theorem zle0znsub (a b: nat): $ 0 <=Z a -ZN b <-> b <= a $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
bitr4 |
(0 <=Z a -ZN b <-> ~a < b) -> (b <= a <-> ~a < b) -> (0 <=Z a -ZN b <-> b <= a) |
| 2 |
|
noteq |
(a -ZN b <Z 0 <-> a < b) -> (~a -ZN b <Z 0 <-> ~a < b) |
| 3 |
2 |
conv zle |
(a -ZN b <Z 0 <-> a < b) -> (0 <=Z a -ZN b <-> ~a < b) |
| 4 |
|
zltznsub0 |
a -ZN b <Z 0 <-> a < b |
| 5 |
3, 4 |
ax_mp |
0 <=Z a -ZN b <-> ~a < b |
| 6 |
1, 5 |
ax_mp |
(b <= a <-> ~a < b) -> (0 <=Z a -ZN b <-> b <= a) |
| 7 |
|
lenlt |
b <= a <-> ~a < b |
| 8 |
6, 7 |
ax_mp |
0 <=Z a -ZN b <-> b <= 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)