theorem add01 (a: nat): $ 0 + a = a $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
addeq2 |
x = a -> 0 + x = 0 + a |
| 2 |
|
id |
x = a -> x = a |
| 3 |
1, 2 |
eqeqd |
x = a -> (0 + x = x <-> 0 + a = a) |
| 4 |
|
addeq2 |
x = 0 -> 0 + x = 0 + 0 |
| 5 |
|
id |
x = 0 -> x = 0 |
| 6 |
4, 5 |
eqeqd |
x = 0 -> (0 + x = x <-> 0 + 0 = 0) |
| 7 |
|
addeq2 |
x = y -> 0 + x = 0 + y |
| 8 |
|
id |
x = y -> x = y |
| 9 |
7, 8 |
eqeqd |
x = y -> (0 + x = x <-> 0 + y = y) |
| 10 |
|
addeq2 |
x = suc y -> 0 + x = 0 + suc y |
| 11 |
|
id |
x = suc y -> x = suc y |
| 12 |
10, 11 |
eqeqd |
x = suc y -> (0 + x = x <-> 0 + suc y = suc y) |
| 13 |
|
add0 |
0 + 0 = 0 |
| 14 |
|
addS |
0 + suc y = suc (0 + y) |
| 15 |
|
suceq |
0 + y = y -> suc (0 + y) = suc y |
| 16 |
14, 15 |
syl5eq |
0 + y = y -> 0 + suc y = suc y |
| 17 |
3, 6, 9, 12, 13, 16 |
ind |
0 + a = 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_peano
(peano2,
peano5,
addeq,
add0,
addS)