Theorem mod01 ≪ | index | src | ≫

theorem mod01 (a: nat): $ 0 % a = 0 $;

Step	Hyp	Ref	Expression
1		mod0	0 % 0 = 0
2		modeq2	a = 0 -> 0 % a = 0 % 0
3	1, 2	syl6eq	a = 0 -> 0 % a = 0
4		bi2	(0 < a <-> ~a = 0) -> ~a = 0 -> 0 < a
5		lt01	0 < a <-> a != 0
6	5	conv ne	0 < a <-> ~a = 0
7	4, 6	ax_mp	~a = 0 -> 0 < a
8		eqtr	a * 0 + 0 = a * 0 -> a * 0 = 0 -> a * 0 + 0 = 0
9		add0	a * 0 + 0 = a * 0
10	8, 9	ax_mp	a * 0 = 0 -> a * 0 + 0 = 0
11		mul0	a * 0 = 0
12	10, 11	ax_mp	a * 0 + 0 = 0
13	12	a1i	~a = 0 -> a * 0 + 0 = 0
14	7, 13	eqdivmod	~a = 0 -> 0 // a = 0 /\ 0 % a = 0
15	14	anrd	~a = 0 -> 0 % a = 0
16	3, 15	cases	0 % 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)