pub theorem modlt (a b: nat): $ b != 0 -> a % b < b $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
divlem3 |
b != 0 -> E. q E. r (r < b /\ b * q + r = a) |
| 2 |
|
anrl |
b != 0 /\ (r < b /\ b * q + r = a) -> r < b |
| 3 |
|
anrr |
b != 0 /\ (r < b /\ b * q + r = a) -> b * q + r = a |
| 4 |
2, 3 |
eqdivmod |
b != 0 /\ (r < b /\ b * q + r = a) -> a // b = q /\ a % b = r |
| 5 |
4 |
anrd |
b != 0 /\ (r < b /\ b * q + r = a) -> a % b = r |
| 6 |
5 |
lteq1d |
b != 0 /\ (r < b /\ b * q + r = a) -> (a % b < b <-> r < b) |
| 7 |
6, 2 |
mpbird |
b != 0 /\ (r < b /\ b * q + r = a) -> a % b < b |
| 8 |
7 |
eexda |
b != 0 -> E. r (r < b /\ b * q + r = a) -> a % b < b |
| 9 |
8 |
eexd |
b != 0 -> E. q E. r (r < b /\ b * q + r = a) -> a % b < b |
| 10 |
1, 9 |
mpd |
b != 0 -> a % b < b |
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),
axs_peano
(peano1,
peano2,
peano5,
addeq,
muleq,
add0,
addS,
mul0,
mulS)