Theorem zltsubadd ≪ | index | src | ≫ 

theorem zltsubadd (a b c: nat): $ a -Z b  a 
    

      
        
StepHypRefExpression

        

          1
          
          bitr2
          
(a <Z c +Z b <-> a +Z -uZ b <Z c +Z b +Z -uZ b) -> (a +Z -uZ b <Z c +Z b +Z -uZ b <-> a -Z b <Z c) -> (a -Z b <Z c <-> a <Z c +Z b)

        

        

          2
          
          zltadd1
          
a <Z c +Z b <-> a +Z -uZ b <Z c +Z b +Z -uZ b

        

        

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

        

        

          4
          
          zlteq2
          
c +Z b +Z -uZ b = c -> (a +Z -uZ b <Z c +Z b +Z -uZ b <-> a +Z -uZ b <Z c)

        

        

          5
          4
          conv zsub
          
c +Z b +Z -uZ b = c -> (a +Z -uZ b <Z c +Z b +Z -uZ b <-> a -Z b <Z c)

        

        

          6
          
          zpncan
          
c +Z b -Z b = c

        

        

          7
          6
          conv zsub
          
c +Z b +Z -uZ b = c

        

        

          8
          5, 7
          ax_mp
          
a +Z -uZ b <Z c +Z b +Z -uZ b <-> a -Z b <Z c

        

        

          9
          3, 8
          ax_mp
          
a -Z b <Z c <-> a <Z c +Z b

        

      
    

    
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)