theorem znpnpcan2 (a b c: nat): $ a + b -ZN (a + c) = b -ZN c $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
zneqb |
a + b -ZN (a + c) = b -ZN c <-> a + b + c = b + (a + c) |
| 2 |
|
eqtr |
a + b + c = a + (b + c) -> a + (b + c) = b + (a + c) -> a + b + c = b + (a + c) |
| 3 |
|
addass |
a + b + c = a + (b + c) |
| 4 |
2, 3 |
ax_mp |
a + (b + c) = b + (a + c) -> a + b + c = b + (a + c) |
| 5 |
|
addlass |
a + (b + c) = b + (a + c) |
| 6 |
4, 5 |
ax_mp |
a + b + c = b + (a + c) |
| 7 |
1, 6 |
mpbir |
a + b -ZN (a + c) = b -ZN c |
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)