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Theorem oaliv 1003
 Description: Orthoarguesian law. Godowski/Greechie, Eq. IV. (Contributed by NM, 25-Nov-1998.)
Assertion
Ref Expression
oaliv (b ∩ ((a2 b) ∪ ((a2 c) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c)))))) ≤ ((b ∩ (a2 b)) ∪ (c ∩ (a2 c)))

Proof of Theorem oaliv
StepHypRef Expression
1 lea 160 . . . 4 (b ∩ ((a2 b) ∪ ((a2 c) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c)))))) ≤ b
2 oalii 1002 . . . 4 (b ∩ ((a2 b) ∪ ((a2 c) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c)))))) ≤ a
31, 2ler2an 173 . . 3 (b ∩ ((a2 b) ∪ ((a2 c) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c)))))) ≤ (ba )
4 df-i2 45 . . . . . . 7 (a2 b) = (b ∪ (ab ))
5 ancom 74 . . . . . . . 8 (ab ) = (ba )
65lor 70 . . . . . . 7 (b ∪ (ab )) = (b ∪ (ba ))
74, 6ax-r2 36 . . . . . 6 (a2 b) = (b ∪ (ba ))
87lan 77 . . . . 5 (b ∩ (a2 b)) = (b ∩ (b ∪ (ba )))
9 omlan 448 . . . . 5 (b ∩ (b ∪ (ba ))) = (ba )
108, 9ax-r2 36 . . . 4 (b ∩ (a2 b)) = (ba )
1110ax-r1 35 . . 3 (ba ) = (b ∩ (a2 b))
123, 11lbtr 139 . 2 (b ∩ ((a2 b) ∪ ((a2 c) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c)))))) ≤ (b ∩ (a2 b))
13 leo 158 . 2 (b ∩ (a2 b)) ≤ ((b ∩ (a2 b)) ∪ (c ∩ (a2 c)))
1412, 13letr 137 1 (b ∩ ((a2 b) ∪ ((a2 c) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c)))))) ≤ ((b ∩ (a2 b)) ∪ (c ∩ (a2 c)))
 Colors of variables: term Syntax hints:   ≤ wle 2  ⊥ wn 4   ∪ wo 6   ∩ wa 7   →2 wi2 13 This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38  ax-r3 439  ax-3oa 998 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 This theorem is referenced by: (None)
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