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Theorem nom20 313
 Description: Part of Lemma 3.3(14) from "Non-Orthomodular Models..." paper.
Assertion
Ref Expression
nom20 (a0 (ab)) = (a1 b)

Proof of Theorem nom20
StepHypRef Expression
1 lea 160 . . . . . 6 (ab) ≤ a
2 leor 159 . . . . . 6 a ≤ (ba)
31, 2letr 137 . . . . 5 (ab) ≤ (ba)
43lelor 166 . . . 4 (a ∪ (ab)) ≤ (a ∪ (ba))
5 ax-a3 32 . . . . . 6 ((ab ) ∪ a) = (a ∪ (ba))
65ax-r1 35 . . . . 5 (a ∪ (ba)) = ((ab ) ∪ a)
7 oran3 93 . . . . . 6 (ab ) = (ab)
87ax-r5 38 . . . . 5 ((ab ) ∪ a) = ((ab)a)
96, 8ax-r2 36 . . . 4 (a ∪ (ba)) = ((ab)a)
104, 9lbtr 139 . . 3 (a ∪ (ab)) ≤ ((ab)a)
1110df2le2 136 . 2 ((a ∪ (ab)) ∩ ((ab)a)) = (a ∪ (ab))
12 df-id0 49 . 2 (a0 (ab)) = ((a ∪ (ab)) ∩ ((ab)a))
13 df-i1 44 . 2 (a1 b) = (a ∪ (ab))
1411, 12, 133tr1 63 1 (a0 (ab)) = (a1 b)
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ wo 6   ∩ wa 7   →1 wi1 12   ≡0 wid0 17 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-id0 49  df-le1 130  df-le2 131 This theorem is referenced by:  nom30  319  nom50  331
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