Theorem muladdmod1lt ≪ | index | src | ≫

theorem muladdmod1lt (a b c: nat): $ b != 0 /\ c < b -> (a * b + c) % b = c $;

Step	Hyp	Ref	Expression
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		muladdmod2lt	b != 0 /\ c < b -> (b * a + c) % b = c
7	5, 6	syl5eq	b != 0 /\ c < b -> (a * b + c) % b = c

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)