Theorem zdvd01 ≪ | index | src | ≫ 

theorem zdvd01 (a: nat): $ 0 |Z a <-> a = 0 $;
StepHypRefExpression
1 bitr3 
(b0 0 |Z a <-> 0 |Z a) -> (b0 0 |Z a <-> a = 0) -> (0 |Z a <-> a = 0)
2 zdvdeq1 
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 <-> a = 0) -> (0 |Z a <-> a = 0)
6 bitr 
(b0 0 |Z a <-> 0 || zabs a) -> (0 || zabs a <-> a = 0) -> (b0 0 |Z a <-> a = 0)
7 zdvdb01 
b0 0 |Z a <-> 0 || zabs a
8 6, 7 ax_mp 
(0 || zabs a <-> a = 0) -> (b0 0 |Z a <-> a = 0)
9 bitr 
(0 || zabs a <-> zabs a = 0) -> (zabs a = 0 <-> a = 0) -> (0 || zabs a <-> a = 0)
10 dvd01 
0 || zabs a <-> zabs a = 0
11 9, 10 ax_mp 
(zabs a = 0 <-> a = 0) -> (0 || zabs a <-> a = 0)
12 zabseq0 
zabs a = 0 <-> a = 0
13 11, 12 ax_mp 
0 || zabs a <-> a = 0
14 8, 13 ax_mp 
b0 0 |Z a <-> a = 0
15 5, 14 ax_mp 
0 |Z a <-> 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)