Theorem dvdbgcd ≪ | index | src | ≫ 

theorem dvdbgcd (a b d: nat): $ d || a /\ d || b -> d || bgcd a b $;
StepHypRefExpression
1 bgcdbezout 
E. x E. y x * a = y * b + bgcd a b
2 dvdadd1 
d || y * b -> (d || bgcd a b <-> d || y * b + bgcd a b)
3 dvdmul12 
d || b -> d || y * b
4 anlr 
d || a /\ d || b /\ x * a = y * b + bgcd a b -> d || b
5 3, 4 syl 
d || a /\ d || b /\ x * a = y * b + bgcd a b -> d || y * b
6 2, 5 syl 
d || a /\ d || b /\ x * a = y * b + bgcd a b -> (d || bgcd a b <-> d || y * b + bgcd a b)
7 dvdeq2 
x * a = y * b + bgcd a b -> (d || x * a <-> d || y * b + bgcd a b)
8 7 anwr 
d || a /\ d || b /\ x * a = y * b + bgcd a b -> (d || x * a <-> d || y * b + bgcd a b)
9 dvdmul12 
d || a -> d || x * a
10 9 anwll 
d || a /\ d || b /\ x * a = y * b + bgcd a b -> d || x * a
11 8, 10 mpbid 
d || a /\ d || b /\ x * a = y * b + bgcd a b -> d || y * b + bgcd a b
12 6, 11 mpbird 
d || a /\ d || b /\ x * a = y * b + bgcd a b -> d || bgcd a b
13 12 eexda 
d || a /\ d || b -> E. y x * a = y * b + bgcd a b -> d || bgcd a b
14 13 eexd 
d || a /\ d || b -> E. x E. y x * a = y * b + bgcd a b -> d || bgcd a b
15 1, 14 mpi 
d || a /\ d || b -> d || bgcd 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)