Theorem dvdbgcdb ≪ | index | src | ≫ 

theorem dvdbgcdb (a b d: nat):
  $ a != 0 -> (d || bgcd a b <-> d || a /\ d || b) $;
StepHypRefExpression
1
 
d || bgcd a b -> bgcd a b || a -> d || a
2
 
a != 0 /\ d || bgcd a b -> d || bgcd a b
3
 
bgcd a b || a
4
 
a != 0 /\ d || bgcd a b -> bgcd a b || a
5
1, 2, 4
 
a != 0 /\ d || bgcd a b -> d || a
6
 
d || bgcd a b -> bgcd a b || b -> d || b
7
 
a != 0 -> bgcd a b || b
8
 
a != 0 /\ d || bgcd a b -> bgcd a b || b
9
6, 2, 8
 
a != 0 /\ d || bgcd a b -> d || b
10
5, 9
 
a != 0 /\ d || bgcd a b -> d || a /\ d || b
11
 
a != 0 -> d || bgcd a b -> d || a /\ d || b
12
 
d || a /\ d || b -> d || bgcd a b
13
 
a != 0 -> d || a /\ d || b -> d || bgcd a b
14
 
a != 0 -> (d || bgcd a b <-> d || a /\ d || 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)