Theorem zeqmneg | index | src |

theorem zeqmneg (a b n: nat): $ modZ(n): -uZ a = -uZ b <-> modZ(n): a = b $;
StepHypRefExpression
1 bitr
(modZ(n): -uZ a = -uZ b <-> modZ(n): -uZ b = -uZ a) -> (modZ(n): -uZ b = -uZ a <-> modZ(n): a = b) -> (modZ(n): -uZ a = -uZ b <-> modZ(n): a = b)
2 zeqmcomb
modZ(n): -uZ a = -uZ b <-> modZ(n): -uZ b = -uZ a
3 1, 2 ax_mp
(modZ(n): -uZ b = -uZ a <-> modZ(n): a = b) -> (modZ(n): -uZ a = -uZ b <-> modZ(n): a = b)
4 zdvdeq2
-uZ b -Z -uZ a = a -Z b -> (b0 n |Z -uZ b -Z -uZ a <-> b0 n |Z a -Z b)
5 4 conv zeqm
-uZ b -Z -uZ a = a -Z b -> (modZ(n): -uZ b = -uZ a <-> modZ(n): a = b)
6 znegsub2
-uZ b -Z -uZ a = a -Z b
7 5, 6 ax_mp
modZ(n): -uZ b = -uZ a <-> modZ(n): a = b
8 3, 7 ax_mp
modZ(n): -uZ a = -uZ b <-> modZ(n): a = 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)