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Theorem oadistb 1020
Description: Distributive law derived from OAL.
Hypotheses
Ref Expression
oadistb.2 d =< (a ->2 b)
oadistb.1 e =< ((b v c) ->0 ((a ->2 b) ^ (a ->2 c)))
Assertion
Ref Expression
oadistb (d ^ (e v ((a ->2 b) ^ (a ->2 c)))) = ((d ^ e) v (d ^ ((a ->2 b) ^ (a ->2 c))))
Proof of Theorem oadistb
StepHypRef Expression
1 oadistb.2 . . . . . . 7 d =< (a ->2 b)
21df2le2 136 . . . . . 6 (d ^ (a ->2 b)) = d
32ran 78 . . . . 5 ((d ^ (a ->2 b)) ^ (e v ((a ->2 b) ^ (a ->2 c)))) = (d ^ (e v ((a ->2 b) ^ (a ->2 c))))
43ax-r1 35 . . . 4 (d ^ (e v ((a ->2 b) ^ (a ->2 c)))) = ((d ^ (a ->2 b)) ^ (e v ((a ->2 b) ^ (a ->2 c))))
5 anass 76 . . . . 5 ((d ^ (a ->2 b)) ^ (e v ((a ->2 b) ^ (a ->2 c)))) = (d ^ ((a ->2 b) ^ (e v ((a ->2 b) ^ (a ->2 c)))))
6 oadistb.1 . . . . . . 7 e =< ((b v c) ->0 ((a ->2 b) ^ (a ->2 c)))
76oagen1 1014 . . . . . 6 ((a ->2 b) ^ (e v ((a ->2 b) ^ (a ->2 c)))) = ((a ->2 b) ^ (a ->2 c))
87lan 77 . . . . 5 (d ^ ((a ->2 b) ^ (e v ((a ->2 b) ^ (a ->2 c))))) = (d ^ ((a ->2 b) ^ (a ->2 c)))
95, 8ax-r2 36 . . . 4 ((d ^ (a ->2 b)) ^ (e v ((a ->2 b) ^ (a ->2 c)))) = (d ^ ((a ->2 b) ^ (a ->2 c)))
104, 9ax-r2 36 . . 3 (d ^ (e v ((a ->2 b) ^ (a ->2 c)))) = (d ^ ((a ->2 b) ^ (a ->2 c)))
11 leor 159 . . 3 (d ^ ((a ->2 b) ^ (a ->2 c))) =< ((d ^ e) v (d ^ ((a ->2 b) ^ (a ->2 c))))
1210, 11bltr 138 . 2 (d ^ (e v ((a ->2 b) ^ (a ->2 c)))) =< ((d ^ e) v (d ^ ((a ->2 b) ^ (a ->2 c))))
13 ledi 174 . 2 ((d ^ e) v (d ^ ((a ->2 b) ^ (a ->2 c)))) =< (d ^ (e v ((a ->2 b) ^ (a ->2 c))))
1412, 13lebi 145 1 (d ^ (e v ((a ->2 b) ^ (a ->2 c)))) = ((d ^ e) v (d ^ ((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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