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Theorem oalii 1002
 Description: Orthoarguesian law. Godowski/Greechie, Eq. II. This proof references oaliii 1001 only.
Assertion
Ref Expression
oalii (b ∩ ((a2 b) ∪ ((a2 c) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c)))))) ≤ a

Proof of Theorem oalii
StepHypRef Expression
1 orabs 120 . . . . 5 ((a2 b) ∪ ((a2 b) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c))))) = (a2 b)
2 oaliii 1001 . . . . . 6 ((a2 b) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c)))) = ((a2 c) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c))))
32lor 70 . . . . 5 ((a2 b) ∪ ((a2 b) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c))))) = ((a2 b) ∪ ((a2 c) ∩ ((bc) ∪ ((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) ∪ ((a2 b) ∩ (a2 c))))) = (b ∪ (ba ))
98lan 77 . . 3 (b ∩ ((a2 b) ∪ ((a2 c) ∩ ((bc) ∪ ((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) ∪ ((a2 b) ∩ (a2 c)))))) = (ba )
12 lear 161 . 2 (ba ) ≤ a
1311, 12bltr 138 1 (b ∩ ((a2 b) ∪ ((a2 c) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c)))))) ≤ a
 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:  oaliv  1003  oalem1  1005
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