Theorem zltneg0 | index | src |

theorem zltneg0 (a: nat): $ -uZ a  0 
    
StepHypRefExpression
1 bitr3
(0 <Z -uZ -uZ a <-> -uZ a <Z 0) -> (0 <Z -uZ -uZ a <-> 0 <Z a) -> (-uZ a <Z 0 <-> 0 <Z a)
2 zlt0neg
0 <Z -uZ -uZ a <-> -uZ a <Z 0
3 1, 2 ax_mp
(0 <Z -uZ -uZ a <-> 0 <Z a) -> (-uZ a <Z 0 <-> 0 <Z a)
4 zlteq2
-uZ -uZ a = a -> (0 <Z -uZ -uZ a <-> 0 <Z a)
5 znegneg
-uZ -uZ a = a
6 4, 5 ax_mp
0 <Z -uZ -uZ a <-> 0 <Z a
7 3, 6 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)