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Theorem nom15 312
 Description: Part of Lemma 3.3(14) from "Non-Orthomodular Models..." paper. (Contributed by NM, 7-Feb-1999.)
Assertion
Ref Expression
nom15 (a5 (ab)) = (a1 b)

Proof of Theorem nom15
StepHypRef Expression
1 anass 76 . . . . . . . 8 ((aa) ∩ b) = (a ∩ (ab))
21ax-r1 35 . . . . . . 7 (a ∩ (ab)) = ((aa) ∩ b)
3 anidm 111 . . . . . . . 8 (aa) = a
43ran 78 . . . . . . 7 ((aa) ∩ b) = (ab)
52, 4ax-r2 36 . . . . . 6 (a ∩ (ab)) = (ab)
65ax-r5 38 . . . . 5 ((a ∩ (ab)) ∪ (a ∩ (ab))) = ((ab) ∪ (a ∩ (ab)))
7 ax-a2 31 . . . . 5 ((ab) ∪ (a ∩ (ab))) = ((a ∩ (ab)) ∪ (ab))
8 lear 161 . . . . . 6 (a ∩ (ab)) ≤ (ab)
98df-le2 131 . . . . 5 ((a ∩ (ab)) ∪ (ab)) = (ab)
106, 7, 93tr 65 . . . 4 ((a ∩ (ab)) ∪ (a ∩ (ab))) = (ab)
11 oran3 93 . . . . . . 7 (ab ) = (ab)
1211lan 77 . . . . . 6 (a ∩ (ab )) = (a ∩ (ab) )
1312ax-r1 35 . . . . 5 (a ∩ (ab) ) = (a ∩ (ab ))
14 anabs 121 . . . . 5 (a ∩ (ab )) = a
1513, 14ax-r2 36 . . . 4 (a ∩ (ab) ) = a
1610, 152or 72 . . 3 (((a ∩ (ab)) ∪ (a ∩ (ab))) ∪ (a ∩ (ab) )) = ((ab) ∪ a )
17 ax-a2 31 . . 3 ((ab) ∪ a ) = (a ∪ (ab))
1816, 17ax-r2 36 . 2 (((a ∩ (ab)) ∪ (a ∩ (ab))) ∪ (a ∩ (ab) )) = (a ∪ (ab))
19 df-i5 48 . 2 (a5 (ab)) = (((a ∩ (ab)) ∪ (a ∩ (ab))) ∪ (a ∩ (ab) ))
20 df-i1 44 . 2 (a1 b) = (a ∪ (ab))
2118, 19, 203tr1 63 1 (a5 (ab)) = (a1 b)
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ wo 6   ∩ wa 7   →1 wi1 12   →5 wi5 16 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-i5 48  df-le1 130  df-le2 131 This theorem is referenced by:  nom45  330  lem3.3.7i5e3  1074
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