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Theorem modexp 1150
 Description: Expansion by modular law. (Contributed by NM, 10-Apr-2012.)
Assertion
Ref Expression
modexp (a ∩ (bc)) = (a ∩ (b ∪ (c ∩ (ab))))

Proof of Theorem modexp
StepHypRef Expression
1 anass 76 . 2 ((a ∩ (ab)) ∩ (bc)) = (a ∩ ((ab) ∩ (bc)))
2 anabs 121 . . 3 (a ∩ (ab)) = a
32ran 78 . 2 ((a ∩ (ab)) ∩ (bc)) = (a ∩ (bc))
4 ancom 74 . . . 4 ((ab) ∩ (bc)) = ((bc) ∩ (ab))
5 leor 159 . . . . 5 b ≤ (ab)
65mlduali 1126 . . . 4 ((bc) ∩ (ab)) = (b ∪ (c ∩ (ab)))
74, 6tr 62 . . 3 ((ab) ∩ (bc)) = (b ∪ (c ∩ (ab)))
87lan 77 . 2 (a ∩ ((ab) ∩ (bc))) = (a ∩ (b ∪ (c ∩ (ab))))
91, 3, 83tr2 64 1 (a ∩ (bc)) = (a ∩ (b ∪ (c ∩ (ab))))
 Colors of variables: term Syntax hints:   = wb 1   ∪ wo 6   ∩ wa 7 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-ml 1120 This theorem depends on definitions:  df-a 40  df-t 41  df-f 42  df-le1 130  df-le2 131 This theorem is referenced by: (None)
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