QLE Home Quantum Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  QLE Home  >  Th. List  >  lem3.4.6 Unicode version
Theorem lem3.4.6 1079
Description: Equation 3.14 of [PavMeg1999] p. 9. (Contributed by Roy F. Longton, 3-Jul-05.)
Hypothesis
Ref Expression
lem3.4.6.1 (a ==5 b) = 1
Assertion
Ref Expression
lem3.4.6 ((a v c) ==5 (b v c)) = 1
Proof of Theorem lem3.4.6
StepHypRef Expression
1 lem3.3.6 1056 . . . 4 (a ->2 (b v c)) = ((a v c) ->2 (b v c))
21ax-r1 35 . . 3 ((a v c) ->2 (b v c)) = (a ->2 (b v c))
3 lem3.4.6.1 . . . 4 (a ==5 b) = 1
43lem3.4.5 1078 . . 3 (a ->2 (b v c)) = 1
52, 4ax-r2 36 . 2 ((a v c) ->2 (b v c)) = 1
6 lem3.3.6 1056 . . . 4 (b ->2 (a v c)) = ((b v c) ->2 (a v c))
76ax-r1 35 . . 3 ((b v c) ->2 (a v c)) = (b ->2 (a v c))
8 df-id5 1047 . . . . 5 (b ==5 a) = ((b ^ a) v (b' ^ a'))
9 ancom 74 . . . . . . 7 (b ^ a) = (a ^ b)
10 ancom 74 . . . . . . 7 (b' ^ a') = (a' ^ b')
119, 102or 72 . . . . . 6 ((b ^ a) v (b' ^ a')) = ((a ^ b) v (a' ^ b'))
12 df-id5 1047 . . . . . . 7 (a ==5 b) = ((a ^ b) v (a' ^ b'))
1312ax-r1 35 . . . . . 6 ((a ^ b) v (a' ^ b')) = (a ==5 b)
1411, 13, 33tr 65 . . . . 5 ((b ^ a) v (b' ^ a')) = 1
158, 14ax-r2 36 . . . 4 (b ==5 a) = 1
1615lem3.4.5 1078 . . 3 (b ->2 (a v c)) = 1
177, 16ax-r2 36 . 2 ((b v c) ->2 (a v c)) = 1
185, 17lem3.4.4 1077 1 ((a v c) ==5 (b v c)) = 1
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   v wo 6   ^ wa 7  1wt 8   ->2 wi2 13   ==5 wid5 22
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-wom 361
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  df-id5 1047  df-b1 1048
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator