Theorem zlt0sub ≪ | index | src | ≫ 

theorem zlt0sub (a b: nat): $ 0  b 
    

      
        
StepHypRefExpression

        

          1
          
          bitr3
          
(0 +Z b <Z a <-> 0 <Z a -Z b) -> (0 +Z b <Z a <-> b <Z a) -> (0 <Z a -Z b <-> b <Z a)

        

        

          2
          
          zltaddsub
          
0 +Z b <Z a <-> 0 <Z a -Z b

        

        

          3
          1, 2
          ax_mp
          
(0 +Z b <Z a <-> b <Z a) -> (0 <Z a -Z b <-> b <Z a)

        

        

          4
          
          zlteq1
          
0 +Z b = b -> (0 +Z b <Z a <-> b <Z a)

        

        

          5
          
          zadd01
          
0 +Z b = b

        

        

          6
          4, 5
          ax_mp
          
0 +Z b <Z a <-> b <Z a

        

        

          7
          3, 6
          ax_mp
          
0 <Z a -Z b <-> b <Z a

        

      
    

    
Axiom use

    axs_prop_calc
     (ax_1,
      ax_2,
      ax_3,
      ax_mp,
      itru),
    axs_pred_calc
     (ax_gen,
      ax_4,
      ax_5,
      ax_6,
      ax_7,
      ax_10,
      ax_11,
      ax_12),
    axs_set
     (elab,
      ax_8),
    axs_the
     (theid,
      the0),
    axs_peano
     (peano1,
      peano2,
      peano5,
      addeq,
      muleq,
      add0,
      addS,
      mul0,
      mulS)