QLE Home Quantum Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  QLE Home  >  Th. List  >  oatr Unicode version
Theorem oatr 928
Description: Reverse transformation lemma for studying the orthoarguesian law.
Hypothesis
Ref Expression
oatr.1 b =< (a' ->1 c)
Assertion
Ref Expression
oatr (a' ^ (a v b)) =< c
Proof of Theorem oatr
StepHypRef Expression
1 leo 158 . . . . 5 a =< (a v (a' ^ c))
2 oatr.1 . . . . . 6 b =< (a' ->1 c)
3 df-i1 44 . . . . . . 7 (a' ->1 c) = (a'' v (a' ^ c))
4 ax-a1 30 . . . . . . . . 9 a = a''
54ax-r5 38 . . . . . . . 8 (a v (a' ^ c)) = (a'' v (a' ^ c))
65ax-r1 35 . . . . . . 7 (a'' v (a' ^ c)) = (a v (a' ^ c))
73, 6ax-r2 36 . . . . . 6 (a' ->1 c) = (a v (a' ^ c))
82, 7lbtr 139 . . . . 5 b =< (a v (a' ^ c))
91, 8lel2or 170 . . . 4 (a v b) =< (a v (a' ^ c))
109lelan 167 . . 3 (a' ^ (a v b)) =< (a' ^ (a v (a' ^ c)))
11 omlan 448 . . 3 (a' ^ (a v (a' ^ c))) = (a' ^ c)
1210, 11lbtr 139 . 2 (a' ^ (a v b)) =< (a' ^ c)
13 lear 161 . 2 (a' ^ c) =< c
1412, 13letr 137 1 (a' ^ (a v b)) =< c
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-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38  ax-r3 439
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
This theorem is referenced by:  oa4dtoc  969
  Copyright terms: Public domain W3C validator