Theorem zeqmid | index | src |

theorem zeqmid (a n: nat): $ modZ(n): a = a $;
StepHypRefExpression
1 zdvdeq2
a -Z a = 0 -> (b0 n |Z a -Z a <-> b0 n |Z 0)
2 1 conv zeqm
a -Z a = 0 -> (modZ(n): a = a <-> b0 n |Z 0)
3 zsubid
a -Z a = 0
4 2, 3 ax_mp
modZ(n): a = a <-> b0 n |Z 0
5 zdvd02
b0 n |Z 0
6 4, 5 mpbir
modZ(n): a = 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)