Theorem dvd12 ≪ | index | src | ≫ 

theorem dvd12 (a: nat): $ a || 1 <-> a = 1 $;
StepHypRefExpression
1 d1ne0 
1 != 0
2 1 a1i 
a || 1 -> 1 != 0
3 id 
a || 1 -> a || 1
4 2, 3 dvdle 
a || 1 -> a <= 1
5 le11 
1 <= a <-> a != 0
6 1 conv ne 
~1 = 0
7 dvd01 
0 || 1 <-> 1 = 0
8 dvdeq1 
a = 0 -> (a || 1 <-> 0 || 1)
9 8 anwr 
a || 1 /\ a = 0 -> (a || 1 <-> 0 || 1)
10 anl 
a || 1 /\ a = 0 -> a || 1
11 9, 10 mpbid 
a || 1 /\ a = 0 -> 0 || 1
12 7, 11 sylib 
a || 1 /\ a = 0 -> 1 = 0
13 6, 12 mtani 
a || 1 -> ~a = 0
14 13 conv ne 
a || 1 -> a != 0
15 5, 14 sylibr 
a || 1 -> 1 <= a
16 4, 15 leasymd 
a || 1 -> a = 1
17 dvdid 
1 || 1
18 dvdeq1 
a = 1 -> (a || 1 <-> 1 || 1)
19 17, 18 mpbiri 
a = 1 -> a || 1
20 16, 19 ibii 
a || 1 <-> a = 1

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_peano (peano1, peano2, peano5, addeq, muleq, add0, addS, mul0, mulS)