QLE Home Quantum Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  QLE Home  >  Th. List  >  lel Unicode version
Theorem lel 151
Description: Add conjunct to left of l.e.
Hypothesis
Ref Expression
le.1 a =< b
Assertion
Ref Expression
lel (a ^ c) =< b
Proof of Theorem lel
StepHypRef Expression
1 an32 83 . . 3 ((a ^ c) ^ b) = ((a ^ b) ^ c)
2 le.1 . . . . 5 a =< b
32df2le2 136 . . . 4 (a ^ b) = a
43ran 78 . . 3 ((a ^ b) ^ c) = (a ^ c)
51, 4ax-r2 36 . 2 ((a ^ c) ^ b) = (a ^ c)
65df2le1 135 1 (a ^ c) =< b
Colors of variables: term
Syntax hints:   =< wle 2   ^ wa 7
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
This theorem depends on definitions:  df-a 40  df-le1 130  df-le2 131
This theorem is referenced by:  negantlem9  859  neg3antlem2  865  marsdenlem3  882  cancellem  891  oa3moa3  1029  lem3.4.3  1076
  Copyright terms: Public domain W3C validator