theorem takemin (l n: nat): $ take l (min (len l) n) = take l n $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
eor |
(n <= len l -> take l (min (len l) n) = take l n) ->
(len l <= n -> take l (min (len l) n) = take l n) ->
n <= len l \/ len l <= n ->
take l (min (len l) n) = take l n |
| 2 |
|
eqmin2 |
n <= len l -> min (len l) n = n |
| 3 |
2 |
takeeq2d |
n <= len l -> take l (min (len l) n) = take l n |
| 4 |
1, 3 |
ax_mp |
(len l <= n -> take l (min (len l) n) = take l n) -> n <= len l \/ len l <= n -> take l (min (len l) n) = take l n |
| 5 |
|
eqmin1 |
len l <= n -> min (len l) n = len l |
| 6 |
5 |
takeeq2d |
len l <= n -> take l (min (len l) n) = take l (len l) |
| 7 |
|
takeall |
len l <= len l -> take l (len l) = l |
| 8 |
|
leid |
len l <= len l |
| 9 |
7, 8 |
ax_mp |
take l (len l) = l |
| 10 |
|
takeall |
len l <= n -> take l n = l |
| 11 |
9, 10 |
syl6eqr |
len l <= n -> take l n = take l (len l) |
| 12 |
6, 11 |
eqtr4d |
len l <= n -> take l (min (len l) n) = take l n |
| 13 |
4, 12 |
ax_mp |
n <= len l \/ len l <= n -> take l (min (len l) n) = take l n |
| 14 |
|
leorle |
n <= len l \/ len l <= n |
| 15 |
13, 14 |
ax_mp |
take l (min (len l) n) = take l n |
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)