Theorem dvdzeqm ≪ | index | src | ≫ 

theorem dvdzeqm (G: wff) (a b m n: nat):
  $ G -> m || n $ >
  $ G -> modZ(n): a = b $ >
  $ G -> modZ(m): a = b $;
StepHypRefExpression
1 dvdtr 
zabs (b0 m) || zabs (b0 n) -> zabs (b0 n) || zabs (a -Z b) -> zabs (b0 m) || zabs (a -Z b)
2 1 conv zdvd, zeqm 
zabs (b0 m) || zabs (b0 n) -> zabs (b0 n) || zabs (a -Z b) -> modZ(m): a = b
3 dvdeq 
zabs (b0 m) = m -> zabs (b0 n) = n -> (zabs (b0 m) || zabs (b0 n) <-> m || n)
4 zabsb0 
zabs (b0 m) = m
5 3, 4 ax_mp 
zabs (b0 n) = n -> (zabs (b0 m) || zabs (b0 n) <-> m || n)
6 zabsb0 
zabs (b0 n) = n
7 5, 6 ax_mp 
zabs (b0 m) || zabs (b0 n) <-> m || n
8 hyp h1 
G -> m || n
9 7, 8 sylibr 
G -> zabs (b0 m) || zabs (b0 n)
10 hyp h2 
G -> modZ(n): a = b
11 10 conv zdvd, zeqm 
G -> zabs (b0 n) || zabs (a -Z b)
12 2, 9, 11 sylc 
G -> modZ(m): 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)