QLE Home Quantum Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  QLE Home  >  Th. List  >  oa6 Unicode version

Theorem oa6 1036
Description: Derivation of 6-variable orthoarguesian law from 4-variable version.
Hypotheses
Ref Expression
oa6.1 a =< b'
oa6.2 c =< d'
oa6.3 e =< f'
Assertion
Ref Expression
oa6 (((a v b) ^ (c v d)) ^ (e v f)) =< (b v (a ^ (c v (((a v c) ^ (b v d)) ^ (((a v e) ^ (b v f)) v ((c v e) ^ (d v f)))))))

Proof of Theorem oa6
StepHypRef Expression
1 oa6.1 . 2 a =< b'
2 oa6.2 . 2 c =< d'
3 oa6.3 . 2 e =< f'
4 id 59 . 2 (((a' ^ b') v (c' ^ d')) v (e' ^ f')) = (((a' ^ b') v (c' ^ d')) v (e' ^ f'))
5 id 59 . 2 a' = a'
6 id 59 . 2 c' = c'
7 id 59 . 2 e' = e'
8 axoa4b 1035 . 2 ((a' ->1 (((a' ^ b') v (c' ^ d')) v (e' ^ f'))) ^ (a' v (c' ^ (((a' ^ c') v ((a' ->1 (((a' ^ b') v (c' ^ d')) v (e' ^ f'))) ^ (c' ->1 (((a' ^ b') v (c' ^ d')) v (e' ^ f'))))) v (((a' ^ e') v ((a' ->1 (((a' ^ b') v (c' ^ d')) v (e' ^ f'))) ^ (e' ->1 (((a' ^ b') v (c' ^ d')) v (e' ^ f'))))) ^ ((c' ^ e') v ((c' ->1 (((a' ^ b') v (c' ^ d')) v (e' ^ f'))) ^ (e' ->1 (((a' ^ b') v (c' ^ d')) v (e' ^ f')))))))))) =< (((a' ^ b') v (c' ^ d')) v (e' ^ f'))
91, 2, 3, 4, 5, 6, 7, 8oa4to6 965 1 (((a v b) ^ (c v d)) ^ (e v f)) =< (b v (a ^ (c v (((a v c) ^ (b v d)) ^ (((a v e) ^ (b v f)) v ((c v e) ^ (d v f)))))))
Colors of variables: term
Syntax hints:   =< wle 2  'wn 4   v wo 6   ^ wa 7
This theorem is referenced by:  axoa4a  1037
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
  Copyright terms: Public domain W3C validator