QLE Home Quantum Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  QLE Home  >  Th. List  >  oaidlem1 Unicode version
Theorem oaidlem1 294
Description: Lemma for OA identity-like law.
Hypothesis
Ref Expression
oaidlem1.1 (a ^ b) =< c
Assertion
Ref Expression
oaidlem1 (a' v (b ->1 c)) = 1
Proof of Theorem oaidlem1
StepHypRef Expression
1 df-i1 44 . . 3 (b ->1 c) = (b' v (b ^ c))
21lor 70 . 2 (a' v (b ->1 c)) = (a' v (b' v (b ^ c)))
3 oran3 93 . . . 4 (a' v b') = (a ^ b)'
43ax-r5 38 . . 3 ((a' v b') v (b ^ c)) = ((a ^ b)' v (b ^ c))
5 ax-a3 32 . . 3 ((a' v b') v (b ^ c)) = (a' v (b' v (b ^ c)))
6 lear 161 . . . . 5 (a ^ b) =< b
7 oaidlem1.1 . . . . 5 (a ^ b) =< c
86, 7ler2an 173 . . . 4 (a ^ b) =< (b ^ c)
98sklem 230 . . 3 ((a ^ b)' v (b ^ c)) = 1
104, 5, 93tr2 64 . 2 (a' v (b' v (b ^ c))) = 1
112, 10ax-r2 36 1 (a' v (b ->1 c)) = 1
Colors of variables: term
Syntax hints:   = wb 1   =< wle 2  'wn 4   v wo 6   ^ wa 7  1wt 8   ->1 wi1 12
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
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: (None)
  Copyright terms: Public domain W3C validator