theorem zmulsub (a b c: nat): $ a *Z (b -Z c) = a *Z b -Z a *Z c $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
eqcom |
a *Z b -Z a *Z c = a *Z (b -Z c) -> a *Z (b -Z c) = a *Z b -Z a *Z c |
| 2 |
|
eqzsub |
a *Z b -Z a *Z c = a *Z (b -Z c) <-> a *Z (b -Z c) +Z a *Z c = a *Z b |
| 3 |
|
eqtr3 |
a *Z (b -Z c +Z c) = a *Z (b -Z c) +Z a *Z c -> a *Z (b -Z c +Z c) = a *Z b -> a *Z (b -Z c) +Z a *Z c = a *Z b |
| 4 |
|
zmuladd |
a *Z (b -Z c +Z c) = a *Z (b -Z c) +Z a *Z c |
| 5 |
3, 4 |
ax_mp |
a *Z (b -Z c +Z c) = a *Z b -> a *Z (b -Z c) +Z a *Z c = a *Z b |
| 6 |
|
zmuleq2 |
b -Z c +Z c = b -> a *Z (b -Z c +Z c) = a *Z b |
| 7 |
|
znpcan |
b -Z c +Z c = b |
| 8 |
6, 7 |
ax_mp |
a *Z (b -Z c +Z c) = a *Z b |
| 9 |
5, 8 |
ax_mp |
a *Z (b -Z c) +Z a *Z c = a *Z b |
| 10 |
2, 9 |
mpbir |
a *Z b -Z a *Z c = a *Z (b -Z c) |
| 11 |
1, 10 |
ax_mp |
a *Z (b -Z c) = a *Z b -Z a *Z 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)