theorem div01 (a: nat): $ 0 // a = 0 $;
Step | Hyp | Ref | Expression |
1 |
|
div0 |
0 // 0 = 0 |
2 |
|
diveq2 |
a = 0 -> 0 // a = 0 // 0 |
3 |
1, 2 |
syl6eq |
a = 0 -> 0 // a = 0 |
4 |
|
bi2 |
(0 < a <-> ~a = 0) -> ~a = 0 -> 0 < a |
5 |
|
lt01 |
0 < a <-> a != 0 |
6 |
5 |
conv ne |
0 < a <-> ~a = 0 |
7 |
4, 6 |
ax_mp |
~a = 0 -> 0 < a |
8 |
|
eqtr |
a * 0 + 0 = a * 0 -> a * 0 = 0 -> a * 0 + 0 = 0 |
9 |
|
add0 |
a * 0 + 0 = a * 0 |
10 |
8, 9 |
ax_mp |
a * 0 = 0 -> a * 0 + 0 = 0 |
11 |
|
mul0 |
a * 0 = 0 |
12 |
10, 11 |
ax_mp |
a * 0 + 0 = 0 |
13 |
12 |
a1i |
~a = 0 -> a * 0 + 0 = 0 |
14 |
7, 13 |
eqdivmod |
~a = 0 -> 0 // a = 0 /\ 0 % a = 0 |
15 |
14 |
anld |
~a = 0 -> 0 // a = 0 |
16 |
3, 15 |
cases |
0 // a = 0 |
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)