QLE Home Quantum Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  QLE Home  >  Th. List  >  k1-6 Unicode version
Theorem k1-6 353
Description: Statement (6) in proof of Theorem 1 of Kalmbach, Orthomodular Lattices, p. 21.
Hypothesis
Ref Expression
k1-6.1 x = ((x ^ c) v (x ^ c'))
Assertion
Ref Expression
k1-6 (x' ^ c) = ((x' v c') ^ c)
Proof of Theorem k1-6
StepHypRef Expression
1 anor3 90 . . . . 5 ((x ^ c)' ^ (x ^ c')') = ((x ^ c) v (x ^ c'))'
21cm 61 . . . 4 ((x ^ c) v (x ^ c'))' = ((x ^ c)' ^ (x ^ c')')
3 k1-6.1 . . . . 5 x = ((x ^ c) v (x ^ c'))
43con4 69 . . . 4 x' = ((x ^ c) v (x ^ c'))'
5 oran3 93 . . . . 5 (x' v c') = (x ^ c)'
6 oran2 92 . . . . 5 (x' v c) = (x ^ c')'
75, 62an 79 . . . 4 ((x' v c') ^ (x' v c)) = ((x ^ c)' ^ (x ^ c')')
82, 4, 73tr1 63 . . 3 x' = ((x' v c') ^ (x' v c))
98ran 78 . 2 (x' ^ c) = (((x' v c') ^ (x' v c)) ^ c)
10 anass 76 . 2 (((x' v c') ^ (x' v c)) ^ c) = ((x' v c') ^ ((x' v c) ^ c))
11 ancom 74 . . . 4 ((x' v c) ^ c) = (c ^ (x' v c))
12 ax-a2 31 . . . . 5 (x' v c) = (c v x')
1312lan 77 . . . 4 (c ^ (x' v c)) = (c ^ (c v x'))
14 anabs 121 . . . 4 (c ^ (c v x')) = c
1511, 13, 143tr 65 . . 3 ((x' v c) ^ c) = c
1615lan 77 . 2 ((x' v c') ^ ((x' v c) ^ c)) = ((x' v c') ^ c)
179, 10, 163tr 65 1 (x' ^ c) = ((x' v c') ^ c)
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   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
This theorem depends on definitions:  df-a 40
This theorem is referenced by:  k1-8a  355
  Copyright terms: Public domain W3C validator