| Step |
Hyp |
Ref |
Expression |
| 1 |
|
modsubi.1 |
|- N e. NN |
| 2 |
|
modsubi.2 |
|- A e. NN |
| 3 |
|
modsubi.3 |
|- B e. NN0 |
| 4 |
|
modsubi.4 |
|- M e. NN0 |
| 5 |
|
modsubi.6 |
|- ( A mod N ) = ( K mod N ) |
| 6 |
|
modsubi.5 |
|- ( M + B ) = K |
| 7 |
2
|
nnrei |
|- A e. RR |
| 8 |
4 3
|
nn0addcli |
|- ( M + B ) e. NN0 |
| 9 |
8
|
nn0rei |
|- ( M + B ) e. RR |
| 10 |
6 9
|
eqeltrri |
|- K e. RR |
| 11 |
7 10
|
pm3.2i |
|- ( A e. RR /\ K e. RR ) |
| 12 |
3
|
nn0rei |
|- B e. RR |
| 13 |
12
|
renegcli |
|- -u B e. RR |
| 14 |
|
nnrp |
|- ( N e. NN -> N e. RR+ ) |
| 15 |
1 14
|
ax-mp |
|- N e. RR+ |
| 16 |
13 15
|
pm3.2i |
|- ( -u B e. RR /\ N e. RR+ ) |
| 17 |
|
modadd1 |
|- ( ( ( A e. RR /\ K e. RR ) /\ ( -u B e. RR /\ N e. RR+ ) /\ ( A mod N ) = ( K mod N ) ) -> ( ( A + -u B ) mod N ) = ( ( K + -u B ) mod N ) ) |
| 18 |
11 16 5 17
|
mp3an |
|- ( ( A + -u B ) mod N ) = ( ( K + -u B ) mod N ) |
| 19 |
2
|
nncni |
|- A e. CC |
| 20 |
3
|
nn0cni |
|- B e. CC |
| 21 |
19 20
|
negsubi |
|- ( A + -u B ) = ( A - B ) |
| 22 |
21
|
oveq1i |
|- ( ( A + -u B ) mod N ) = ( ( A - B ) mod N ) |
| 23 |
10
|
recni |
|- K e. CC |
| 24 |
23 20
|
negsubi |
|- ( K + -u B ) = ( K - B ) |
| 25 |
4
|
nn0cni |
|- M e. CC |
| 26 |
23 20 25
|
subadd2i |
|- ( ( K - B ) = M <-> ( M + B ) = K ) |
| 27 |
6 26
|
mpbir |
|- ( K - B ) = M |
| 28 |
24 27
|
eqtri |
|- ( K + -u B ) = M |
| 29 |
28
|
oveq1i |
|- ( ( K + -u B ) mod N ) = ( M mod N ) |
| 30 |
18 22 29
|
3eqtr3i |
|- ( ( A - B ) mod N ) = ( M mod N ) |