QLE Home Quantum Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  QLE Home  >  Th. List  >  oadistd Unicode version

Theorem oadistd 1023
Description: OA distributive law.
Hypotheses
Ref Expression
oadistd.1 d =< (a ->2 b)
oadistd.2 e =< ((b v c) ->0 ((a ->2 b) ^ (a ->2 c)))
oadistd.3 f =< ((b v c) ->0 ((a ->2 b) ^ (a ->2 c)))
oadistd.4 (d ^ (a ->2 c)) =< f
Assertion
Ref Expression
oadistd (d ^ (e v f)) = ((d ^ e) v (d ^ f))

Proof of Theorem oadistd
StepHypRef Expression
1 oadistd.2 . . . . . . . . . 10 e =< ((b v c) ->0 ((a ->2 b) ^ (a ->2 c)))
2 oadistd.3 . . . . . . . . . 10 f =< ((b v c) ->0 ((a ->2 b) ^ (a ->2 c)))
31, 2le2or 168 . . . . . . . . 9 (e v f) =< (((b v c) ->0 ((a ->2 b) ^ (a ->2 c))) v ((b v c) ->0 ((a ->2 b) ^ (a ->2 c))))
4 oridm 110 . . . . . . . . 9 (((b v c) ->0 ((a ->2 b) ^ (a ->2 c))) v ((b v c) ->0 ((a ->2 b) ^ (a ->2 c)))) = ((b v c) ->0 ((a ->2 b) ^ (a ->2 c)))
53, 4lbtr 139 . . . . . . . 8 (e v f) =< ((b v c) ->0 ((a ->2 b) ^ (a ->2 c)))
65lelan 167 . . . . . . 7 (d ^ (e v f)) =< (d ^ ((b v c) ->0 ((a ->2 b) ^ (a ->2 c))))
76df2le2 136 . . . . . 6 ((d ^ (e v f)) ^ (d ^ ((b v c) ->0 ((a ->2 b) ^ (a ->2 c))))) = (d ^ (e v f))
87ax-r1 35 . . . . 5 (d ^ (e v f)) = ((d ^ (e v f)) ^ (d ^ ((b v c) ->0 ((a ->2 b) ^ (a ->2 c)))))
9 df-i0 43 . . . . . . . 8 ((b v c) ->0 ((a ->2 b) ^ (a ->2 c))) = ((b v c)' v ((a ->2 b) ^ (a ->2 c)))
109lan 77 . . . . . . 7 (d ^ ((b v c) ->0 ((a ->2 b) ^ (a ->2 c)))) = (d ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c))))
11 oadistd.1 . . . . . . . 8 d =< (a ->2 b)
12 leo 158 . . . . . . . . 9 (b v c)' =< ((b v c)' v ((a ->2 b) ^ (a ->2 c)))
139ax-r1 35 . . . . . . . . 9 ((b v c)' v ((a ->2 b) ^ (a ->2 c))) = ((b v c) ->0 ((a ->2 b) ^ (a ->2 c)))
1412, 13lbtr 139 . . . . . . . 8 (b v c)' =< ((b v c) ->0 ((a ->2 b) ^ (a ->2 c)))
1511, 14oagen1b 1015 . . . . . . 7 (d ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c)))) = (d ^ (a ->2 c))
1610, 15ax-r2 36 . . . . . 6 (d ^ ((b v c) ->0 ((a ->2 b) ^ (a ->2 c)))) = (d ^ (a ->2 c))
1716lan 77 . . . . 5 ((d ^ (e v f)) ^ (d ^ ((b v c) ->0 ((a ->2 b) ^ (a ->2 c))))) = ((d ^ (e v f)) ^ (d ^ (a ->2 c)))
188, 17ax-r2 36 . . . 4 (d ^ (e v f)) = ((d ^ (e v f)) ^ (d ^ (a ->2 c)))
19 lear 161 . . . . 5 ((d ^ (e v f)) ^ (d ^ (a ->2 c))) =< (d ^ (a ->2 c))
20 oadistd.4 . . . . . . . . 9 (d ^ (a ->2 c)) =< f
2120df2le2 136 . . . . . . . 8 ((d ^ (a ->2 c)) ^ f) = (d ^ (a ->2 c))
2221ax-r1 35 . . . . . . 7 (d ^ (a ->2 c)) = ((d ^ (a ->2 c)) ^ f)
23 an32 83 . . . . . . 7 ((d ^ (a ->2 c)) ^ f) = ((d ^ f) ^ (a ->2 c))
2422, 23ax-r2 36 . . . . . 6 (d ^ (a ->2 c)) = ((d ^ f) ^ (a ->2 c))
25 lea 160 . . . . . 6 ((d ^ f) ^ (a ->2 c)) =< (d ^ f)
2624, 25bltr 138 . . . . 5 (d ^ (a ->2 c)) =< (d ^ f)
2719, 26letr 137 . . . 4 ((d ^ (e v f)) ^ (d ^ (a ->2 c))) =< (d ^ f)
2818, 27bltr 138 . . 3 (d ^ (e v f)) =< (d ^ f)
29 leor 159 . . 3 (d ^ f) =< ((d ^ e) v (d ^ f))
3028, 29letr 137 . 2 (d ^ (e v f)) =< ((d ^ e) v (d ^ f))
31 ledi 174 . 2 ((d ^ e) v (d ^ f)) =< (d ^ (e v f))
3230, 31lebi 145 1 (d ^ (e v f)) = ((d ^ e) v (d ^ f))
Colors of variables: term
Syntax hints:   = wb 1   =< wle 2  'wn 4   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
  Copyright terms: Public domain W3C validator