theorem zeqmznsub1 (a b c n: nat):
$ modZ(n): a -ZN c = b -ZN c <-> mod(n): a = b $;
Step | Hyp | Ref | Expression |
1 |
|
bitr3 |
(modZ(n): b0 a -Z b0 c = b0 b -Z b0 c <-> modZ(n): a -ZN c = b -ZN c) ->
(modZ(n): b0 a -Z b0 c = b0 b -Z b0 c <-> mod(n): a = b) ->
(modZ(n): a -ZN c = b -ZN c <-> mod(n): a = b) |
2 |
|
zeqmeq |
n = n -> b0 a -Z b0 c = a -ZN c -> b0 b -Z b0 c = b -ZN c -> (modZ(n): b0 a -Z b0 c = b0 b -Z b0 c <-> modZ(n): a -ZN c = b -ZN c) |
3 |
|
eqid |
n = n |
4 |
2, 3 |
ax_mp |
b0 a -Z b0 c = a -ZN c -> b0 b -Z b0 c = b -ZN c -> (modZ(n): b0 a -Z b0 c = b0 b -Z b0 c <-> modZ(n): a -ZN c = b -ZN c) |
5 |
|
zsubb0 |
b0 a -Z b0 c = a -ZN c |
6 |
4, 5 |
ax_mp |
b0 b -Z b0 c = b -ZN c -> (modZ(n): b0 a -Z b0 c = b0 b -Z b0 c <-> modZ(n): a -ZN c = b -ZN c) |
7 |
|
zsubb0 |
b0 b -Z b0 c = b -ZN c |
8 |
6, 7 |
ax_mp |
modZ(n): b0 a -Z b0 c = b0 b -Z b0 c <-> modZ(n): a -ZN c = b -ZN c |
9 |
1, 8 |
ax_mp |
(modZ(n): b0 a -Z b0 c = b0 b -Z b0 c <-> mod(n): a = b) -> (modZ(n): a -ZN c = b -ZN c <-> mod(n): a = b) |
10 |
|
bitr |
(modZ(n): b0 a -Z b0 c = b0 b -Z b0 c <-> modZ(n): b0 a = b0 b) ->
(modZ(n): b0 a = b0 b <-> mod(n): a = b) ->
(modZ(n): b0 a -Z b0 c = b0 b -Z b0 c <-> mod(n): a = b) |
11 |
|
zeqmsub1 |
modZ(n): b0 a -Z b0 c = b0 b -Z b0 c <-> modZ(n): b0 a = b0 b |
12 |
10, 11 |
ax_mp |
(modZ(n): b0 a = b0 b <-> mod(n): a = b) -> (modZ(n): b0 a -Z b0 c = b0 b -Z b0 c <-> mod(n): a = b) |
13 |
|
zeqmeqm |
modZ(n): b0 a = b0 b <-> mod(n): a = b |
14 |
12, 13 |
ax_mp |
modZ(n): b0 a -Z b0 c = b0 b -Z b0 c <-> mod(n): a = b |
15 |
9, 14 |
ax_mp |
modZ(n): a -ZN c = b -ZN c <-> mod(n): 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,
the0),
axs_peano
(peano1,
peano2,
peano5,
addeq,
muleq,
add0,
addS,
mul0,
mulS)