Theorem zmul2neg | index | src |

theorem zmul2neg (a b: nat): $ -uZ a *Z -uZ b = a *Z b $;
StepHypRefExpression
1 eqtr
-uZ a *Z -uZ b = -uZ (a *Z -uZ b) -> -uZ (a *Z -uZ b) = a *Z b -> -uZ a *Z -uZ b = a *Z b
2 zmulneg1
-uZ a *Z -uZ b = -uZ (a *Z -uZ b)
3 1, 2 ax_mp
-uZ (a *Z -uZ b) = a *Z b -> -uZ a *Z -uZ b = a *Z b
4 znegeqcom
-uZ (a *Z b) = a *Z -uZ b <-> -uZ (a *Z -uZ b) = a *Z b
5 eqcom
a *Z -uZ b = -uZ (a *Z b) -> -uZ (a *Z b) = a *Z -uZ b
6 zmulneg2
a *Z -uZ b = -uZ (a *Z b)
7 5, 6 ax_mp
-uZ (a *Z b) = a *Z -uZ b
8 4, 7 mpbi
-uZ (a *Z -uZ b) = a *Z b
9 3, 8 ax_mp
-uZ a *Z -uZ b = a *Z 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)