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Theorem nom21 314
 Description: Part of Lemma 3.3(14) from "Non-Orthomodular Models..." paper.
Assertion
Ref Expression
nom21 (a1 (ab)) = (a1 b)

Proof of Theorem nom21
StepHypRef Expression
1 ancom 74 . . 3 ((a ∪ (ab )) ∩ (a ∪ (ab))) = ((a ∪ (ab)) ∩ (a ∪ (ab )))
2 or12 80 . . . . 5 (a ∪ (ab )) = (a ∪ (ab ))
3 oran3 93 . . . . . 6 (ab ) = (ab)
43lor 70 . . . . 5 (a ∪ (ab )) = (a ∪ (ab) )
52, 4ax-r2 36 . . . 4 (a ∪ (ab )) = (a ∪ (ab) )
6 anidm 111 . . . . . . . 8 (aa) = a
76ran 78 . . . . . . 7 ((aa) ∩ b) = (ab)
87ax-r1 35 . . . . . 6 (ab) = ((aa) ∩ b)
9 anass 76 . . . . . 6 ((aa) ∩ b) = (a ∩ (ab))
108, 9ax-r2 36 . . . . 5 (ab) = (a ∩ (ab))
1110lor 70 . . . 4 (a ∪ (ab)) = (a ∪ (a ∩ (ab)))
125, 112an 79 . . 3 ((a ∪ (ab )) ∩ (a ∪ (ab))) = ((a ∪ (ab) ) ∩ (a ∪ (a ∩ (ab))))
13 lea 160 . . . . . 6 (ab) ≤ a
14 leo 158 . . . . . 6 a ≤ (ab )
1513, 14letr 137 . . . . 5 (ab) ≤ (ab )
1615lelor 166 . . . 4 (a ∪ (ab)) ≤ (a ∪ (ab ))
1716df2le2 136 . . 3 ((a ∪ (ab)) ∩ (a ∪ (ab ))) = (a ∪ (ab))
181, 12, 173tr2 64 . 2 ((a ∪ (ab) ) ∩ (a ∪ (a ∩ (ab)))) = (a ∪ (ab))
19 df-id1 50 . 2 (a1 (ab)) = ((a ∪ (ab) ) ∩ (a ∪ (a ∩ (ab))))
20 df-i1 44 . 2 (a1 b) = (a ∪ (ab))
2118, 19, 203tr1 63 1 (a1 (ab)) = (a1 b)
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ wo 6   ∩ wa 7   →1 wi1 12   ≡1 wid1 18 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-id1 50  df-le1 130  df-le2 131 This theorem is referenced by:  nom34  323  nom52  333
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