Theorem eqmdvdsub3 ≪ | index | src | ≫ 

theorem eqmdvdsub3 (a n: nat): $ mod(n): zfst a = zsnd a <-> n || zabs a $;
StepHypRefExpression
1 bitr 
(mod(n): zfst a = zsnd a <-> mod(n): zsnd a = zfst a) -> (mod(n): zsnd a = zfst a <-> n || zabs a) -> (mod(n): zfst a = zsnd a <-> n || zabs a)
2 eqmcomb 
mod(n): zfst a = zsnd a <-> mod(n): zsnd a = zfst a
3 1, 2 ax_mp 
(mod(n): zsnd a = zfst a <-> n || zabs a) -> (mod(n): zfst a = zsnd a <-> n || zabs a)
4 bitr 
(mod(n): zsnd a = zfst a <-> n || zabs (zfst a -ZN zsnd a)) -> (n || zabs (zfst a -ZN zsnd a) <-> n || zabs a) -> (mod(n): zsnd a = zfst a <-> n || zabs a)
5 eqmdvdsub2 
mod(n): zsnd a = zfst a <-> n || zabs (zfst a -ZN zsnd a)
6 4, 5 ax_mp 
(n || zabs (zfst a -ZN zsnd a) <-> n || zabs a) -> (mod(n): zsnd a = zfst a <-> n || zabs a)
7 dvdeq2 
zabs (zfst a -ZN zsnd a) = zabs a -> (n || zabs (zfst a -ZN zsnd a) <-> n || zabs a)
8 zabseq 
zfst a -ZN zsnd a = a -> zabs (zfst a -ZN zsnd a) = zabs a
9 zfstsnd 
zfst a -ZN zsnd a = a
10 8, 9 ax_mp 
zabs (zfst a -ZN zsnd a) = zabs a
11 7, 10 ax_mp 
n || zabs (zfst a -ZN zsnd a) <-> n || zabs a
12 6, 11 ax_mp 
mod(n): zsnd a = zfst a <-> n || zabs a
13 3, 12 ax_mp 
mod(n): zfst a = zsnd a <-> n || zabs 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)