theorem eqmax2 (a b: nat): $ a <= b -> max a b = b $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
leloe |
a <= b <-> a < b \/ a = b |
| 2 |
|
eor |
(a < b -> max a b = b) -> (a = b -> max a b = b) -> a < b \/ a = b -> max a b = b |
| 3 |
|
ifpos |
a < b -> if (a < b) b a = b |
| 4 |
3 |
conv max |
a < b -> max a b = b |
| 5 |
2, 4 |
ax_mp |
(a = b -> max a b = b) -> a < b \/ a = b -> max a b = b |
| 6 |
|
ifid |
if (a < b) b b = b |
| 7 |
|
id |
a = b -> a = b |
| 8 |
7 |
ifeq3d |
a = b -> if (a < b) b a = if (a < b) b b |
| 9 |
8 |
conv max |
a = b -> max a b = if (a < b) b b |
| 10 |
6, 9 |
syl6eq |
a = b -> max a b = b |
| 11 |
5, 10 |
ax_mp |
a < b \/ a = b -> max a b = b |
| 12 |
1, 11 |
sylbi |
a <= b -> max a b = 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,
add0,
addS)