Theorem zsndeq0 ≪ | index | src | ≫

theorem zsndeq0 (a: nat): $ zsnd a = 0 <-> 0 <=Z a $;

Step	Hyp	Ref	Expression
1		zle0b0	0 <=Z b0 (zfst a)
2		znsub02	zfst a -ZN 0 = b0 (zfst a)
3		zfstsnd	zfst a -ZN zsnd a = a
4		znsubeq2	zsnd a = 0 -> zfst a -ZN zsnd a = zfst a -ZN 0
5	3, 4	syl5eqr	zsnd a = 0 -> a = zfst a -ZN 0
6	2, 5	syl6eq	zsnd a = 0 -> a = b0 (zfst a)
7	6	zleeq2d	zsnd a = 0 -> (0 <=Z a <-> 0 <=Z b0 (zfst a))
8	1, 7	mpbiri	zsnd a = 0 -> 0 <=Z a
9		zsndb0	zsnd (b0 (a // 2)) = 0
10		zle02eq	0 <=Z a <-> a = b0 (a // 2)
11		zsndeq	a = b0 (a // 2) -> zsnd a = zsnd (b0 (a // 2))
12	10, 11	sylbi	0 <=Z a -> zsnd a = zsnd (b0 (a // 2))
13	9, 12	syl6eq	0 <=Z a -> zsnd a = 0
14	8, 13	ibii	zsnd a = 0 <-> 0 <=Z 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 (peano1, peano2, peano5, addeq, muleq, add0, addS, mul0, mulS)