QLE Home Quantum Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  QLE Home  >  Th. List  >  oacom Unicode version
Theorem oacom 1011
Description: Commutation law requiring OA.
Hypotheses
Ref Expression
oacom.1 d C ((b v c) ->0 ((a ->2 b) ^ (a ->2 c)))
oacom.2 (d ^ ((b v c) ->0 ((a ->2 b) ^ (a ->2 c)))) C (a ->2 b)
Assertion
Ref Expression
oacom d C ((a ->2 b) ^ (a ->2 c))
Proof of Theorem oacom
StepHypRef Expression
1 oacom.1 . . . . 5 d C ((b v c) ->0 ((a ->2 b) ^ (a ->2 c)))
21comcom 453 . . . 4 ((b v c) ->0 ((a ->2 b) ^ (a ->2 c))) C d
3 ancom 74 . . . . . 6 (((b v c) ->0 ((a ->2 b) ^ (a ->2 c))) ^ d) = (d ^ ((b v c) ->0 ((a ->2 b) ^ (a ->2 c))))
4 oacom.2 . . . . . 6 (d ^ ((b v c) ->0 ((a ->2 b) ^ (a ->2 c)))) C (a ->2 b)
53, 4bctr 181 . . . . 5 (((b v c) ->0 ((a ->2 b) ^ (a ->2 c))) ^ d) C (a ->2 b)
65comcom 453 . . . 4 (a ->2 b) C (((b v c) ->0 ((a ->2 b) ^ (a ->2 c))) ^ d)
72, 6gsth2 490 . . 3 ((a ->2 b) ^ ((b v c) ->0 ((a ->2 b) ^ (a ->2 c)))) C d
87comcom 453 . 2 d C ((a ->2 b) ^ ((b v c) ->0 ((a ->2 b) ^ (a ->2 c))))
9 df-i0 43 . . . 4 ((b v c) ->0 ((a ->2 b) ^ (a ->2 c))) = ((b v c)' v ((a ->2 b) ^ (a ->2 c)))
109lan 77 . . 3 ((a ->2 b) ^ ((b v c) ->0 ((a ->2 b) ^ (a ->2 c)))) = ((a ->2 b) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c))))
11 oath1 1004 . . 3 ((a ->2 b) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c)))) = ((a ->2 b) ^ (a ->2 c))
1210, 11ax-r2 36 . 2 ((a ->2 b) ^ ((b v c) ->0 ((a ->2 b) ^ (a ->2 c)))) = ((a ->2 b) ^ (a ->2 c))
138, 12cbtr 182 1 d C ((a ->2 b) ^ (a ->2 c))
Colors of variables: term
Syntax hints:   C wc 3  '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-a4 33  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38  ax-r3 439  ax-3oa 998
This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i0 43  df-i1 44  df-i2 45  df-le1 130  df-le2 131  df-c1 132  df-c2 133
This theorem is referenced by:  oacom2  1012
  Copyright terms: Public domain W3C validator