theorem zfstznsub (m n: nat): $ zfst (m -ZN n) = m - n $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
zfstb1 |
zfst (b1 (n - suc m)) = 0 |
| 2 |
|
znsubneg |
m < n -> m -ZN n = b1 (n - suc m) |
| 3 |
2 |
zfsteqd |
m < n -> zfst (m -ZN n) = zfst (b1 (n - suc m)) |
| 4 |
1, 3 |
syl6eq |
m < n -> zfst (m -ZN n) = 0 |
| 5 |
|
ltsubeq0 |
m < n -> m - n = 0 |
| 6 |
4, 5 |
eqtr4d |
m < n -> zfst (m -ZN n) = m - n |
| 7 |
|
zfstb0 |
zfst (b0 (m - n)) = m - n |
| 8 |
|
ifneg |
~m < n -> if (m < n) (b1 (n - suc m)) (b0 (m - n)) = b0 (m - n) |
| 9 |
8 |
conv znsub |
~m < n -> m -ZN n = b0 (m - n) |
| 10 |
9 |
zfsteqd |
~m < n -> zfst (m -ZN n) = zfst (b0 (m - n)) |
| 11 |
7, 10 |
syl6eq |
~m < n -> zfst (m -ZN n) = m - n |
| 12 |
6, 11 |
cases |
zfst (m -ZN n) = m - n |
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)