Theorem zeqm03 | index | src |

theorem zeqm03 (a n: nat): $ modZ(n): a = 0 <-> b0 n |Z a $;
StepHypRefExpression
1 zdvdeq2
a -Z 0 = a -> (b0 n |Z a -Z 0 <-> b0 n |Z a)
2 1 conv zeqm
a -Z 0 = a -> (modZ(n): a = 0 <-> b0 n |Z a)
3 zsub02
a -Z 0 = a
4 2, 3 ax_mp
modZ(n): a = 0 <-> b0 n |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)