Theorem zltaddsub | index | src |

theorem zltaddsub (a b c: nat): $ a +Z b  a 
    
StepHypRefExpression
1 bitr2
(a <Z c -Z b <-> a +Z b <Z c -Z b +Z b) -> (a +Z b <Z c -Z b +Z 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 b <Z c -Z b +Z b
3 1, 2 ax_mp
(a +Z b <Z c -Z b +Z b <-> a +Z b <Z c) -> (a +Z b <Z c <-> a <Z c -Z b)
4 zlteq2
c -Z b +Z b = c -> (a +Z b <Z c -Z b +Z b <-> a +Z b <Z c)
5 znpcan
c -Z b +Z b = c
6 4, 5 ax_mp
a +Z b <Z c -Z b +Z b <-> a +Z b <Z c
7 3, 6 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)