Theorem div12 ≪ | index | src | ≫ 

theorem div12 (a: nat): $ a // 1 = a $;
StepHypRefExpression
1 eqtr3 
1 * (a // 1) + a % 1 = a // 1 -> 1 * (a // 1) + a % 1 = a -> a // 1 = a
2 eqtr 
1 * (a // 1) + a % 1 = a // 1 + 0 -> a // 1 + 0 = a // 1 -> 1 * (a // 1) + a % 1 = a // 1
3 addeq 
1 * (a // 1) = a // 1 -> a % 1 = 0 -> 1 * (a // 1) + a % 1 = a // 1 + 0
4 mul11 
1 * (a // 1) = a // 1
5 3, 4 ax_mp 
a % 1 = 0 -> 1 * (a // 1) + a % 1 = a // 1 + 0
6 mod12 
a % 1 = 0
7 5, 6 ax_mp 
1 * (a // 1) + a % 1 = a // 1 + 0
8 2, 7 ax_mp 
a // 1 + 0 = a // 1 -> 1 * (a // 1) + a % 1 = a // 1
9 add02 
a // 1 + 0 = a // 1
10 8, 9 ax_mp 
1 * (a // 1) + a % 1 = a // 1
11 1, 10 ax_mp 
1 * (a // 1) + a % 1 = a -> a // 1 = a
12 divmod 
1 * (a // 1) + a % 1 = a
13 11, 12 ax_mp 
a // 1 = 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)