theorem zmulzn (a b c d: nat):
$ (a -ZN c) *Z (b -ZN d) = a * b + c * d -ZN (a * d + b * c) $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
zneqb |
zfst (a -ZN c) * zfst (b -ZN d) + zsnd (a -ZN c) * zsnd (b -ZN d) -ZN (zfst (a -ZN c) * zsnd (b -ZN d) + zsnd (a -ZN c) * zfst (b -ZN d)) =
a * b + c * d -ZN (a * d + b * c) <->
zfst (a -ZN c) * zfst (b -ZN d) + zsnd (a -ZN c) * zsnd (b -ZN d) + (a * d + b * c) =
a * b + c * d + (zfst (a -ZN c) * zsnd (b -ZN d) + zsnd (a -ZN c) * zfst (b -ZN d)) |
| 2 |
1 |
conv zmul |
(a -ZN c) *Z (b -ZN d) = a * b + c * d -ZN (a * d + b * c) <->
zfst (a -ZN c) * zfst (b -ZN d) + zsnd (a -ZN c) * zsnd (b -ZN d) + (a * d + b * c) =
a * b + c * d + (zfst (a -ZN c) * zsnd (b -ZN d) + zsnd (a -ZN c) * zfst (b -ZN d)) |
| 3 |
|
zneqb |
zfst (a -ZN c) -ZN zsnd (a -ZN c) = a -ZN c <-> zfst (a -ZN c) + c = a + zsnd (a -ZN c) |
| 4 |
|
zfstsnd |
zfst (a -ZN c) -ZN zsnd (a -ZN c) = a -ZN c |
| 5 |
3, 4 |
mpbi |
zfst (a -ZN c) + c = a + zsnd (a -ZN c) |
| 6 |
|
zneqb |
zfst (b -ZN d) -ZN zsnd (b -ZN d) = b -ZN d <-> zfst (b -ZN d) + d = b + zsnd (b -ZN d) |
| 7 |
|
zfstsnd |
zfst (b -ZN d) -ZN zsnd (b -ZN d) = b -ZN d |
| 8 |
6, 7 |
mpbi |
zfst (b -ZN d) + d = b + zsnd (b -ZN d) |
| 9 |
5, 8 |
zmulznlem |
zfst (a -ZN c) * zfst (b -ZN d) + zsnd (a -ZN c) * zsnd (b -ZN d) + (a * d + b * c) =
a * b + c * d + (zfst (a -ZN c) * zsnd (b -ZN d) + zsnd (a -ZN c) * zfst (b -ZN d)) |
| 10 |
2, 9 |
mpbir |
(a -ZN c) *Z (b -ZN d) = a * b + c * d -ZN (a * d + b * 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)