Theorem divltmul2 | index | src |

theorem divltmul2 (a b c: nat): $ b != 0 -> (a // b < c <-> a < c * b) $;
StepHypRefExpression
1 lteq2
b * c = c * b -> (a < b * c <-> a < c * b)
2 mulcom
b * c = c * b
3 1, 2 ax_mp
a < b * c <-> a < c * b
4 divltmul1
b != 0 -> (a // b < c <-> a < b * c)
5 3, 4 syl6bb
b != 0 -> (a // b < c <-> a < c * 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)