theorem lemax1 (a b: nat): $ a <= max a b $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
ifpos |
a < b -> if (a < b) b a = b |
| 2 |
1 |
conv max |
a < b -> max a b = b |
| 3 |
2 |
leeq2d |
a < b -> (a <= max a b <-> a <= b) |
| 4 |
|
ltle |
a < b -> a <= b |
| 5 |
3, 4 |
mpbird |
a < b -> a <= max a b |
| 6 |
|
eqler |
max a b = a -> a <= max a b |
| 7 |
|
ifneg |
~a < b -> if (a < b) b a = a |
| 8 |
7 |
conv max |
~a < b -> max a b = a |
| 9 |
6, 8 |
syl |
~a < b -> a <= max a b |
| 10 |
5, 9 |
cases |
a <= max a 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),
axs_peano
(peano2,
peano5,
addeq,
add0,
addS)