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Theorem oadist 1019
Description: Distributive law derived from OAL.
Hypothesis
Ref Expression
oadist.1 d =< ((b v c) ->0 ((a ->2 b) ^ (a ->2 c)))
Assertion
Ref Expression
oadist ((a ->2 b) ^ (d v ((a ->2 b) ^ (a ->2 c)))) = (((a ->2 b) ^ d) v ((a ->2 b) ^ ((a ->2 b) ^ (a ->2 c))))
Proof of Theorem oadist
StepHypRef Expression
1 oadist.1 . . . . 5 d =< ((b v c) ->0 ((a ->2 b) ^ (a ->2 c)))
21oagen1 1014 . . . 4 ((a ->2 b) ^ (d v ((a ->2 b) ^ (a ->2 c)))) = ((a ->2 b) ^ (a ->2 c))
32bile 142 . . 3 ((a ->2 b) ^ (d v ((a ->2 b) ^ (a ->2 c)))) =< ((a ->2 b) ^ (a ->2 c))
4 anidm 111 . . . . . . 7 ((a ->2 b) ^ (a ->2 b)) = (a ->2 b)
54ax-r1 35 . . . . . 6 (a ->2 b) = ((a ->2 b) ^ (a ->2 b))
65ran 78 . . . . 5 ((a ->2 b) ^ (a ->2 c)) = (((a ->2 b) ^ (a ->2 b)) ^ (a ->2 c))
7 anass 76 . . . . 5 (((a ->2 b) ^ (a ->2 b)) ^ (a ->2 c)) = ((a ->2 b) ^ ((a ->2 b) ^ (a ->2 c)))
86, 7ax-r2 36 . . . 4 ((a ->2 b) ^ (a ->2 c)) = ((a ->2 b) ^ ((a ->2 b) ^ (a ->2 c)))
9 leor 159 . . . 4 ((a ->2 b) ^ ((a ->2 b) ^ (a ->2 c))) =< (((a ->2 b) ^ d) v ((a ->2 b) ^ ((a ->2 b) ^ (a ->2 c))))
108, 9bltr 138 . . 3 ((a ->2 b) ^ (a ->2 c)) =< (((a ->2 b) ^ d) v ((a ->2 b) ^ ((a ->2 b) ^ (a ->2 c))))
113, 10letr 137 . 2 ((a ->2 b) ^ (d v ((a ->2 b) ^ (a ->2 c)))) =< (((a ->2 b) ^ d) v ((a ->2 b) ^ ((a ->2 b) ^ (a ->2 c))))
12 ledi 174 . 2 (((a ->2 b) ^ d) v ((a ->2 b) ^ ((a ->2 b) ^ (a ->2 c)))) =< ((a ->2 b) ^ (d v ((a ->2 b) ^ (a ->2 c))))
1311, 12lebi 145 1 ((a ->2 b) ^ (d v ((a ->2 b) ^ (a ->2 c)))) = (((a ->2 b) ^ d) v ((a ->2 b) ^ ((a ->2 b) ^ (a ->2 c))))
Colors of variables: term
Syntax hints:   = wb 1   =< wle 2   v wo 6   ^ wa 7   ->0 wi0 11   ->2 wi2 13
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-3oa 998
This theorem depends on definitions:  df-a 40  df-t 41  df-f 42  df-i0 43  df-i1 44  df-i2 45  df-le1 130  df-le2 131
This theorem is referenced by: (None)
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