QLE Home Quantum Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  QLE Home  >  Th. List  >  axoa4a Unicode version
Theorem axoa4a 1037
Description: Proper 4-variable OA law variant.
Assertion
Ref Expression
axoa4a ((a ->1 d) ^ (a v (b ^ (((a ^ b) v ((a ->1 d) ^ (b ->1 d))) v (((a ^ c) v ((a ->1 d) ^ (c ->1 d))) ^ ((b ^ c) v ((b ->1 d) ^ (c ->1 d)))))))) =< (((a ^ d) v (b ^ d)) v (c ^ d))
Proof of Theorem axoa4a
StepHypRef Expression
1 id 59 . 2 (a ->1 d)' = (a ->1 d)'
2 id 59 . 2 (b ->1 d)' = (b ->1 d)'
3 id 59 . 2 (c ->1 d)' = (c ->1 d)'
4 leo 158 . . . 4 a' =< (a' v (a ^ d))
5 df-i1 44 . . . . . 6 (a ->1 d) = (a' v (a ^ d))
65ax-r1 35 . . . . 5 (a' v (a ^ d)) = (a ->1 d)
7 ax-a1 30 . . . . 5 (a ->1 d) = (a ->1 d)''
86, 7ax-r2 36 . . . 4 (a' v (a ^ d)) = (a ->1 d)''
94, 8lbtr 139 . . 3 a' =< (a ->1 d)''
10 leo 158 . . . 4 b' =< (b' v (b ^ d))
11 df-i1 44 . . . . . 6 (b ->1 d) = (b' v (b ^ d))
1211ax-r1 35 . . . . 5 (b' v (b ^ d)) = (b ->1 d)
13 ax-a1 30 . . . . 5 (b ->1 d) = (b ->1 d)''
1412, 13ax-r2 36 . . . 4 (b' v (b ^ d)) = (b ->1 d)''
1510, 14lbtr 139 . . 3 b' =< (b ->1 d)''
16 leo 158 . . . 4 c' =< (c' v (c ^ d))
17 df-i1 44 . . . . . 6 (c ->1 d) = (c' v (c ^ d))
1817ax-r1 35 . . . . 5 (c' v (c ^ d)) = (c ->1 d)
19 ax-a1 30 . . . . 5 (c ->1 d) = (c ->1 d)''
2018, 19ax-r2 36 . . . 4 (c' v (c ^ d)) = (c ->1 d)''
2116, 20lbtr 139 . . 3 c' =< (c ->1 d)''
229, 15, 21oa6 1036 . 2 (((a' v (a ->1 d)') ^ (b' v (b ->1 d)')) ^ (c' v (c ->1 d)')) =< ((a ->1 d)' v (a' ^ (b' v (((a' v b') ^ ((a ->1 d)' v (b ->1 d)')) ^ (((a' v c') ^ ((a ->1 d)' v (c ->1 d)')) v ((b' v c') ^ ((b ->1 d)' v (c ->1 d)')))))))
231, 2, 3, 22oa6to4 958 1 ((a ->1 d) ^ (a v (b ^ (((a ^ b) v ((a ->1 d) ^ (b ->1 d))) v (((a ^ c) v ((a ->1 d) ^ (c ->1 d))) ^ ((b ^ c) v ((b ->1 d) ^ (c ->1 d)))))))) =< (((a ^ d) v (b ^ d)) v (c ^ d))
Colors of variables: term
Syntax hints:   =< wle 2  'wn 4   v wo 6   ^ wa 7   ->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  ax-r3 439  ax-4oa 1033
This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i1 44  df-le1 130  df-le2 131  df-c1 132  df-c2 133
This theorem is referenced by:  4oa  1039
  Copyright terms: Public domain W3C validator