ∪ ((a2 c) ∩ ((bc) →1 ((a2 b) ∩ (a2 c)))))) = (b ∩ (b ∪ (ba )))
10 omlan 448 . . 3 (b ∩ (b ∪ (ba ))) = (ba )
119, 10ax-r2 36 . 2 (b ∩ ((a2 b) ∪ ((a2 c) ∩ ((bc) →1 ((a2 b) ∩ (a2 c)))))) = (ba )
12 lear 161 . 2 (ba ) ≤ a
1311, 12bltr 138 1 (b ∩ ((a2 b) ∪ ((a2 c) ∩ ((bc) →1 ((a2 b) ∩ (a2 c)))))) ≤ a
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: (None)
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