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Theorem nom22 315
 Description: Part of Lemma 3.3(14) from "Non-Orthomodular Models..." paper.
Assertion
Ref Expression
nom22 (a2 (ab)) = (a1 b)

Proof of Theorem nom22
StepHypRef Expression
1 oran3 93 . . . . . . 7 (ab ) = (ab)
21lor 70 . . . . . 6 (a ∪ (ab )) = (a ∪ (ab) )
32ax-r1 35 . . . . 5 (a ∪ (ab) ) = (a ∪ (ab ))
4 or12 80 . . . . 5 (a ∪ (ab )) = (a ∪ (ab ))
53, 4ax-r2 36 . . . 4 (a ∪ (ab) ) = (a ∪ (ab ))
6 ax-a2 31 . . . . 5 ((ab) ∪ (a ∩ (ab) )) = ((a ∩ (ab) ) ∪ (ab))
71lan 77 . . . . . . . 8 (a ∩ (ab )) = (a ∩ (ab) )
87ax-r1 35 . . . . . . 7 (a ∩ (ab) ) = (a ∩ (ab ))
9 anabs 121 . . . . . . 7 (a ∩ (ab )) = a
108, 9ax-r2 36 . . . . . 6 (a ∩ (ab) ) = a
1110ax-r5 38 . . . . 5 ((a ∩ (ab) ) ∪ (ab)) = (a ∪ (ab))
126, 11ax-r2 36 . . . 4 ((ab) ∪ (a ∩ (ab) )) = (a ∪ (ab))
135, 122an 79 . . 3 ((a ∪ (ab) ) ∩ ((ab) ∪ (a ∩ (ab) ))) = ((a ∪ (ab )) ∩ (a ∪ (ab)))
14 ancom 74 . . 3 ((a ∪ (ab )) ∩ (a ∪ (ab))) = ((a ∪ (ab)) ∩ (a ∪ (ab )))
15 lea 160 . . . . . 6 (ab) ≤ a
16 leo 158 . . . . . 6 a ≤ (ab )
1715, 16letr 137 . . . . 5 (ab) ≤ (ab )
1817lelor 166 . . . 4 (a ∪ (ab)) ≤ (a ∪ (ab ))
1918df2le2 136 . . 3 ((a ∪ (ab)) ∩ (a ∪ (ab ))) = (a ∪ (ab))
2013, 14, 193tr 65 . 2 ((a ∪ (ab) ) ∩ ((ab) ∪ (a ∩ (ab) ))) = (a ∪ (ab))
21 df-id2 51 . 2 (a2 (ab)) = ((a ∪ (ab) ) ∩ ((ab) ∪ (a ∩ (ab) )))
22 df-i1 44 . 2 (a1 b) = (a ∪ (ab))
2320, 21, 223tr1 63 1 (a2 (ab)) = (a1 b)
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ wo 6   ∩ wa 7   →1 wi1 12   ≡2 wid2 19 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 This theorem depends on definitions:  df-a 40  df-t 41  df-f 42  df-i1 44  df-id2 51  df-le1 130  df-le2 131 This theorem is referenced by:  nom33  322  nom51  332
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