Theorem divleid ≪ | index | src | ≫ 

theorem divleid (a b: nat): $ a // b <= a $;
StepHypRefExpression
1 le01 
0 <= a
2 div0 
a // 0 = 0
3 diveq2 
b = 0 -> a // b = a // 0
4 2, 3 syl6eq 
b = 0 -> a // b = 0
5 4 leeq1d 
b = 0 -> (a // b <= a <-> 0 <= a)
6 1, 5 mpbiri 
b = 0 -> a // b <= a
7 leeq1 
1 * (a // b) = a // b -> (1 * (a // b) <= b * (a // b) <-> a // b <= b * (a // b))
8 mul11 
1 * (a // b) = a // b
9 7, 8 ax_mp 
1 * (a // b) <= b * (a // b) <-> a // b <= b * (a // b)
10 le11 
1 <= b <-> b != 0
11 10 conv ne 
1 <= b <-> ~b = 0
12 lemul1a 
1 <= b -> 1 * (a // b) <= b * (a // b)
13 11, 12 sylbir 
~b = 0 -> 1 * (a // b) <= b * (a // b)
14 9, 13 sylib 
~b = 0 -> a // b <= b * (a // b)
15 muldivle 
b * (a // b) <= a
16 15 a1i 
~b = 0 -> b * (a // b) <= a
17 14, 16 letrd 
~b = 0 -> a // b <= a
18 6, 17 cases 
a // b <= 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)