theorem mulsub (a b c: nat): $ a * (b - c) = a * b - a * c $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
eor |
(c <= b -> a * (b - c) = a * b - a * c) -> (b <= c -> a * (b - c) = a * b - a * c) -> c <= b \/ b <= c -> a * (b - c) = a * b - a * c |
| 2 |
|
eqsub1 |
a * (b - c) + a * c = a * b -> a * b - a * c = a * (b - c) |
| 3 |
|
muladd |
a * (b - c + c) = a * (b - c) + a * c |
| 4 |
|
npcan |
c <= b -> b - c + c = b |
| 5 |
4 |
muleq2d |
c <= b -> a * (b - c + c) = a * b |
| 6 |
3, 5 |
syl5eqr |
c <= b -> a * (b - c) + a * c = a * b |
| 7 |
2, 6 |
syl |
c <= b -> a * b - a * c = a * (b - c) |
| 8 |
7 |
eqcomd |
c <= b -> a * (b - c) = a * b - a * c |
| 9 |
1, 8 |
ax_mp |
(b <= c -> a * (b - c) = a * b - a * c) -> c <= b \/ b <= c -> a * (b - c) = a * b - a * c |
| 10 |
|
mul02 |
a * 0 = 0 |
| 11 |
|
lesubeq0 |
b <= c <-> b - c = 0 |
| 12 |
|
muleq2 |
b - c = 0 -> a * (b - c) = a * 0 |
| 13 |
11, 12 |
sylbi |
b <= c -> a * (b - c) = a * 0 |
| 14 |
10, 13 |
syl6eq |
b <= c -> a * (b - c) = 0 |
| 15 |
|
lesubeq0 |
a * b <= a * c <-> a * b - a * c = 0 |
| 16 |
|
lemul2a |
b <= c -> a * b <= a * c |
| 17 |
15, 16 |
sylib |
b <= c -> a * b - a * c = 0 |
| 18 |
14, 17 |
eqtr4d |
b <= c -> a * (b - c) = a * b - a * c |
| 19 |
9, 18 |
ax_mp |
c <= b \/ b <= c -> a * (b - c) = a * b - a * c |
| 20 |
|
leorle |
c <= b \/ b <= c |
| 21 |
19, 20 |
ax_mp |
a * (b - c) = a * b - a * c |
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)