Theorem muladdmod1 | index | src |

theorem muladdmod1 (a b c: nat): $ b != 0 -> (a * b + c) % b = c % b $;
StepHypRefExpression
1 modeq1
a * b + c = b * a + c -> (a * b + c) % b = (b * a + c) % b
2 addeq1
a * b = b * a -> a * b + c = b * a + c
3 mulcom
a * b = b * a
4 2, 3 ax_mp
a * b + c = b * a + c
5 1, 4 ax_mp
(a * b + c) % b = (b * a + c) % b
6 muladdmod2
b != 0 -> (b * a + c) % b = c % b
7 5, 6 syl5eq
b != 0 -> (a * b + c) % b = 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)