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Theorem modexp 1150
Description: Expansion by modular law.
Assertion
Ref Expression
modexp (a ^ (b v c)) = (a ^ (b v (c ^ (a v b))))
Proof of Theorem modexp
StepHypRef Expression
1 anass 76 . 2 ((a ^ (a v b)) ^ (b v c)) = (a ^ ((a v b) ^ (b v c)))
2 anabs 121 . . 3 (a ^ (a v b)) = a
32ran 78 . 2 ((a ^ (a v b)) ^ (b v c)) = (a ^ (b v c))
4 ancom 74 . . . 4 ((a v b) ^ (b v c)) = ((b v c) ^ (a v b))
5 leor 159 . . . . 5 b =< (a v b)
65mlduali 1126 . . . 4 ((b v c) ^ (a v b)) = (b v (c ^ (a v b)))
74, 6tr 62 . . 3 ((a v b) ^ (b v c)) = (b v (c ^ (a v b)))
87lan 77 . 2 (a ^ ((a v b) ^ (b v c))) = (a ^ (b v (c ^ (a v b))))
91, 3, 83tr2 64 1 (a ^ (b v c)) = (a ^ (b v (c ^ (a v b))))
Colors of variables: term
Syntax hints:   = wb 1   v 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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