theorem zaddpos (a b: nat): $ 0 <=Z a -> 0 <=Z b -> 0 <=Z a +Z b $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
zle02eq |
0 <=Z a <-> a = b0 (a // 2) |
| 2 |
|
zle02eq |
0 <=Z b <-> b = b0 (b // 2) |
| 3 |
|
zle0b0 |
0 <=Z b0 (a // 2 + b // 2) |
| 4 |
|
zaddb0 |
b0 (a // 2) +Z b0 (b // 2) = b0 (a // 2 + b // 2) |
| 5 |
|
id |
a = b0 (a // 2) -> a = b0 (a // 2) |
| 6 |
5 |
anwl |
a = b0 (a // 2) /\ b = b0 (b // 2) -> a = b0 (a // 2) |
| 7 |
|
id |
b = b0 (b // 2) -> b = b0 (b // 2) |
| 8 |
7 |
anwr |
a = b0 (a // 2) /\ b = b0 (b // 2) -> b = b0 (b // 2) |
| 9 |
6, 8 |
zaddeqd |
a = b0 (a // 2) /\ b = b0 (b // 2) -> a +Z b = b0 (a // 2) +Z b0 (b // 2) |
| 10 |
4, 9 |
syl6eq |
a = b0 (a // 2) /\ b = b0 (b // 2) -> a +Z b = b0 (a // 2 + b // 2) |
| 11 |
10 |
zleeq2d |
a = b0 (a // 2) /\ b = b0 (b // 2) -> (0 <=Z a +Z b <-> 0 <=Z b0 (a // 2 + b // 2)) |
| 12 |
3, 11 |
mpbiri |
a = b0 (a // 2) /\ b = b0 (b // 2) -> 0 <=Z a +Z b |
| 13 |
12 |
exp |
a = b0 (a // 2) -> b = b0 (b // 2) -> 0 <=Z a +Z b |
| 14 |
2, 13 |
syl5bi |
a = b0 (a // 2) -> 0 <=Z b -> 0 <=Z a +Z b |
| 15 |
1, 14 |
sylbi |
0 <=Z a -> 0 <=Z b -> 0 <=Z a +Z 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)