Theorem zeqmadd2 | index | src |

theorem zeqmadd2 (a b c n: nat):
  $ modZ(n): a +Z b = a +Z c <-> modZ(n): b = c $;
StepHypRefExpression
1 zdvdeq2
a +Z b -Z (a +Z c) = b -Z c -> (b0 n |Z a +Z b -Z (a +Z c) <-> b0 n |Z b -Z c)
2 1 conv zeqm
a +Z b -Z (a +Z c) = b -Z c -> (modZ(n): a +Z b = a +Z c <-> modZ(n): b = c)
3 zpnpcan2
a +Z b -Z (a +Z c) = b -Z c
4 2, 3 ax_mp
modZ(n): a +Z b = a +Z c <-> modZ(n): b = c

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)