Theorem zeqm01 ≪ | index | src | ≫ 

theorem zeqm01 (a b: nat): $ modZ(0): a = b <-> a = b $;
StepHypRefExpression
1 bitr 
(modZ(0): a = b <-> 0 |Z a -Z b) -> (0 |Z a -Z b <-> a = b) -> (modZ(0): a = b <-> a = b)
2 zdvdeq1 
b0 0 = 0 -> (b0 0 |Z a -Z b <-> 0 |Z a -Z b)
3 2 conv zeqm 
b0 0 = 0 -> (modZ(0): a = b <-> 0 |Z a -Z b)
4 b00 
b0 0 = 0
5 3, 4 ax_mp 
modZ(0): a = b <-> 0 |Z a -Z b
6 1, 5 ax_mp 
(0 |Z a -Z b <-> a = b) -> (modZ(0): a = b <-> a = b)
7 bitr 
(0 |Z a -Z b <-> a -Z b = 0) -> (a -Z b = 0 <-> a = b) -> (0 |Z a -Z b <-> a = b)
8 zdvd01 
0 |Z a -Z b <-> a -Z b = 0
9 7, 8 ax_mp 
(a -Z b = 0 <-> a = b) -> (0 |Z a -Z b <-> a = b)
10 zsubeq0 
a -Z b = 0 <-> a = b
11 9, 10 ax_mp 
0 |Z a -Z b <-> a = b
12 6, 11 ax_mp 
modZ(0): a = b <-> a = b

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)