Theorem zleneg0 | index | src |

theorem zleneg0 (a: nat): $ -uZ a <=Z 0 <-> 0 <=Z a $;
StepHypRefExpression
1 noteq
(0 <Z -uZ a <-> a <Z 0) -> (~0 <Z -uZ a <-> ~a <Z 0)
2 1 conv zle
(0 <Z -uZ a <-> a <Z 0) -> (-uZ a <=Z 0 <-> 0 <=Z a)
3 zlt0neg
0 <Z -uZ a <-> a <Z 0
4 2, 3 ax_mp
-uZ a <=Z 0 <-> 0 <=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)