QLE Home Quantum Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  QLE Home  >  Th. List  >  u1lemle2 Unicode version
Theorem u1lemle2 715
Description: Sasaki implication to l.e.
Hypothesis
Ref Expression
u1lemle2.1 (a ->1 b) = 1
Assertion
Ref Expression
u1lemle2 a =< b
Proof of Theorem u1lemle2
StepHypRef Expression
1 anidm 111 . . . . . . . . 9 (a ^ a) = a
21ran 78 . . . . . . . 8 ((a ^ a) ^ b) = (a ^ b)
32ax-r1 35 . . . . . . 7 (a ^ b) = ((a ^ a) ^ b)
4 anass 76 . . . . . . 7 ((a ^ a) ^ b) = (a ^ (a ^ b))
53, 4ax-r2 36 . . . . . 6 (a ^ b) = (a ^ (a ^ b))
6 dff 101 . . . . . 6 0 = (a ^ a')
75, 62or 72 . . . . 5 ((a ^ b) v 0) = ((a ^ (a ^ b)) v (a ^ a'))
8 ax-a2 31 . . . . . . . 8 (a' v (a ^ b)) = ((a ^ b) v a')
98lan 77 . . . . . . 7 (a ^ (a' v (a ^ b))) = (a ^ ((a ^ b) v a'))
10 coman1 185 . . . . . . . 8 (a ^ b) C a
1110comcom2 183 . . . . . . . 8 (a ^ b) C a'
1210, 11fh2 470 . . . . . . 7 (a ^ ((a ^ b) v a')) = ((a ^ (a ^ b)) v (a ^ a'))
139, 12ax-r2 36 . . . . . 6 (a ^ (a' v (a ^ b))) = ((a ^ (a ^ b)) v (a ^ a'))
1413ax-r1 35 . . . . 5 ((a ^ (a ^ b)) v (a ^ a')) = (a ^ (a' v (a ^ b)))
157, 14ax-r2 36 . . . 4 ((a ^ b) v 0) = (a ^ (a' v (a ^ b)))
16 df-i1 44 . . . . . . 7 (a ->1 b) = (a' v (a ^ b))
1716ax-r1 35 . . . . . 6 (a' v (a ^ b)) = (a ->1 b)
18 u1lemle2.1 . . . . . 6 (a ->1 b) = 1
1917, 18ax-r2 36 . . . . 5 (a' v (a ^ b)) = 1
2019lan 77 . . . 4 (a ^ (a' v (a ^ b))) = (a ^ 1)
2115, 20ax-r2 36 . . 3 ((a ^ b) v 0) = (a ^ 1)
22 or0 102 . . 3 ((a ^ b) v 0) = (a ^ b)
23 an1 106 . . 3 (a ^ 1) = a
2421, 22, 233tr2 64 . 2 (a ^ b) = a
2524df2le1 135 1 a =< b
Colors of variables: term
Syntax hints:   = wb 1   =< wle 2  'wn 4   v wo 6   ^ wa 7  1wt 8  0wf 9   ->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
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:  3vded11  814  3vded12  815
  Copyright terms: Public domain W3C validator