QLE Home Quantum Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  QLE Home  >  Th. List  >  go1 Unicode version
Theorem go1 343
Description: Lemma for proof of Mayet 8-variable "full" equation from 4-variable Godowski equation.
Assertion
Ref Expression
go1 ((a ^ b) ^ (a ->1 b')) = 0
Proof of Theorem go1
StepHypRef Expression
1 df-i1 44 . . 3 (a ->1 b') = (a' v (a ^ b'))
21lan 77 . 2 ((a ^ b) ^ (a ->1 b')) = ((a ^ b) ^ (a' v (a ^ b')))
3 lear 161 . . . . . 6 (a ^ b') =< b'
43lelor 166 . . . . 5 (a' v (a ^ b')) =< (a' v b')
54lelan 167 . . . 4 ((a ^ b) ^ (a' v (a ^ b'))) =< ((a ^ b) ^ (a' v b'))
6 oran3 93 . . . . . 6 (a' v b') = (a ^ b)'
76lan 77 . . . . 5 ((a ^ b) ^ (a' v b')) = ((a ^ b) ^ (a ^ b)')
8 dff 101 . . . . . 6 0 = ((a ^ b) ^ (a ^ b)')
98ax-r1 35 . . . . 5 ((a ^ b) ^ (a ^ b)') = 0
107, 9ax-r2 36 . . . 4 ((a ^ b) ^ (a' v b')) = 0
115, 10lbtr 139 . . 3 ((a ^ b) ^ (a' v (a ^ b'))) =< 0
12 le0 147 . . 3 0 =< ((a ^ b) ^ (a' v (a ^ b')))
1311, 12lebi 145 . 2 ((a ^ b) ^ (a' v (a ^ b'))) = 0
142, 13ax-r2 36 1 ((a ^ b) ^ (a ->1 b')) = 0
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   v wo 6   ^ wa 7  0wf 9   ->1 wi1 12
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-i1 44  df-le1 130  df-le2 131
This theorem is referenced by:  gomaex4  900
  Copyright terms: Public domain W3C validator