theorem znegzn (a b: nat): $ -uZ (a -ZN b) = b -ZN a $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
zneqb |
zsnd (a -ZN b) -ZN zfst (a -ZN b) = b -ZN a <-> zsnd (a -ZN b) + a = b + zfst (a -ZN b) |
| 2 |
1 |
conv zneg |
-uZ (a -ZN b) = b -ZN a <-> zsnd (a -ZN b) + a = b + zfst (a -ZN b) |
| 3 |
|
eqtr |
zsnd (a -ZN b) + a = a + zsnd (a -ZN b) -> a + zsnd (a -ZN b) = b + zfst (a -ZN b) -> zsnd (a -ZN b) + a = b + zfst (a -ZN b) |
| 4 |
|
addcom |
zsnd (a -ZN b) + a = a + zsnd (a -ZN b) |
| 5 |
3, 4 |
ax_mp |
a + zsnd (a -ZN b) = b + zfst (a -ZN b) -> zsnd (a -ZN b) + a = b + zfst (a -ZN b) |
| 6 |
|
eqtr3 |
zfst (a -ZN b) + b = a + zsnd (a -ZN b) -> zfst (a -ZN b) + b = b + zfst (a -ZN b) -> a + zsnd (a -ZN b) = b + zfst (a -ZN b) |
| 7 |
|
zneqb |
zfst (a -ZN b) -ZN zsnd (a -ZN b) = a -ZN b <-> zfst (a -ZN b) + b = a + zsnd (a -ZN b) |
| 8 |
|
zfstsnd |
zfst (a -ZN b) -ZN zsnd (a -ZN b) = a -ZN b |
| 9 |
7, 8 |
mpbi |
zfst (a -ZN b) + b = a + zsnd (a -ZN b) |
| 10 |
6, 9 |
ax_mp |
zfst (a -ZN b) + b = b + zfst (a -ZN b) -> a + zsnd (a -ZN b) = b + zfst (a -ZN b) |
| 11 |
|
addcom |
zfst (a -ZN b) + b = b + zfst (a -ZN b) |
| 12 |
10, 11 |
ax_mp |
a + zsnd (a -ZN b) = b + zfst (a -ZN b) |
| 13 |
5, 12 |
ax_mp |
zsnd (a -ZN b) + a = b + zfst (a -ZN b) |
| 14 |
2, 13 |
mpbir |
-uZ (a -ZN b) = b -ZN 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)