Theorem takenth0 ≪ | index | src | ≫

theorem takenth0 (i l n: nat): $ n <= i -> nth i (take l n) = 0 $;

Step	Hyp	Ref	Expression
1		ntheq0	nth i (take l n) = 0 <-> len (take l n) <= i
2		letr	len (take l n) <= n -> n <= i -> len (take l n) <= i
3		leeq1	len (take l n) = min (len l) n -> (len (take l n) <= n <-> min (len l) n <= n)
4		takelen	len (take l n) = min (len l) n
5	3, 4	ax_mp	len (take l n) <= n <-> min (len l) n <= n
6		minle2	min (len l) n <= n
7	5, 6	mpbir	len (take l n) <= n
8	2, 7	ax_mp	n <= i -> len (take l n) <= i
9	1, 8	sylibr	n <= i -> nth i (take l n) = 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)