Theorem zmod01 ≪ | index | src | ≫

theorem zmod01 (a: nat): $ 0 %Z a = 0 $;

Step	Hyp	Ref	Expression
1		eqtr3	b0 0 %Z a = 0 %Z a -> b0 0 %Z a = 0 -> 0 %Z a = 0
2		zmodeq1	b0 0 = 0 -> b0 0 %Z a = 0 %Z a
3		b00	b0 0 = 0
4	2, 3	ax_mp	b0 0 %Z a = 0 %Z a
5	1, 4	ax_mp	b0 0 %Z a = 0 -> 0 %Z a = 0
6		eqtr	b0 0 %Z a = 0 % a -> 0 % a = 0 -> b0 0 %Z a = 0
7		zmodb0	b0 0 %Z a = 0 % a
8	6, 7	ax_mp	0 % a = 0 -> b0 0 %Z a = 0
9		mod01	0 % a = 0
10	8, 9	ax_mp	b0 0 %Z a = 0
11	5, 10	ax_mp	0 %Z 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)