theorem znsubodd (a b: nat): $ odd (a -ZN b) <-> a < b $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
ax_3 |
(~a < b -> ~odd (a -ZN b)) -> odd (a -ZN b) -> a < b |
| 2 |
|
b0odd |
~odd (b0 (a - b)) |
| 3 |
|
ifneg |
~a < b -> if (a < b) (b1 (b - suc a)) (b0 (a - b)) = b0 (a - b) |
| 4 |
3 |
conv znsub |
~a < b -> a -ZN b = b0 (a - b) |
| 5 |
4 |
oddeqd |
~a < b -> (odd (a -ZN b) <-> odd (b0 (a - b))) |
| 6 |
5 |
noteqd |
~a < b -> (~odd (a -ZN b) <-> ~odd (b0 (a - b))) |
| 7 |
2, 6 |
mpbiri |
~a < b -> ~odd (a -ZN b) |
| 8 |
1, 7 |
ax_mp |
odd (a -ZN b) -> a < b |
| 9 |
|
b1odd |
odd (b1 (b - suc a)) |
| 10 |
|
ifpos |
a < b -> if (a < b) (b1 (b - suc a)) (b0 (a - b)) = b1 (b - suc a) |
| 11 |
10 |
conv znsub |
a < b -> a -ZN b = b1 (b - suc a) |
| 12 |
11 |
oddeqd |
a < b -> (odd (a -ZN b) <-> odd (b1 (b - suc a))) |
| 13 |
9, 12 |
mpbiri |
a < b -> odd (a -ZN b) |
| 14 |
8, 13 |
ibii |
odd (a -ZN b) <-> a < 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,
the0),
axs_peano
(peano1,
peano2,
peano5,
addeq,
muleq,
add0,
addS,
mul0,
mulS)