Theorem modid ≪ | index | src | ≫

theorem modid (a: nat): $ a % a = 0 $;

Step	Hyp	Ref	Expression
1		eqtr3	a * 1 % a = a % a -> a * 1 % a = 0 -> a % a = 0
2		modeq1	a * 1 = a -> a * 1 % a = a % a
3		mul12	a * 1 = a
4	2, 3	ax_mp	a * 1 % a = a % a
5	1, 4	ax_mp	a * 1 % a = 0 -> a % a = 0
6		mulmod1	a * 1 % a = 0
7	5, 6	ax_mp	a % a = 0

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)