Theorem mod0 ≪ | index | src | ≫

theorem mod0 (a: nat): $ a % 0 = a $;

Step	Hyp	Ref	Expression
1		eqtr	a % 0 = a - 0 -> a - 0 = a -> a % 0 = a
2		subeq2	0 * (a // 0) = 0 -> a - 0 * (a // 0) = a - 0
3	2	conv mod	0 * (a // 0) = 0 -> a % 0 = a - 0
4		mul01	0 * (a // 0) = 0
5	3, 4	ax_mp	a % 0 = a - 0
6	1, 5	ax_mp	a - 0 = a -> a % 0 = a
7		sub02	a - 0 = a
8	6, 7	ax_mp	a % 0 = a

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 (peano2, peano5, addeq, muleq, add0, addS, mul0, mulS)