QLE Home Quantum Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  QLE Home  >  Th. List  >  lem3.3.7i5e1 Unicode version
Theorem lem3.3.7i5e1 1072
Description: Equation 3.7 of [PavMeg1999] p. 9. The variable i in the paper is set to 5, and this is the first part of the equation. (Contributed by Roy F. Longton, 3-Jul-05.)
Assertion
Ref Expression
lem3.3.7i5e1 (a ->5 (a ^ b)) = (a ==5 (a ^ b))
Proof of Theorem lem3.3.7i5e1
StepHypRef Expression
1 lear 161 . . . . . 6 (a ^ (a ^ b)) =< (a ^ b)
2 lea 160 . . . . . . 7 (a ^ b) =< a
3 leid 148 . . . . . . 7 (a ^ b) =< (a ^ b)
42, 3ler2an 173 . . . . . 6 (a ^ b) =< (a ^ (a ^ b))
51, 4lebi 145 . . . . 5 (a ^ (a ^ b)) = (a ^ b)
62lecon 154 . . . . . 6 a' =< (a ^ b)'
76ortha 438 . . . . 5 (a' ^ (a ^ b)) = 0
85, 72or 72 . . . 4 ((a ^ (a ^ b)) v (a' ^ (a ^ b))) = ((a ^ b) v 0)
98ax-r5 38 . . 3 (((a ^ (a ^ b)) v (a' ^ (a ^ b))) v (a' ^ (a ^ b)')) = (((a ^ b) v 0) v (a' ^ (a ^ b)'))
10 or0 102 . . . 4 ((a ^ b) v 0) = (a ^ b)
116df2le2 136 . . . 4 (a' ^ (a ^ b)') = a'
1210, 112or 72 . . 3 (((a ^ b) v 0) v (a' ^ (a ^ b)')) = ((a ^ b) v a')
134, 1lebi 145 . . . 4 (a ^ b) = (a ^ (a ^ b))
1411ax-r1 35 . . . 4 a' = (a' ^ (a ^ b)')
1513, 142or 72 . . 3 ((a ^ b) v a') = ((a ^ (a ^ b)) v (a' ^ (a ^ b)'))
169, 12, 153tr 65 . 2 (((a ^ (a ^ b)) v (a' ^ (a ^ b))) v (a' ^ (a ^ b)')) = ((a ^ (a ^ b)) v (a' ^ (a ^ b)'))
17 df-i5 48 . 2 (a ->5 (a ^ b)) = (((a ^ (a ^ b)) v (a' ^ (a ^ b))) v (a' ^ (a ^ b)'))
18 df-id5 1047 . 2 (a ==5 (a ^ b)) = ((a ^ (a ^ b)) v (a' ^ (a ^ b)'))
1916, 17, 183tr1 63 1 (a ->5 (a ^ b)) = (a ==5 (a ^ b))
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   v wo 6   ^ wa 7  0wf 9   ->5 wi5 16   ==5 wid5 22
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  df-t 41  df-f 42  df-i5 48  df-le1 130  df-le2 131  df-id5 1047
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator