theorem shrss (a b n: nat): $ a C_ b -> shr a n C_ shr b n $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
elshr |
x e. shr a n <-> x + n e. a |
| 2 |
|
elshr |
x e. shr b n <-> x + n e. b |
| 3 |
|
ssel |
a C_ b -> x + n e. a -> x + n e. b |
| 4 |
2, 3 |
syl6ibr |
a C_ b -> x + n e. a -> x e. shr b n |
| 5 |
1, 4 |
syl5bi |
a C_ b -> x e. shr a n -> x e. shr b n |
| 6 |
5 |
iald |
a C_ b -> A. x (x e. shr a n -> x e. shr b n) |
| 7 |
6 |
conv subset |
a C_ b -> shr a n C_ shr b 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)